Q JEE Main 2024

The number of real solutions of the equation $x|x+5|+2|x+7|-2=0$ is $\_\_\_\_$ .

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Q JEE Main 2024

Let a line perpendicular to the line $2 x-y=10$ touch the parabola $y^2=4(x-9)$ at the point $P$. The distance of...

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Q JEE Main 2024

Let the maximum and minimum values of $\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in R$ be $M$ and $m$ respectively. Then $\mathrm{M}^2-\mathrm{m}^2$ is...

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Q JEE Main 2024

Let a>0 be a root of the equation $2 \mathrm{x}^2+\mathrm{x}-2=0$. If $\lim _{x \rightarrow \frac{1}{a}} \frac{16\left(1-\cos \left(2+x-2 x^2\right)\right)}{\left(1-a x^2\right)}=\alpha+\beta \sqrt{17}$,...

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Q JEE Main 2024

If $1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\cdots$. upto $\infty=2\left(\sqrt{\frac{b}{a}}+1\right) \log _e\left(\frac{a}{b}\right)$, where a and b are integers with $...

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Q JEE Main 2024

Let the point $(-1, \alpha, \beta)$ lie on the line of the shortest distance between the lines $\frac{x+2}{-3}=\frac{y-2}{4}=\frac{z-5}{2}$ and $\frac{x+2}{-1}=\frac{y+6}{2}=\frac{z-1}{0}$....

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Q JEE Main 2024

The number of solutions of $\sin ^2 x+\left(2+2 x-x^2\right) \sin x-3(x-1)^2=0$, where $-\pi \leq \mathrm{x} \leq \pi$, is $\_\_\_\_$ .

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Q JEE Main 2024

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+\frac{2 x}{\left(1+x^2\right)^2} y=x e^{\frac{1}{\left(1+x^2\right)}} ; y(0)=0$. Then the area...

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Q JEE Main 2024

Let the mean and the standard deviation of the probability distribution be $\mu$ and $\sigma$, respectively. If $\sigma-\mu=2$, then $\sigma+\mu$...

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For $x \geq 0$, the least value of $K$, for which $4^{1+x}+4^{1-x}, \frac{K}{2}, 16^x+16^{-x}$ are three consecutive terms of an...

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Q JEE Main 2024

If $y(\theta)=\frac{2 \cos \theta+\cos 2 \theta}{\cos 3 \theta+4 \cos 2 \theta+5 \cos \theta+2}$, then at $\theta=\frac{\pi}{2}, y^{\prime \prime}+y^{\prime}+y$ is equal...

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Q JEE Main 2024

Let $\alpha \beta \neq 0$ and $...

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Q JEE Main 2024

Let $\beta(m, n)=\int_0^1 x^{m-1}(1-x)^{n-1} d x, m, n>0$. If $\int_0^1\left(1-x^{10}\right)^{20} d x=a \times \beta(b, c)$, then $100(a+b+x)$ equals

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Q JEE Main 2024

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line...

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Q JEE Main 2024

The coefficients $a, b, c$ in the quadratic equation $a x^2+b x+c=0$ are from the set $\{1,2,3,4,5,6\}$. If the probability...

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Q JEE Main 2024

The values of $m, n$, for which the system of equations $...

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Q JEE Main 2024

Let the set $S=\{2,4,8,16, \ldots, 512\}$ be partitioned into 3 sets $\mathrm{A}, \mathrm{B}, \mathrm{C}$ with equal number of elements such...

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Q JEE Main 2024

Let the circle $C_1: x^2+y^2-2(x+y)+1=0$ and $C_2$ be a circle having centre at $(-1,0)$ and radius 2 . If the...

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Q JEE Main 2024

Let $f, \mathrm{~g}: \mathrm{R} \rightarrow \mathrm{R}$ be defined as : $f(\mathrm{x})=|\mathrm{x}-1|$ and $...

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Q JEE Main 2024

If the constant term in the expansion of $\left(\frac{\sqrt[5]{3}}{x}+\frac{2 x}{\sqrt[3]{5}}\right)^{12}, x \neq 0$, is $\alpha \times 2^8 \times \sqrt[5]{3}$, then...

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Q JEE Main 2024

Let $(\alpha, \beta, \gamma)$ be the point $(8,5,7)$ in the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{5}$. Then $\alpha+\beta+\gamma$ is equal to

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Q JEE Main 2024

Let $\mathrm{A}(-1,1)$ and $\mathrm{B}(2,3)$ be two points and P be a variable point above the line $A B$ such that...

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Q JEE Main 2024

Consider three vectors $\vec{a}, \vec{b}, \vec{c}$. Let $|\vec{a}|=2,|\vec{b}|=3$ and $\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}$. If $\alpha \in\left[0, \frac{\pi}{3}\right]$ is the angle between...

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Q JEE Main 2024

Let $\vec{a}=2 \hat{\imath}+5 \hat{\jmath}-\hat{k}, \vec{b}=2 \hat{\imath}-2 \hat{\jmath}+2 \hat{k}$ and $\overrightarrow{\mathrm{c}}$ be three vectors such that $(\vec{c}+\hat{\imath}) \times(\vec{a}+\vec{b}+\hat{\imath})=\vec{a} \times(\vec{c}+\hat{\imath})$. $...

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Q JEE Main 2024

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words...

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Q JEE Main 2024

The area enclosed between the curves $y=x|x|$ and $\mathrm{y}=\mathrm{x}-|\mathrm{x}|$ is :

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Q JEE Main 2024

Let $S_1=\{z \in C:|z| \leq 5\}, S_2=\left\{z \in C: \operatorname{Im}\left(\frac{z+1-\sqrt{3} i}{1-\sqrt{3} i}\right) \geq 0\right\}$ and $...

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Q JEE Main 2024

The differential equation of the family of circles passing the origin and having center at the line $\mathrm{y}=\mathrm{x}$ is :

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Q JEE Main 2024

Let $f:[-1,2] \rightarrow \mathrm{R}$ be given by $f(\mathrm{x})=2 \mathrm{x}^2+\mathrm{x}+\left[\mathrm{x}^2\right]-[\mathrm{x}]$, where $[\mathrm{t}]$ denotes the greatest integer less than or equal to...

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Q JEE Main 2024

Consider the matrices : $\mathrm{A}=\left[\begin{array}{rr}2 & -5 \\ 3 & \mathrm{~m}\end{array}\right], \mathrm{B}=\left[\begin{array}{c}20 \\ \mathrm{~m}\end{array}\right]$ and $X=\left[\begin{array}{l}x \\ y\end{array}\right]$. Let the...

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Q JEE Main 2024

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $...

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Q JEE Main 2024

If $\left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\cdots .+\frac{1}{\alpha+1012}\right)-\left(\frac{1}{2 \cdot 1}+\frac{1}{4 \cdot 3}+\frac{1}{6 \cdot 5}+\cdots .+\frac{1}{2024 \cdot 2023}\right)=\frac{1}{2024}$, then $\alpha$ is equal to-

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Q JEE Main 2024

The square of the distance of the image of the point $(6,1,5)$ in the line $\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}$, from the origin is

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Q JEE Main 2024

Let $\mathrm{A}, \mathrm{B}$ and C be three points on the parabola $y^2=6 x$ and let the line segment $A B$...

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Q JEE Main 2024

Let $A=\{(x, y): 2 x+3 y=23, x, y \in N\}$ and $\mathrm{B}=\{\mathrm{x}:(\mathrm{x}, \mathrm{y}) \in \mathrm{A}\}$. Then the number of one-one...

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Q JEE Main 2024

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is

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Q JEE Main 2024

For a differentiable function $\mathrm{f}: I R \rightarrow I R$, suppose $f^{\prime}(\mathrm{x})=3 f(\mathrm{x})+\alpha$, where $\alpha \in \mathrm{IR}, f(0)=1$ and $...

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Q JEE Main 2024

Let the set of all values of $p$, for which $f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right)+2(2-p) x+7$ does...

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Q JEE Main 2024

Consider the circle $C: x^2+y^2=4$ and the parabola $P: y^2=8 x$. If the set of all values of $\alpha$, for...

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Q JEE Main 2024

Let $\alpha, \beta ; \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2} x-\sqrt{3}=0$. Let $P_n=\alpha^n-\beta^n, n \in N$. Then $...

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Q JEE Main 2024

The sum of the coefficient of $x^{2 / 3}$ and $x^{-2 / 5}$ in the binomial expansion of $...

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Q JEE Main 2024

The value of the integral $\int_{-1}^2 \log _e\left(x+\sqrt{x^2+1}\right) d x$ is :

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Q JEE Main 2024

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{\text {th }}$...

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Q JEE Main 2024

Let a, ar, $a r^{2^2} \ldots \ldots .$. be an infinite G.P. If $\sum_{n=0}^{\infty} a^n=57$ and $\sum_{n=0}^{\infty} a^3 r^{3 n}=9747$,...

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Q JEE Main 2024

The integral $\int_{1 / 4}^{3 / 4} \cos \left(2 \cot ^{-1} \sqrt{\frac{1-\mathrm{x}}{1+\mathrm{x}}}\right) \mathrm{dx}$ is equal to:

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Q JEE Main 2024

If $\log _e y=3 \sin ^{-1} x$, then $(1-x)^2 y^{\prime \prime}-x y^{\prime}$ at $x=\frac{1}{2}$ is equal to :

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Q JEE Main 2024

Let $B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$ and $A$ be a $2 \times 2$ matrix such that $\mathrm{AB}^{-1}=\mathrm{A}^{-1}$. If...

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Q JEE Main 2024

$\lim _{x \rightarrow \frac{x}{2}}\left(\frac{\int_{x^3}^{(\pi / 2)^3}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) d t}{\left(x-\frac{\pi}{2}\right)^2}\right)$ is equal to :

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Q JEE Main 2024

Between the following two statements : Statement-I : Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ and $\vec{b}=2 \hat{\imath}+\hat{\jmath}-\hat{k}$. Then the vector $\overrightarrow{\mathrm{r}}$ satisfying...

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Q JEE Main 2024

Let the range of the function $f(x)=\frac{1}{2+\sin 3 x+\cos 3 x}, x \in \mathrm{IR}$ be $[a, b]$. If $\alpha$ and...

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Q JEE Main 2024

If the variance of the frequency distribution is 160 , then the value of $c \in N$ is

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Q JEE Main 2024

Two vertices of a triangle ABC are $\mathrm{A}(3,-1)$ and $\mathrm{B}(-2,3)$, and its orthocentre is $\mathrm{P}(1,1)$. If the coordinates of the...

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Q JEE Main 2024

Let the foci of a hyperbola H coincide with the foci of the ellipse $E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$ and the eccentricity of...

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Q JEE Main 2024

The area (in square units) of the region enclosed by the ellipse $x^2+3 y^2=18$ in the first quadrant below the...

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Q JEE Main 2024

Let z be a complex number such that the real part of $\frac{z-2 i}{z+2 i}$ is zero. Then, the maximum...

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Q JEE Main 2024

Let $\int_0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} \mathrm{dt}=\int_0^{\mathrm{x}} \mathrm{y}(\mathrm{t}) \mathrm{dt}, 0 \leq \mathrm{x} \leq 3, \mathrm{y} \geq 0, y(0)=0$. Then at $x=2, y^{\prime \prime}+y+1$...

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Q JEE Main 2024

Consider the line L passing through the points $(1,2,3)$ and $(2,3,5)$. The distance of the point $\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)$ from...

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Q JEE Main 2024

$\operatorname{Lim}_{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x}$ is equal to :

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Q JEE Main 2024

Let y=y(x) be the solution of the differential equation $(x+y+2)^2 d x=d y, y(0)=-2$. Let the maximum and minimum values...

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Q JEE Main 2024

Consider a line L passing through the points $\mathrm{P}(1,2,1)$ and $\mathrm{Q}(2,1,-1)$. If the mirror image of the point $\mathrm{A}(2,2,2)$ in...

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Q JEE Main 2024

Consider a triangle ABC having the vertices $\mathrm{A}(1,2), \mathrm{B}(\alpha, \beta)$ and $\mathrm{C}(\gamma, \delta)$ and angles $\angle \mathrm{ABC}=\frac{\pi}{6}$ and $...

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Q JEE Main 2024

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac{1}{3}$ and $\frac{2}{3}$...

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Q JEE Main 2024

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If...

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Q JEE Main 2024

Let 𝐴 be a 2 × 2 symmetric matrix such that $A\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\right]$ and the determinant of...

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Q JEE Main 2024

Consider the function $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$. If the composition of $...

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Q JEE Main 2024

Let f: ℝ → ℝ be a thrice differentiable function such that f(0) = 0, f(1) = 1, f(2) =...

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Q JEE Main 2024

If $\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cosec}^2 x+\frac{3}{2}\right)+\beta \log _e\left|\tan \frac{x}{2}\right|+C$ where $\alpha, \beta \in \mathbb{R}$ and...

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Q JEE Main 2024

Let P the point of intersection of the lines $\frac{x-2}{1}=\frac{y-4}{5}=\frac{z-2}{1}$ and $\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-3}{2}$. Then, the shortest distance of $P$ from the...

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Q JEE Main 2024

Let $y=y(x)$ be the solution of the differential equation $\left(x^2+4\right)^2 d y+\left(2 x^3 y+8 x y-2\right) d x=0$. If $y(0)=$...

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Q JEE Main 2024

Given the inverse trigonometric function assumes principal values only. Let $x, y$ be any two real numbers in $[-1,1]$ such...

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Q JEE Main 2024

Let P Q be a chord of the parabola $y^2=12 x$ and the midpoint of PQ be at $(4,1)$. Then,...

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Q JEE Main 2024

If the mean of the following probability distribution of a random variable X; is $\frac{46}{9}$, then the variance of the...

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Q JEE Main 2024

Consider a hyperbola $H$ having centre at the origin and foci and the $x$-axis. Let $C_1$ be the circle touching...

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Q JEE Main 2024

If the coefficients of $x^4, x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the...

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Q JEE Main 2024

Let f(x) = 3 $ \sqrt{\mathrm{x}-2}+\sqrt{4-\mathrm{x}}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and...

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Q JEE Main 2024

If the value of the integral $\int_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x$ is $\frac{2}{\pi}$. Then, a value of $\alpha$ is

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Q JEE Main 2024

The area (in sq. units) of the region $S=\{z \in \mathbb{C} ;|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{lm}(z) \geq 0\}$...

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Q JEE Main 2024

The area (in sq. units) of the region described by $\left\{(x, y): y^2 \leq 2 x\right.$, and $...

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Q JEE Main 2024

Let $f(x)=\int_0^x\left(t+\sin \left(1-e^t\right)\right) d t, x \in \mathbb{R}$. Then $\lim _{x \rightarrow 0} \frac{f(x)}{x^3}$ is equal to

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Q JEE Main 2024

The value of $\frac{1 \times 2^2+2 \times 3^2+\cdots+100 \times(101)^2}{1^2 \times 2+2^2 \times 3+\cdots+100^2 \times 101}$ is

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Q JEE MAIN 2024

Let ABC be an isosceles triangle in which A is at $(-1,0), \angle A=\frac{2 \pi}{3}, A B=A C$ and $B$...

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Q JEE MAIN 2024

If $\frac{d x}{d y}=\frac{1+x-y^2}{y}, x(1)=1$, then $5 x(2)$ is equal to :

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Q JEE Main 2024

Let $A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$ and $B=I+\operatorname{adj}(A)+(\operatorname{adj} A)^2+\cdots+(\operatorname{adj} A)^{10}$. Then, the sum of all the elements of...

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Q JEE Main 2024

Let three real numbers 𝑎, 𝑏, 𝑐 be in arithmetic progression and a +1, b, c +3 be in geometric...

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Q JEE MAIN 2024

The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas...

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Q JEE Main 2024

Let a relation R on N x N be defined as : $\left(x_1, y_1\right) R\left(x_2, y_2\right)$ if and only if...

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Q JEE MAIN 2024

Three points $\mathrm{O}(0,0), \mathrm{P}\left(\mathrm{a}, \mathrm{a}^2\right), \mathrm{Q}\left(-\mathrm{b}, \mathrm{b}^2\right), \mathrm{a}>0, \mathrm{~b}>0$, are on the parabola $y=x^2$. Let $S_1$ be the area of...

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The lines $\mathrm{L}_1, \mathrm{~L}_2, \ldots, \mathrm{I}_{20}$ are distinct. For $\mathrm{n}=1,2,3, \ldots, 10$ all the lines $\mathrm{L}_{2 \mathrm{n}-1}$ are parallel to...

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Q JEE MAIN 2024

Let $\vec{a}=\hat{\imath}+\hat{\jmath}+\hat{k}, \vec{b}=-\hat{\imath}-8 \hat{\jmath}+2 \hat{k}$ and $\overrightarrow{\mathrm{c}}=4 \hat{\imath}+\mathrm{c}_2 \hat{\jmath}+\mathrm{c}_3 \hat{\mathrm{k}}$ be three vectors such that $\vec{b} \times \vec{a}=\vec{c} \times \vec{a}$....

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Q JEE Main 2024

Let C be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersects the...

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Q JEE Main 2024

If $\lambda>0$, let $\theta$ be the angle between the vectors $\vec{a}=\hat{\imath}+\lambda \hat{\jmath}-3 \hat{k}$ and $\vec{b}=3 \hat{\imath}-\hat{\jmath}+2 \hat{k}$. If the vectors...

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Q JEE Main 2024

If the function $f(x)=\left\{\begin{array}{ll}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}} & , x \neq 0 \\ \log _e 2 \log _e 3 & , x=0\end{array}\right.$...

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Q JEE MAIN 2024

If $y=\frac{(\sqrt{x}+1)\left(x^2-\sqrt{x}\right)}{x \sqrt{x}+x+\sqrt{x}}+\frac{1}{15}\left(3 \cos ^2 x-5\right) \cos ^3 x$, then $96 y^{\prime}\left(\frac{\pi}{6}\right)$ is equal to :

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Let $f:(0, \infty) \rightarrow R$ and $F(x)=\int_0^x t f(t) d t$. If $F\left(x^2\right)=\mathrm{x}^4+\mathrm{x}^5$, then $\sum_{\mathrm{r}=1}^{12} \mathrm{f}\left(\mathrm{r}^2\right)$ is equal to :

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Q JEE MAIN 2024

Let $A=I_2-2 M^{\mathrm{T}}$, where M is real matrix of order $2 \times 1$ such that the relation $\mathrm{M}^{\mathrm{T}} \mathrm{M}=\mathrm{I}_1$ holds....

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Q Jee Main 2024

Let the slope of the line $45 x+5 y+3=0$ be $27 r_1+\frac{9 r_2}{2}$ for some $r_1, r_2 \in R$. Then...

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Let the set $C=\left\{(x, y) \mid x^2-2^y=2023, x, y \in \mathbb{N}\right\}$. Then $\sum_{(x, y) \in C}(x+y)$ is equal to

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Q Jee Main 2024

Remainder when $64^{32^{32}}$ is divided by 9 is equal to

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Let $f(x)=\sqrt{\lim _{r \rightarrow x}\left\{\frac{2 r^2\left[(f(r))^2-f(x) f(r)\right]}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right\}}$ be differentiable in $(-\infty, 0) \cup(0, \infty)$ and $f(1)=1$. Then the value...

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Q JEE MAIN 2024

The number of symmetric relations defined on the set $\{1,2,3,4\}$ which are not reflexive is

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Q Jee Main 2024

Let $O$ be the origin, and $M$ and $N$ be the points on the lines $\frac{x-5}{4}=\frac{y-4}{1}=\frac{z-5}{3}$ and $\frac{x+8}{12}=\frac{y+2}{5}=\frac{z+11}{9}$ respectively such...

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Let the area of the region $\left\{(x, y): 0 \leq x \leq 3,0 \leq y \leq \min \left\{\mathrm{x}^2+2,2 \mathrm{x}+2\right\}\right\}$ be...

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Q JEE MAIN 2024

The variance $\sigma^2$ of the data

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If $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1-\sin 2 x} d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}$, where $\alpha, \beta$ and $\gamma$ are rational numbers, then $...

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Q JEE MAIN 2024

The area of the region enclosed by the parabola $(y-2)^2=x-1$, the line $x-2 y+4=0$ and the positive coordinate axes is

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Q JEE MAIN 2024

The number of real solutions of the equation $x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$ is

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Q Jee Main 2024

Let $\mathrm{P}(\alpha, \beta)$ be a point on the parabola $\mathrm{y}^2=4 \mathrm{x}$. If P also lies on the chord of the...

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Q Jee Main 2024

Let for any three distinct consecutive terms $a, b, c$ of an A.P, the lines $a x+b y+c=0$ be concurrent...

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Q JEE MAIN 2024

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into...

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Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{6} x+3=0$ such that $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Let $a, b$ be integers not...

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Q Jee Main 2024

Let a unit vector $\hat{\mathrm{u}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}$ make angles $\frac{\pi}{2}, \frac{\pi}{3}$ and $\frac{2 \pi}{3}$ with the vectors $...

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Q Jee Main 2024

An integer is chosen at random from the integers $1,2,3, \ldots, 50$. The probability that the chosen integer is a...

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Q Jee Main 2024

If $R$ is the smallest equivalence relation on the set $\{1,2,3,4\}$ such that $\{(1,2),(1,3)\} \subset R$, then the number of...

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Q Jee Main 2024

Let $\mathrm{y}=\log _{\mathrm{e}}\left(\frac{1-\mathrm{x}^2}{1+\mathrm{x}^2}\right),-1

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Q JEE MAIN 2024

Let $S_n$ be the sum to $n$-terms of an arithmetic progression $3,7,11, \ldots \ldots$. If $40

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Q Jee Main 2024

The function $f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}$

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Q JEE MAIN 2024

Let $\alpha=\sum_{\mathrm{k}=0}^{\mathrm{n}}\left(\frac{\left({ }^{\mathrm{n}} \mathrm{C}_{\mathrm{k}}\right)^2}{\mathrm{k}+1}\right)$ and $\beta=\sum_{\mathrm{k}=0}^{\mathrm{n}-1}\left(\frac{{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{k}}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{k}+1}}{\mathrm{k}+2}\right)$. If $5 \alpha=6 \beta$, then $n$ equals

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Q Jee Main 2024

Let $x=\frac{m}{n}\left(\mathrm{~m}, n\right.$ are co-prime natural numbers) be a solution of the equation $\cos \left(2 \sin ^{-1} x\right)=\frac{1}{9}$ and let...

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Q JEE MAIN 2024

If three successive terms of a G.P. with common ratio $r(r>1)$ are the lengths of the sides of a triangle...

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Q JEE MAIN 2024

Consider two circles $\mathrm{C}_1: \mathrm{x}^2+\mathrm{y}^2=25$ and $\mathrm{C}_2:(\mathrm{x}-\alpha)^2+y^2=16$, where $\alpha \in(5,9)$. Let the angle between the two radii (one to each...

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Q Jee Main 2024

Let A be the point of intersection of the lines $3 \mathrm{x}+2 y=14,5 x-y=6$ and $B$ be the point of...

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Q Jee Main 2024

If each term of a geometric progression $a_1, a_2, a_3, \ldots$ with $a_1=\frac{1}{8}$ and $a_2 \neq a_1$, is the arithmetic...

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Q JEE MAIN 2024

Let a line passing through the point $(-1,2,3)$ intersect the lines $\mathrm{L}_1: \frac{\mathrm{x}-1}{3}=\frac{\mathrm{y}-2}{2}=\frac{\mathrm{z}+1}{-2}$ at $M(\alpha, \beta, \gamma)$ and $L_2: \frac{x+2}{-3}=\frac{y-2}{-2}=\frac{z-1}{4}$...

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Q JEE MAIN 2024

Let the locus of the mid points of the chords of circle $x^2+(y-1)^2=1$ drawn from the origin intersect the line...

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Q JEE-Main 2024

Let the complex numbers $\alpha$ and $\frac{1}{\bar{\alpha}}$ lie on the circles $\left|z-z_0\right|^2=4$ and $\left|z-z_0\right|^2=16$ respectively, where $\mathrm{z}_0=1+\mathrm{i}$. Then, the value...

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Q JEE-Main 2024

The lines $\frac{x-2}{2}=\frac{y}{-2}=\frac{z-7}{16}$ and $\frac{x+3}{4}=\frac{y+2}{3}=\frac{z+2}{1}$ intersect at the point $P$. If the distance of P from the line $\frac{x+1}{2}= \frac{\mathrm{y}-1}{3}=\frac{\mathrm{z}-1}{1}$...

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Consider a circle $(x-\alpha)^2+(y-\beta)^2=50$, where $\alpha, \beta>0$. If the circle touches the line $y+x=0$ at the point P , whose...

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Q JEE MAIN 2024

Let $Y=Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $\mathrm{Y}-\mathrm{y}=\mathrm{Y}^{\prime}(\mathrm{x})(\mathrm{X}- \mathrm{x})$...

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Q JEE-Main 2024

If the solution curve, of the differential equation $\frac{d y}{d x}=\frac{x+y-2}{x-y}$ passing through the point $(2,1)$ is $...

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Let $f(x)=\int_0^x g(t) \log _e\left(\frac{1-t}{1+t}\right) d t$, where $g$ is a continuous odd function. If $...

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Q JEE MAIN 2024

Let $\mathrm{R}=\left(\begin{array}{lll}\mathrm{x} & 0 & 0 \\ 0 & \mathrm{y} & 0 \\ 0 & 0 & \mathrm{z}\end{array}\right)$ be a...

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Q JEE MAIN 2024

Let Ajay will not appear in JEE exam with probability $\mathrm{p}=\frac{2}{7}$, while both Ajay and Vijay will appear in the...

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Q Jee Main 2024

If $\sin \left(\frac{y}{x}\right)=\log _e|x|+\frac{\alpha}{2}$ is the solution of the differential equation $x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x$ and...

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Q JEE-Main 2024

The coefficient of $x^{2012}$ in the expansion of $(1-x)^{2008}\left(1+x+x^2\right)^{2007}$ is equal to

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Q JEE-Main 2024

If the sum of squares of all real values of $\alpha$, for which the lines $2 \mathrm{x}-\mathrm{y}+3=0,6 \mathrm{x}+3 \mathrm{y}+1=0$ and...

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Q JEE MAIN 2024

Let $f: R \rightarrow R$ be a function defined $f(x)=\frac{x}{\left(1+x^4\right)^{1 / 4}}$ and $g(x)=f(f(f(f(x))))$ then $$ 18 \int_0^{\sqrt{2 \sqrt{5}}} x^2...

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Q JEE MAIN 2024

consider 10 observation $\mathrm{x}_1, \mathrm{x}_{2, \ldots}, \mathrm{x}_{10}$. such that $\sum_{i=1}^{10}\left(x_i-\alpha\right)=2$ and $\sum_{i=1}^{10}\left(x_i-\beta\right)^2=40$, where $\alpha, \beta$ are positive integers. Let the...

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Q JEE MAIN 2024

If the domain of the function $f(x)=\log _e\left(\frac{2 x+3}{4 x^2+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right)$ is $(\alpha, \beta]$, then the value of $...

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Q Jee Main 2024

The distance of the point $(2,3)$ from the line $2 x-3 y+28=0$, measured parallel to the line $\sqrt{3} x-y+ 1=0$,...

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Q JEE MAIN 2024

Suppose $2-p, p, 2-\alpha, \alpha$ are the coefficient of four consecutive terms in the expansion of $(1+x)^n$. Then the value...

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Q Jee Main 2024

If $$ \int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x \sin (x-\theta)}} d x=A \sqrt{\cos \theta \tan...

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Q JEE MAIN 2024

If $z$ is a complex number, then the number of common roots of the equation $z^{1985}+z^{100}+1=0$ and $...

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Q JEE MAIN 2024

Let the system of equations $x+2 y+3 z=5,2 x+3 y+z=9,4 x+3 y+\lambda z=\mu$ have infinite number of solutions. Then $...

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Q Jee Main 2024

Let $\mathrm{A}=\{1,2,3, \ldots \ldots \ldots 100\}$. Let R be a relation on $A$ defined by $(x, y) \in R$ if...

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Q JEE MAIN 2024

Let $\mathrm{f}: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$ be a function satisfying $f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$ for all $x, y, f(y) \neq 0$. If $f^{\prime}(1)=2024$, then

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Q Jee Main 2024

Let $y=y(x)$ be the solution of the differential equation $$ \begin{gathered} \sec ^2 x d x+\left(e^{2 y} \tan ^2 x+\tan...

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Q Jee Main 2024

If $\log _e a, \log _e b, \log _e c$ are in an A.P. and $...

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Q Jee Main 2024

A line passes through $\mathrm{A}(4,-6,-2)$ and $\mathrm{B}(16,-2,4)$. The point $\mathrm{P}(\mathrm{a}, \mathrm{b}, \mathrm{c})$ where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are non-negative integers, on...

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Q Jee Main 2024

If $^2$-blog $(1+x)+\quad \lim _{x \rightarrow 0}$ $\_\_\_\_$。 = 1 axes $\cos x$, then $16\left(a^2+b^2+c^2\right)$ is equal to $$ x^2...

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Q JEE MAIN 2024

Let $...

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Q JEE MAIN 2024

Let $a$ and $b$ be real constants such that the function $f$ defined by $...

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Q Jee Main 2024

Let $a=3 \hat{\imath}+2 \hat{\jmath}+k^{\wedge}, b^{+}=2 \hat{\imath}-\hat{\jmath}+3 k^{\wedge}$ and $c$ be a vector such that $...

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Q JEE MAIN 2024

If the mirror image of the point $\mathrm{P}(3,4,9)$ in the line $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-2}{1}$ is $(\alpha, \beta, \gamma)$, then $14(\alpha+\beta+\gamma)$ is :

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Q JEE MAIN 2024

Let $L_1: \vec{r}=(\hat{\imath}-\hat{\jmath}+2 \hat{k})+\lambda(\hat{\imath}-\hat{\jmath}+2 \hat{k}), \lambda \in R \mathrm{~L}_2: \overrightarrow{\mathrm{r}}=(\hat{\jmath}-\hat{\mathrm{k}})+\mu(3 \hat{\imath}+\hat{\jmath}+\mathrm{p} \hat{\mathrm{k}}), \mu \in \mathrm{R}$ and $...

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Q Jee Main 2024

Let $A$ be a $3 \times 3$ matrix and $\operatorname{det}(A)=2$. If $\mathrm{n}=\operatorname{det}(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots \ldots}(\operatorname{adj} A))))$ Then the remainder when n is...

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Q JEE MAIN 2024

Consider the relations $R_1$ and $R_2$ defined as $a R_1 b \Leftrightarrow a^2+b^2=1$ for all $a, b, \in R$ and...

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Q Jee Main 2024

Let the coefficient of $x^r$ in the expansion of $$ \begin{aligned} & (x+3)^{n-1}+(x+3)^{n-2}(x+2)+ \\ & (x+3)^{n-3}(x+2)^2+\cdots \cdots+(x+2)^{n-1} \end{aligned} $$ be...

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Q JEE-Main 2024

Let A be a $2 \times 2$ real matrix and I be the identity matrix of order 2 . If...

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Q JEE-Main 2024

If the area of the region $\left\{(x, y): 0 \leq y \leq \min \left\{2 x, 6 x-x^2\right\}\right\}$ is $A$, then...

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Q JEE-Main 2024

The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was...

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Q JEE-Main 2024

Let the position vectors of the vertices $A, B$ and $C$ of a triangle be $...

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Let $\mathrm{A}(-2,-1), \mathrm{B}(1,0), \mathrm{C}(\alpha, \beta)$ and $\mathrm{D}(\gamma, \delta)$ be the vertices of a parallelogram $A B C D$. If the...

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If the domain of the function $f(x)=\frac{\sqrt{x^2-25}}{\left(4-x^2\right)}+\log _{10}\left(x^2+2 x-15\right)$ is $(-\infty, \alpha) U[\beta, \infty)$, then $\alpha^2+\beta^3$ is equal to :

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Let $\mathrm{e}_1$ be the eccentricity of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $e_2$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=$ 1, $a>b$,...

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Q JEE-Main 2024

Let $\alpha=\frac{(4!)!}{(4!)^{3!}}$ and $\beta=\frac{(5!)!}{(5!)^{4!}}$. Then :

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Q Jee Main 2024

Let $a, b, c$ be the length of three sides of a triangle satisfying the condition $...

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$\left|\frac{120}{\pi^3} \int_0^\pi \frac{x^2 \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x\right|$ is equal to

JEE Main Mathematics Medium

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If $\int_0^{\frac{\pi}{3}} \cos ^4 x d x=a \pi+b \sqrt{3}$, where $a$ and $b$ are rational numbers, then $9 a+8 b$...

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If $y=y(x)$ is the solution curve of the differential equation $\left(x^2-4\right) d y-\left(y^2-3 y\right) d x=0, x>2, y(4)=\frac{3}{2}$ and the...

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Q JEE-Main 2024

The position vectors of the vertices $A, B$ and $C$ of a triangle are $...

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The shortest distance between lines $\mathrm{L}_1$ and $\mathrm{L}_2$, where $\mathrm{L}_1: \frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{-3}=\frac{z+4}{2}$ and $\mathrm{L}_2$ is the line passing through the points...

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If $2 \tan ^2 \theta-5 \sec \theta=1$ has exactly 7 solutions in the interval $\left[0, \frac{n \pi}{2}\right]$, for the least...

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The integral $\int \frac{\left(x^8-x^2\right) d x}{\left(x^{12}+3 x^6+1\right) \tan ^{-1}\left(x^3+\frac{1}{x^3}\right)}$ is equal to :

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Q Jee Main 2024

Let A be a $3 \times 3$ real matrix such that $...

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Q JEE-Main 2024

An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The...

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The values of $\alpha$, for which $...

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A coin is based so that a head is twice as likely to occur as a tail. If the coin...

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If for some $\mathrm{m}, \mathrm{n} ;{ }^6 \mathrm{C}_{\mathrm{m}}+2\left({ }^6 \mathrm{C}_{\mathrm{m}+1}\right)+{ }^6 \mathrm{C}_{\mathrm{m}+2}>{ }^8 \mathrm{C}_3$ and $...

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Q Jee Main 2024

If $a=\sin ^{-1}(\sin (5))$ and $b=\cos ^{-1}(\cos (5))$, then $a^2+b^2$ is equal to

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Q Jee Main 2024

The number of solutions, of the equation $\mathrm{e}^{\sin x}-2 \mathrm{e}^{-\sin x}=2$ is

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Q JEE MAIN 2024

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10}=390$ and the ratio of...

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Q Jee Main 2024

Consider the function $\mathrm{f}:(0, \infty) \rightarrow \mathrm{R}$ defined by $...

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Q Jee Main 2024

Let $\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{OB}}=12 \overrightarrow{\mathrm{a}}+4 \overrightarrow{\mathrm{~b}}$ and $\overrightarrow{\mathrm{OC}}=\overrightarrow{\mathrm{b}}$, where O is the origin. If $S$ is the parallelogram with adjacent sides...

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If the function $\mathrm{f}:(-\infty,-1] \rightarrow(\mathrm{a}, \mathrm{b}]$ defined by $f(x)=e^{x 3-3 x+1}$ is one-one and onto, then the distance of the...

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Let the mean and the variance of 6 observation $a, b, 68,44,48,60$ be 55 and 194 , respectively if $a>b$,...

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Q Jee Main 2024

The sum of the solutions $x \in \mathbb{R}$ of the equation $...

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Q Jee Main 2024

\text { The area of the region enclosed by the parabola } y=4 x-x^2 \text { and } 3 y=(x-4)^2...

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Q JEE MAIN 2024

Consider a $\triangle \mathrm{ABC}$ where $\mathrm{A}(1,2,3), \mathrm{B}(-2,8,0)$ and $\mathrm{C}(3,6,7)$. If the angle bisector of $\angle \mathrm{BAC}$ meets the line $...

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Let $\mathrm{f}: \rightarrow \mathrm{R} \rightarrow(0, \infty)$ be strictly increasing function such that $\lim _{\substack{x \rightarrow \infty \\ f(x)}} \frac{f(7 x)}{2}=1$....

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Q Jee Main 2024

Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z=2-i\left(2 \tan \frac{5 \pi}{8}\right)$, then $...

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Q JEE MAIN 2024

Let $P$ and $Q$ be the points on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ which are at a distance of 6 units from...

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Q Jee Main 2024

Let $2^{\text {nd }}, 8^{\text {th }}$ and $44^{\text {th }}$, terms of a non-constant A.P. be respectively the $...

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Q JEE-Main 2024

Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of...

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Q JEE-Main 2024

If $\alpha, \beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n$, then

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Q JEE MAIN 2024

Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3} X^{\frac{1}{3}}+\frac{1}{2 X^{\frac{2}{3}}}\right)^{18}$....

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Let $\quad f: R \rightarrow R$ be defined $\quad f(x)=a e^{2 x}+b e^x+c x . \quad$ If $...

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Q Jee Main 2024

The temperature $\mathrm{T}(\mathrm{t})$ of a body at time $\mathrm{t}=0$ is $160^{\circ} \mathrm{F}$ and it decreases continuously as per the differential...

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Q Jee Main 2024

The function $f(x)=2 x+3(x)^{\frac{2}{3}}, x \in \mathbb{R}$, has

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If $\lim _{x \rightarrow 0} \frac{3+\alpha \sin x+\beta \cos x+\log _e(1-x)}{3 \tan ^2 x}=\frac{1}{3}$, then $2 \alpha-\beta$ is equal to...

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Let $g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)$ and $f^{\prime \prime}(x)>0$ for all $\mathrm{x} \in(0,3)$. If g is decreasing in $(0, \alpha)$ and increasing in...

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Q Jee Main 2024

Let P be a parabola with vertex $(2,3)$ and directrix $2 x+y=6$. Let an ellipse $E: x_{a_2}{ }^2+y_{b^2}=1, a>b$ of...

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Q JEE MAIN 2024

Bag A contains 3 white, 7 red balls and bag $B$ contains 3 white, 2 red balls. One bag is...

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Let $(\alpha, \beta, \gamma)$ be mirror image of the point $(2,3,5)$ in the line $-\underline{-2} \quad-\underline{-3} x_{-} y_{-} z$. Then...

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Q JEE MAIN 2024

Let P be a point on the hyperbola $\mathrm{H}: \frac{\mathrm{x}^2}{9}-\frac{\mathrm{y}^2}{4}=1$, in the first quadrant such that the area of triangle...

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Q Jee Main 2024

Let $\mathrm{f}, \mathrm{g}:(0, \infty) \rightarrow \mathrm{R}$ be two functions defined by $f(x)=\int_{-x}^x(|t|-t 2) e-t 2 d t$ and $...

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Q JEE-Main 2024

For $0

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Let $\mathrm{f}: \mathrm{R}-\left\{\frac{-1}{2}\right\} \rightarrow \mathrm{R}$ and $\mathrm{g}: \mathrm{R}-\left\{\frac{-5}{2}\right\} \rightarrow \mathrm{R}$ be defined as $f(x)=\frac{2 x+3}{2 x+1}$ and $g(x)=\frac{|x|+1}{2 x+5}$. Then...

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Q Jee Main 2024

Let a variable line passing through the centre of the circle $x^2+y^2-16 x-4 y=0$, meet the positive coordinate axes at...

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Let $P$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let the line passing through $P$ and parallel to $y$-axis meet...

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Q Jee Main 2024

If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of first four observations...

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Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at (1, $\mathrm{f}(1)$ )...

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Q JEE MAIN 2024

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b} \times \vec{a}|=2$. Then $|(\dot{b} \times \vec{a})-\vec{b}|^2$ is equal...

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Q Jee Main 2024

Let $P(3,2,3), Q(4,6,2)$ and $R(7,3,2)$ be the vertices of $\triangle \mathrm{PQR}$. Then, the angle $\angle \mathrm{QPR}$ is

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Q JEE MAIN 2024

If $x^2-y^2+2 h x y+2 g x+2 f y+c=0$ is the locus of a point, which moves such that it...

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Q Jee Main 2024

Let $z_1$ and $z_2$ be two complex number such that $z_1+z_2=5$ and $z_1^3+z_2^3=20+15 i$. Then $\left|z_1^4+z_2^4\right|$ equals-

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Let $A(a, b), B(3,4)$ and $(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of...

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Q JEE MAIN 2024

The value of $\int_0^1\left(2 x^3-3 x^2-x+1\right)^{\frac{1}{3}} d x$ is equal to:

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Q JEE Main 2024

Let $a$ and $b$ be be two distinct positive real numbers. Let $11^{\text {th }}$ term of a GP, whose...

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Let $f(x)=(x+3)^2(x-2)^3, x \in[-4,4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4,4]$, then...

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Q Jee Main 2024

The number of ways in which 21 identical apples can be distributed among three children such that each child gets...

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The $20^{\text {th }}$ term from the end of the progression $20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots,-129 \frac{1}{4}$ is...

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Q JEE-Main 2024

Let R be the interior region between the lines $3 x-y+1=0$ and $x+2 y-5=0$ containing the origin. The set of...

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Q JEE-Main 2024

Let the image of the point $(1,0,7)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ be the point $(\alpha, \beta, \gamma)$. Then which one...

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Q JEE-Main 2024

Consider the function $\mathrm{f}:(0,2) \rightarrow \mathrm{R}$ defined by $f(x)=\frac{x}{2}+\frac{2}{x}$ and the function $g(x)$ defined by $g(x)= \left\{\begin{array}{cc}\min \{f(t)\}, & 0

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Q JEE-Main 2024

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $...

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Q Jee Main 2024

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty...

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Q Jee Main 2024

Let $A=\left[\begin{array}{ccc}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{array}\right]$ and $...

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Q JEE Main 2024

Let $\vec{a}=\hat{\imath}+\alpha \hat{\jmath}+\beta \hat{k}, \alpha, \beta \in R$. Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and...

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Q JEE MAIN 2024

Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5 x+7 y=50$. Let the point $P$ divide...

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Q JEE MAIN 2024

The number of solutions of the equation $...

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Q JEE MAIN 2024

For $\alpha, \beta \in\left(0, \frac{\pi}{2}\right)$, let $3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)$ and a real number k be such that $...

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Q JEE MAIN 2024

Let $\alpha$ and $\beta$ be the roots of the equation $\mathrm{px}^2+\mathrm{qx}-r=0$, where $p \neq 0$. If $p, q$ and $r$...

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Q JEE MAIN 2024

Consider the system of linear equations $$ x+y+z=5, x+2 y+\lambda^2 z=9 $$ $x+3 y+\lambda z=\mu$, where $\lambda, \mu \in R$....

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Q JEE MAIN 2024

Let $f(\mathrm{x})=\left|2 \mathrm{x}^2+5\right| \mathrm{x}|-3|, \mathrm{x} \in \mathrm{R}$. If m and n denote the number of points where $f$ is not...

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Q JEE Main 2026

The number of elements in the set $\left\{x \in\left[0,180^{\circ}\right]: \tan \left(x+100^{\circ}\right)=\tan \left(x+50^{\circ}\right) \tan x \tan \left(x-50^{\circ}\right)\right\}$ is $\_\_\_\_$ .

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Q JEE Main 2026

Let S be a set of 5 elements and $\mathrm{P}(\mathrm{S})$ denote the power set of S . Let E be...

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Q JEE Main 2024

Let z=(1+i)(1+2i)(1+3i)...(1+n i), where $i=\sqrt{-1}$. If $|z|^2=44200$, then $n$ is equal to

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Q JEE Main 2026

Let $(h, k)$ lie on the circle $\mathrm{C}: x^2+y^2=4$ and the point $(2 h+1,3 k+2)$ lie on an ellipse with...

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Q JEE Main 2026

Let [t] denote the greatest integer less than or equal to t. If the function $...

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Q JEE Main 2026

Let $\vec{a}=2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ and $\vec{b}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$. If $\vec{c}$ is a vector such that $2(\vec{a} \times \vec{c})+3(\vec{b} \times \vec{c})=\overrightarrow{0}$...

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Q JEE Main 2026

The smallest positive integral value of $a$, for which all the roots of $x^4-a x^2+9=0$ are real and distinct, is...

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Q JEE Main 2026

Let $f$ be a function such that $3 f(x)+2 f\left(\frac{m}{19 x}\right)=5 x, x \neq 0$, where $m=\sum_{i=1}^9(i)^2$. Then $f(5)-f(2)$ is...

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Q JEE Main 2026

Let $y=y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1)=2$, and $...

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Q JEE Main 2026

Consider the following three statements for the function $f:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=\left|\log _e x\right|-|x-1|$ : (I) $f$...

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Q JEE Advance2015

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least...

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Q JEE Advance2018

Let X be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11, \ldots$., and Y be...

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Q JEE Advace

Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $$ f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)} $$...

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Q JEE MAIN 2021 S2

Which of the following is the negation of the statement "for all $M>0$, there exists $x \in S$ such that...

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Q JEE Main 2019

Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3\left|z_1\right|=4\left|z_2\right|$. If $z=\frac{3 z_1}{2 z_2}+\frac{2 z_2}{3 z_1}$ then

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Q JEE MAIN 2026

If $\sum_{r=1}^{2 s}\left(\frac{r}{r^4+r^2+1}\right)=\frac{p}{q}$, where $p$ and $q$ are positive integers such that $\operatorname{gcd}(p, q)=1$, then $p+q$ is equal to $\_\_\_\_$。

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Q JEE MAIN 2026

If the distance of the point $\mathrm{P}(43, \alpha, \beta), \beta

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Q JEE MAIN 2026

Let $A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ and $B$ be two matrices such that $A^{100}=100 B+I$. Then the sum...

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Q JEE MAIN 2026

Let $f$ be a differentiable function satisfying $f(x)=1-2 x+\int_0^x \mathrm{e}^{(x-t)} f(t) \mathrm{dt}, x \in \mathbf{R}$ and let $...

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Q JEE MAIN 2026

Three persons enter in a lift at the ground floor. The lift will go upto $...

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Q JEE MAIN 2026

Given below are two statements : Statement I: $25^{13}+20^{13}+8^{13}+3^{13}$ is divisible by 7 . Statement II: The integral part of...

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Q JEE MAIN 2026

Let $Q(a, b, c)$ be the image of the point $P(3,2,1)$ in the line $\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}$. Then the distance of $Q$...

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Q JEE MAIN 2026

Let $P Q R$ be a triangle such that $\overrightarrow{P Q}=-2 \hat{i}-\hat{j}+2 \hat{k}$ and $...

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Q JEE MAIN 2026

Let A be the focus of the parabola $y^2=8 x$. Let the line $y=m x+c$ intersect the parabola at two...

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Q JEE MAIN 2026

The value of $\sum_{\mathrm{r}=1}^{20}\left(\left|\sqrt{\pi\left(\int_0^{\mathrm{r}} \mathrm{x}|\sin \pi \mathrm{x}| \mathrm{dx}\right)}\right|\right)$ is $\_\_\_\_$ .

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Q JEE MAIN 2026

Let $$ \begin{aligned} & A=\{z \in \mathbb{C}:|z-2| \leqslant 4\} \text { and } \\ & B=\{z \in \mathbb{C}:|z-2|+|z+2|=5\} \end{aligned} $$...

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Q JEE MAIN 2026

If $k=\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1}\left(\frac{2}{3}\right)\right)+\tan \left(\frac{1}{2} \sin ^{-1}\left(\frac{2}{3}\right)\right)$, then the number of solutions of the equation $...

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Q JEE MAIN 2026

For some $\theta \in\left(0, \frac{\pi}{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $...

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Q JEE MAIN 2026

In a G.P., if the product of the first three terms is 27 and the set of all possible values...

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Q JEE MAIN 2026

Let $y=y(x)$ be the solution of the differential equation $$ x \frac{d y}{d x}-\sin 2 y=x^3\left(2-x^3\right) \cos ^2 y, x...

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Q JEE MAIN 2026

For three unit vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^2+|\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}|^2+|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|^2=9$ and $|2 \overrightarrow{\mathrm{a}}+\mathrm{k} \overrightarrow{\mathrm{b}}+\mathrm{k} \overrightarrow{\mathrm{c}}|=3$, the positive value of k is...

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Q JEE MAIN 2026

If the distances of the point $(1,2, a)$ from the line $\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}$ along the lines $\mathrm{L}_1: \frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b}$ and $\mathrm{L}_2: \frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c}$...

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Q JEE MAIN 2026

The common difference of the A.P: $a_1, a_2, \ldots, a_{\mathrm{m}}$ is 13 more than the common difference of the A.P....

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Q JEE MAIN 2026

Let $\mathrm{A}_2 \mathrm{~B}$ and $C$ be three $2 \times 2$ matrices with real entries such that $B=(I+A)^{-1}$ and $\mathrm{A}+\mathrm{C}=\mathrm{I}$. If...

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Q JEE MAIN 2026

The mean and variance of 10 observations are 9 and 34.2 , respectively. If 8 of these observations are 2,3,5,10,11,13,15,21,...

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Q JEE MAIN 2026

The value of $...

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Q JEE MAIN 2026

The area of the region $\mathrm{R}=\left\{(x, y): x y \leq 8,1 \leq y \leq x^2, x \geq 0\right\}$ is

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Q JEE MAIN 2026

Let $f$ be a polynomial function such that $f\left(\mathrm{x}^2+1\right)=\mathrm{x}^4+5 \mathrm{x}^2+2$, for all $\mathrm{x} \in \mathbb{R}$. Then $\int_0^3 f(\mathrm{x}) \mathrm{dx}$ is...

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Q JEE MAIN 2026

Given below are two statementsi Statement I: The function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x)=\frac{x}{1+|x|}$ is one-one. Statement II:...

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Q JEE MAIN 2026

If $\int\left(\frac{1-5 \cos ^2 x}{\sin ^5 x \cos ^2 x}\right) d x=f(x)+\mathrm{C}$, where C is the constant of integration, then...

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Q JEE MAIN 2026

The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\ldots+x^{1000}$ is :

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Q JEE MAIN 2026

If $\alpha, \beta$, where $\alpha

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Q JEE MAIN 2026

A bag contains 10 balls out of which $k$ are red and $(10-k)$ are black, where $...

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Q JEE MAIN 2026

Let $f(x)=\int \frac{\mathrm{d} x}{x^{\left(\frac{2}{3}\right)}+2 x^{\left(\frac{1}{2}\right)}}$ be such that $f(0)=-26+24 \log _{\mathrm{e}}(2)$. If $f(1)=\mathrm{a}+\mathrm{b} \log _{\mathrm{e}}(3)$, where $\mathrm{a}, \mathrm{b} \in \mathbf{Z}$,...

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Q JEE MAIN 2026

Let $\mathrm{S}=\left\{x^3+a x^2+b x+c: a, b, c \in \mathrm{~N}\right.$ and $\left.a, b, c \leq 20\right\}$ be a set of polynomials....

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Q JEE MAIN 2026

Let $y=y(x)$ be the solution of the differential equation $x \frac{\mathrm{~d} y}{\mathrm{~d} x}-y=x^2 \cot x, x \in(0, \pi)$. If $y\left(\frac{\pi}{2}\right)=\frac{\pi}{2}$,...

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Q JEE MAIN 2026

Let . Let be the number of 9 -digit numbers formed using the digits of the set S such that...

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Q JEE MAIN 2026

Let $\mathrm{P}_1: y=4 x^2$ and $\mathrm{P}_2: y=x^2+27$ be two parabolas. If the area of the bounded region enclosed between $P_1$...

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Q JEE MAIN 2026

Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line $...

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Q JEE MAIN 2026

Let $z$ be a complex number such that $|z-6|=5$ and $|z+2-6 i|=5$. Then the value of $z^3+3 z^2-15 z+141$ is...

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Q JEE MAIN 2026

Let $f(x)=\lim _{\boldsymbol{\theta} \rightarrow 0}\left(\frac{\cos \pi x-x^{\left(\frac{2}{\boldsymbol{\theta}}\right)} \sin (x-1)}{1+x^{\left(\frac{2}{\boldsymbol{\theta}}\right)}(x-1)}\right), x \in \mathbf{R}$. Consider the following two statements : (I) $f(x)$...

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Q JEE MAIN 2026

If $\frac{\tan (\mathrm{A}-\mathrm{B})}{\tan \mathrm{A}}+\frac{\sin ^2 \mathrm{C}}{\sin ^2 \mathrm{~A}}=1, \mathrm{~A}, \mathrm{~B}, \mathrm{C} \in\left(0, \frac{\pi}{2}\right)$, then

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Q JEE MAIN 2026

Let $P$ be a point in the plane of the vectors $\overrightarrow{A B}=3 \hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{A C}=\hat{i}-\hat{j}+3 \hat{k}$ such that...

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Q JEE MAIN 2026

$\frac{6}{3^{26}}+\frac{10 \cdot 1}{3^{25}}+\frac{10 \cdot 2}{3^{24}}+\frac{10 \cdot 2^2}{3^{23}}+\ldots+\frac{10 \cdot 2^{24}}{3}$ is equal to :

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Q JEE MAIN 2026

The sum of all the elements in the range of $$ \begin{aligned} & f(x)=\operatorname{Sgn}(\sin x)+\operatorname{Sgn}(\cos x)+\operatorname{Sgn}(\tan x)+\operatorname{Sgn}(\cot x), x \neq...

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Q JEE MAIN 2026

Let $y=x$ be the equation of a chord of the circle $\mathrm{C}_1$ (in the closed half-plane $x \geq 0$ )...

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Q JEE MAIN 2026

The value of $\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{k(k+1)}{k!}\right)$ is

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Q JEE MAIN 2026

If $g(x)=3 x^2+2 x-3, f(0)=-3$ and $4 g(f(x))=3 x^2-32 x+72$, then $f(g(2))$ is equal to:

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Q JEE MAIN 2026

Let the circle $x^2+y^2=4$ intersect $x$-axis at the points $\mathrm{A}(\mathrm{a}, 0), \mathrm{a}>0$ and $\mathrm{B}(\mathrm{b}, 0)$. Let $...

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Q JEE MAIN 2026

The probability distribution of a random variable X is given below : If $E(X)=\frac{263}{15}$, then $P(X

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Q JEE MAIN 2026

Let the ellipse $\mathrm{E}: \frac{x^2}{144}+\frac{y^2}{169}=1$ and the hyperbola $\mathrm{H}: \frac{x^2}{16}-\frac{y^2}{\lambda^2}=-1$ have the same foci. If e and L respectively denote...

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Q JEE MAIN 2026

An ellipse has its center at $(1,-2)$, one focus at $(3,-2)$ and one vertex at $(5,-2)$. Then the length of...

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Q JEE MAIN 2026

If the solution curve $y=f(x)$ of the differential equation $$ \left(x^2-4\right) y^{\prime}-2 x y+2 x\left(4-x^2\right)^2=0, x>2, $$ passes through the...

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Q JEE MAIN 2026

Considering the principal values of inverse trigonometric functions, the value of the expression $\tan \left(2 \sin ^{-1}\left(\frac{2}{\sqrt{13}}\right)-2 \cos ^{-1}\left(\frac{3}{\sqrt{10}}\right)\right)$ is...

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Q JEE MAIN 2026

Let the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$ be $\frac{5}{16}, a>2$. If $\alpha$ is such that $a, 4, \alpha, b$...

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Q JEE MAIN 2026

Let [.] denote the greatest integer function. Then $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\right) \mathrm{d} x$ is equal to :

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Q JEE MAIN 2026

Let S denote the set of 4-digit numbers abcd such that $a>b>c>d$ and P denote the set of 5-digit numbers...

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Q JEE MAIN 2026

Let $A=\left[\begin{array}{ccc}0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0\end{array}\right]$ and $B$...

JEE Main Mathematics Medium

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Q JEE MAIN 2026

The number of elements in the set $\mathrm{S}=\left\{x: x \in[0,100]\right.$ and $\left.\int_0^x t^2 \sin (x-t) d t=x^2\right\}$ is $\_\_\_\_$ .

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Q JEE MAIN 2026

If the image of the point $\mathrm{P}(a, 2, a)$ in the line $\frac{x}{2}=\frac{y+a}{1}=\frac{z}{1}$ is Q and the image of Q...

JEE Main Mathematics Medium

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Q JEE MAIN 2026

Let $\sum_{k=1}^n a_k=\alpha n^2+\beta n$. If $a_{10}=59$ and $a_6=7 a_1$, then $\alpha+\beta$ is equal to

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Q JEE MAIN 2026

An equilateral triangle OAB is inscribed in the parabola $y^2=4 x$ with the vertex O at the vertex of the...

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Q JEE MAIN 2026

If the mean and the variance of the data are $\mu$ and 19 respectively, then the value of $\lambda+\mu$ is

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Q JEE MAIN 2026

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}, i=\sqrt{-1}$, then $\left(z^{201}-i\right)^8$ is equal to

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Q JEE MAIN 2026

Let $\mathrm{A}=\{0,1,2, \ldots, 9\}$. Let R be a relation on A defined by $(x, y) \in \mathrm{R}$ if and only...

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Q JEE MAIN 2026

The number of ways, in which 16 oranges can be distributed to four children such that each child gets at...

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Q JEE MAIN 2026

Let $\mathrm{I}(x)=\int \frac{3 d x}{(4 x+6)\left(\sqrt{4 x^2+8 x+3}\right)}$ and $\mathrm{I}(0)=\frac{\sqrt{3}}{4}+20$. If $\mathrm{I}\left(\frac{1}{2}\right)=\frac{a \sqrt{2}}{b}+\mathrm{c}$, where $...

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Q JEE MAIN 2026

The least value of $\left(\cos ^2 \theta-6 \sin \theta \cos \theta+3 \sin ^2 \theta+2\right)$ is

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Q JEE MAIN 2026

The area of the region enclosed between the circles $x^2+y^2=4$ and $x^2+(y-2)^2=4$ is:

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Q JEE MAIN 2026

Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One...

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Q JEE MAIN 2026

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $\vec{a} \times \vec{b}=2(\vec{a} \times \vec{c})$. If $|\vec{a}|=1,|\vec{b}|=4,|\vec{c}|=2$, and the angle between...

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Q JEE MAIN 2026

Let PQ be a chord of the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$, perpendicular to the $x$-axis such that OPQ is an equilateral triangle,...

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Q JEE MAIN 2026

If the points of intersection of the ellipses $x^2+2 y^2-6 x-12 y+23=0$ and $4 x^2+2 y^2-20 x-12 y+35=0$ lie on...

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Q JEE MAIN 2026

The sum of all the real solutions of the equation $\log _{(x+3)}\left(6 x^2+28 x+30\right)=5-2 \log _{(6 x+10)}\left(x^2+6 x+9\right)$ is equal...

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Q JEE MAIN 2026

The system of linear equations $$ \begin{aligned} & x+y+z=6 \\ & 2 x+5 y+a z=36 \\ & x+2 y+3 z=b...

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Q JEE MAIN 2026

Consider two sets $\mathrm{A}=\{x \in \mathbb{Z}:(|x-3|-3) \mid \leq 1\}$ and $\mathrm{B}=\left\{x \in \mathbb{R}-\{1,2\}: \frac{(x-2)(x-4)}{x-1} \log _e(|x-2|)=0\right\}$. Then the number of...

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Q JEE MAIN 2026

Let $A(1,2)$ and $C(-3,-6)$ be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to...

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Q JEE MAIN 2026

Let $\vec{a}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \vec{b}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \vec{c}=\lambda \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\vec{v}=\vec{a} \times \vec{b}$. If $\vec{v} \cdot \vec{c}=11$ and the length of...

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Q JEE MAIN 2026

Let $\frac{\pi}{2}

JEE Main Mathematics Medium

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Q JEE MAIN 2026

If $f(x)=\left\{\begin{array}{cc}\frac{a|x|+x^2-2(\sin |x|)(\cos |x|)}{x} & , x \neq 0 \\ b & , x=0\end{array}\right.$ is continuous at $x=0$, then $a+b$...

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Q JEE MAIN 2026

The number of $3 \times 2$ matrices A , which can be formed using the elements of the set $\{-2,-1,0,1,2\}$...

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Q JEE MAIN 2026

Let ( $2 \alpha, \alpha$ ) be the largest interval in which the function $f(t)=\frac{|t+1|}{t^2}, t2$, is $\_\_\_\_$

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Q JEE MAIN 2026

The number of numbers greater than 5000 , less than 9000 and divisible by 3, that can be formed using...

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Q JEE MAIN 2026

Let a line L passing through the point $\mathrm{P}(1,1,1)$ be perpendicular to the lines $\frac{x-4}{4}=\frac{y-1}{1}=\frac{z-1}{1}$ and $\frac{x-17}{1}=\frac{y-71}{1}=\frac{z}{0}$. Let the line...

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Q JEE MAIN 2026

Let a differentiable function f satisfy the equation $\int_0^{36} f\left(\frac{t x}{36}\right) d t=4 \alpha f(x)$. If $y=f(x)$ is a standard...

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Q JEE MAIN 2026

The number of the real solutions of the equation: $x|x+3|+|x-1|-2=0$ is

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Q JEE MAIN 2026

Let $\mathrm{A}_1$ be the bounded area enclosed by the curves $y=x^2+2, x+y=8$ and $y$-axis that lies in the first quadrant....

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Q JEE MAIN 2026

The mean and variance of a data of 10 observations are 10 and 2 , respectively. If an observations $\alpha$...

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Q JEE MAIN 2026

From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then...

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Q JEE MAIN 2026

Let $\alpha, \beta \in \mathbb{R}$ be such that the function $f(x)=\left\{\begin{array}{ll}2 \alpha\left(x^2-2\right)+2 \beta x & , x

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Q JEE MAIN 2026

Consider an A.P.: $a_1, a_2, \ldots, a_{\mathrm{n}} ; a_1>0$. If $a_2-a_1=\frac{-3}{4}, a_{\mathrm{n}}=\frac{1}{4} a_1$, and $\sum_{\mathrm{i}=1}^{\mathrm{n}} a_{\mathrm{i}}=\frac{525}{2}$, then $\sum_{\mathrm{i}=1}^{17} a_1$ is...

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Q JEE MAIN 2026

Let $\mathrm{S}=\frac{1}{25!}+\frac{1}{3!23!}+\frac{1}{5!21!}+\ldots$ up to 13 terms. If $13 \mathrm{~S}=\frac{2^k}{n!}, k \in \mathrm{~N}$, then $n+k$ is equal to

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Q JEE MAIN 2026

Let $A(1,0), B(2,-1)$ and $C\left(\frac{7}{3}, \frac{4}{3}\right)$ be three points. If the equation of the bisector of the angle A B...

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Q JEE MAIN 2026

Let $R$ be a relation defined on the set $\{1,2,3,4\} \times\{1,2,3,4\}$ by $\mathrm{R}=\{((a, b),(c, d)): 2 a+3 b=3 c+4 d\}$....

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Q JEE MAIN 2026

Let $f(t)=\int\left(\frac{1-\sin \left(\log _e t\right)}{1-\cos \left(\log _e t\right)}\right) d t, t>1$. If $f\left(e^{\pi / 2}\right)=-e^{\pi / 2}$ and $...

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Q JEE MAIN 2026

If $\cot x=\frac{5}{12}$ for some $x \in\left(\pi, \frac{3 \pi}{2}\right)$, then $...

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Q JEE MAIN 2026

Let $\mathrm{S}=\left\{z \in \mathbb{C}:\left|\frac{z-6 i}{z-2 i}\right|=1\right.$ and $\left.\left|\frac{z-8+2 i}{z+2 i}\right|=\frac{3}{5}\right\}$. Then $\sum_{z \in \mathrm{~S}}|z|^2$ is equal to

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Q JEE MAIN_2026_

The number of 4 -letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is ______.

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Q JEE MAIN 2026

The value of $\frac{\sqrt{3} \operatorname{cosec} 20^{\circ}-\sec 20^{\circ}}{\cos 20^{\circ} \cos 40^{\circ} \cos 60^{\circ} \cos 80^{\circ}}$ is equal to

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Q JEE MAIN_2026_

From the first 100 natural numbers, two numbers first $a$ and then $b$ are selected randomly without replacement. If the...

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Q JEE MAIN_2026_

Let $f$ be a twice differentiable non-negative function such that $(f(x))^2=25+\int_0^x\left((f(\mathrm{t}))^2+\left(f^{\prime}(\mathrm{t})\right)^2\right) d t$. Then the mean of $...

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Q JEE MAIN_2026_

Let the area of the region bounded by the curve $y=\max \{\sin x, \cos x\}$, lines $x=0, x=\frac{3 \pi}{2}$, and...

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Q JEE MAIN_2026_

Let $|\mathrm{A}|=6$, where A is a $3 \times 3$ matrix. If $...

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Q JEE MAIN 2026

Let each of the two ellipses $\mathrm{E}_1: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$ and $\mathrm{E}_2: \frac{x^2}{\mathrm{~A}^2}+\frac{y^2}{\mathrm{~B}^2}=1,(\mathrm{~A}

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Q JEE MAIN_2026_

Let the direction cosines of two lines satisfy the equations: $4 l+m-n=0$ and $2 m n+10 n l+3 l m=0$....

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Q JEE MAIN 2026

If the domain of the function $$ f(x)=\log _{\left(10 x^2-17 x+7\right)}\left(18 x^2-11 x+1\right) $$ is $...

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Q JEE MAIN_2026_

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone...

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Q JEE MAIN 2026

Let $\vec{a}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}, \vec{b}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$ and $\vec{c}=\vec{a} \times \vec{b}$. Let $\vec{d}$ be a vector such that $|\vec{d}-\vec{a}|=\sqrt{11},|\vec{c} \times \vec{d}|=3$ and...

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Q JEE MAIN 2026

Let $729,81,9,1, \ldots$. be a sequence and $\mathrm{P}_n$ denote the product of the first $n$ terms of this sequence. If...

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Q JEE MAIN_2026

Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function $$ f(\theta)=4\left(\sin ^4\left(\frac{7 \pi}{2}-\theta\right)+\sin ^4(11...

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Q JEE MAIN 2026

Let the lines $\mathrm{L}_1: \vec{r}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}), \lambda \in \mathbb{R} \quad$ and $...

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Q JEE MAIN_2026

Let $\mathrm{S}=\{z: 3 \leq 2 z-3(1+\mathrm{i}) \mid \leq 7\}$ be a set of complex numbers. Then $...

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Q JEE MAIN 2026

Let a circle of radius 4 pass through the origin O , the points $\mathrm{A}(-\sqrt{3} a, 0)$ and $\mathrm{B}(0,-\sqrt{2} b)$,...

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Q JEE MAIN_2026

Number of solutions of $\sqrt{3} \cos 2 \theta+8 \cos \theta+3 \sqrt{3}=0, \theta \in[-3 \pi, 2 \pi]$ is :

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Q JEE MAIN_2026

The vertices $B$ and $C$ of a triangle $A B C$ lie on the line $\frac{x}{1}=\frac{1-y}{-2}=\frac{z-2}{3}$. The coordinates of $A$...

JEE Main Mathematics Medium

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Q JEE MAIN_2026_

The sum of all possible values of $\mathrm{n} \in \mathbf{N}$, so that the coefficients of $x_2 x^2$ and $x^3$ in...

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Q JEE MAIN 2026

If the function $f(x)=\frac{e^x\left(e^{\tan x-x}-1\right)+\log _e(\sec x+\tan x)-x}{\tan x-x}$ is continuous at $x=0$, then the value of $f(0)$ is equal...

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Q JEE MAIN_2026_

Let the mean and variance of 8 numbers $-10,-7,-1, x, y, 9,2,16$ be $\frac{7}{2}$ and $\frac{293}{4}$, respectively. Then the mean...

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Q JEE MAIN_2026_

The value of $\frac{{ }^{100} C_{50}}{51}+\frac{{ }^{100} C_{51}}{52}+\ldots .+\frac{{ }^{100} C_{100}}{101}$ is:

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Q JEE MAIN_2026_

Among the statements :

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Q JEE MAIN 2026

Let $\cos (\alpha+\beta)=-\frac{1}{10}$ and $\sin (\alpha-\beta)=\frac{3}{8}$, where $0

JEE Main Mathematics Easy

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Q JEE MAIN 2026

Let $S$ be the set of the first 11 natural numbers. Then the number of elements in $...

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Q JEE MAIN 2026

Let $[\cdot]$ be the greatest integer function. If $\alpha=\int_0^{64}\left(x^{1 / 3}-\left[x^{1 / 3}\right]\right) \mathrm{d} x$, then $...

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Q JEE MAIN 2026

Let a vector $\overrightarrow{\mathrm{a}}=\sqrt{2} \hat{i}-\hat{j}+\lambda \hat{k}, \lambda>0$, make an obtuse angle with the vector $\overrightarrow{\mathrm{b}}=-\lambda^2 \hat{i}+4 \sqrt{2} \hat{j}+4 \sqrt{2} \hat{k}$...

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Q JEE MAIN 2026

Suppose $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P. and $a^2, 2 b^2, c^2$ are in G.P. If $a

JEE Main Mathematics Medium

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Q JEE MAIN 2026

Let $n$ be the number obtained on rolling a fair die. If the probability that the system $$ \begin{aligned} &...

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Q JEE MAIN 2026

Let $[\bullet]$ denote the greatest integer function, and let $f(x)=\min \left\{\sqrt{2} x, x^2\right\}$. Let $\mathrm{S}=\left\{x \in(-2,2)\right.$ : the function $\mathrm{g}(x)=|x|\left[x^2\right]$...

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Q JEE MAIN 2026

Let L be the line $\frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6}$ and let S be the set of all points $(\mathrm{a}, \mathrm{b}, \mathrm{c})$ on L...

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Q JEE MAIN_2026_

The value of the integral $\int_{\frac{\pi}{24}}^{\frac{5 \pi}{24}} \frac{\mathrm{~d} x}{1+\sqrt[3]{\tan 2 x}}$ is :

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Q JEE MAIN_2026_

Let $f(x)=\left\{\begin{array}{ll}\frac{\mathrm{a} x^2+2 \mathrm{ax}+3}{4 x^2+4 x-3} & , x \neq-\frac{3}{2}, \frac{1}{2} \\ \mathrm{~b} & , x=-\frac{3}{2}, \frac{1}{2}\end{array}\right.$ be continuous at...

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Q JEE MAIN_2026

If $\alpha$ and $\beta(\alpha

JEE Main Mathematics Medium

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Q JEE MAIN_2026_

Let $y=y(x)$ be the solution of the differential equation $x^4 \mathrm{~d} y+\left(4 x^3 y+2 \sin x\right) \mathrm{d} x=0$, $x>0, y\left(\frac{\pi}{2}\right)=0$....

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Q JEE MAIN_2026_

Let $\mathrm{A}=\{-2,-1,0,1,2,3,4\}$. Let R be a relation on A defined by $x \mathrm{Ry}$ if and only if $...

JEE Main Mathematics Medium

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Q JEE MAIN_2026

Let the line $y-x=1$ intersect the ellipse $\frac{x^2}{2}+\frac{y^2}{1}=1$ at the points A and B . Then the angle made by...

JEE Main Mathematics Easy

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Q JEE MAIN 2026

Let the domain of the function $f(x)=\log _3 \log _5\left(7-\log _2\left(x^2-10 x+85\right)\right)+\sin ^{-1}\left(\left|\frac{3 x-7}{17-x}\right|\right)$ be ( $\alpha, \beta$ ]. Then...

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Q JEE MAIN 2026

Let $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\lambda \hat{j}+2 \hat{k}, \lambda \in Z$ be two vectors. Let $\vec{c}=\vec{a} \times \vec{b}$ and $\vec{d}$ be...

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Q JEE MAIN 2026

Let $f(x)=[x]^2-[x+3]-3, x \in \mathbf{R}$, where $[\cdot]$ is the greatest integer function. Then Options

JEE Main Mathematics Easy

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Q JEE MAIN_2026

A rectangle is formed by the lines $x=0, y=0, x=3$ and $y=4$. Let the line L be perpendicular to $...

JEE Main Mathematics Medium

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Q JEE MAIN 2026

Let the locus of the mid-point of the chord through the origin O of the parabola $y^2=4 x$ be the...

JEE Main Mathematics Easy

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Q JEE MAIN 2026

Among the statements (S1) : If $\mathrm{A}(5,-1)$ and $\mathrm{B}(-2,3)$ are two vertices of a triangle, whose orthocentre is $(0,0)$, then...

JEE Main Mathematics Medium

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Q JEE MAIN 2026

The area of the region $\mathrm{A}=\left\{(x, y): 4 x^2+y^2 \leq 8\right.$ and $\left.y^2 \leq 4 x\right\}$ is:

JEE Main Mathematics Easy

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Q JEE MAIN_2026

Let $\overrightarrow{\mathrm{a}}=-\hat{i}+\hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=\hat{i}-\hat{j}-3 \hat{k}, \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}$. Then $(\vec{a}-\vec{b}) \cdot \vec{d}$ is equal to:

JEE Main Mathematics Easy

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Q JEE MAIN 2026

If $\lim _{x \rightarrow 0} \frac{\mathrm{e}^{(\mathrm{a}-1) x}+2 \cos \mathrm{~b} x+(\mathrm{c}-2) \mathrm{e}^{-x}}{x \cos x-\log _{\mathrm{e}}(1+x)}=2$, then $\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2$ is equal to:

JEE Main Mathematics Medium

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Q JEE MAIN 2026

If the mean deviation about the median of the numbers $\mathrm{k}, 2 \mathrm{k}, 3 \mathrm{k}, \ldots \ldots, 1000 \mathrm{k}$ is...

JEE Main Mathematics Easy

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Q JEE MAIN_2026

Let $f(x)=\int \frac{\left(2-x^2\right) \cdot \mathrm{e}^x}{(\sqrt{1+x})(1-x)^{3 / 2}} \mathrm{~d} x$. If $f(0)=0$, then $f\left(\frac{1}{2}\right)$ is equal to :

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Q JEE MAIN 2026

Let $S$ and $S^{\prime}$ be the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $P(\alpha, \beta)$ be a point on the ellipse...

JEE Main Mathematics Medium

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Q JEE MAIN_2026_

Let the domain of the function $f(x)=\log _3 \log _5 \log _7\left(9 x-x^2-13\right)$ be the interval $(\mathrm{m}, \mathrm{n})$. Let the...

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Q JEE MAIN 2026

Let $f$ and $g$ be functions satisfying $f(x+y)=f(x) f(y), f(1)=7$ and $g(x+y)=g(x y), g(1)=1$, for all $x, y \in \mathbf{N}$....

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Q JEE MAIN 2026

If $X=\left[\begin{array}{c}x \\ y \\ z\end{array}\right]$ is a solution of the system of equations $A X=B$, where $...

JEE Main Mathematics Medium

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Q JEE MAIN 2026

If $y=y(x)$ satisfies the differential equation $16(\sqrt{x+9 \sqrt{x}})(4+\sqrt{9+\sqrt{x}}) \cos y \mathrm{~d} y=(1+2 \sin y) \mathrm{d} x, x>0$ and $y(256)=\frac{\pi}{2}, y(49)=\alpha$,...

JEE Main Mathematics Hard

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Q JEE MAIN_2026

Let ABC be a triangle. Consider four points $\mathrm{p}_1, \mathrm{p}_2, \mathrm{p}_3, \mathrm{p}_4$ on the side AB , five points $...

JEE Main Mathematics Easy

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Q JEE MAIN_2026

If $\frac{\cos ^2 48^{\circ}-\sin ^2 12^{\circ}}{\sin ^2 24^{\circ}-\sin ^2 6^{\circ}}=\frac{\alpha+\beta \sqrt{5}}{2}$, where $\alpha, \beta \in \mathbb{N}$, then $\alpha+\beta$ is equal...

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Q JEE MAIN_2026_

If $\int(\sin x)^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}} d x=-\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}}-\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}}-\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}}+\frac{p_4}{q_4}(\cot x)^{\frac{-3}{2}}+\mathrm{C}$, where $p_i$ and $q_i$ are positive integers with $\operatorname{gcd}\left(p_i, q_i\right)=1$...

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Q JEE MAIN_2026

Let $A$ be a $3 \times 3$ matrix such that $A+A^T=O$. If $...

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Q JEE MAIN_2026

Let $\alpha=\frac{-1+i \sqrt{3}}{2}$ and $\beta=\frac{-1-i \sqrt{3}}{2}, i=\sqrt{-1}$. If$(7-7 \alpha+9 \beta)^{20}+(9+7 \alpha-7 \beta)^{20}+(-7+9 \alpha+7 \beta)^{20}+(14+7 \alpha+7 \beta)^{20}=m^{10}$

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Q JEE MAIN_2026_

If the sum of the first four terms of an A.P. is 6 and the sum of its first six...

JEE Main Mathematics Medium

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Q JEE MAIN_2026_

Let the solution curve of the differential equation $x d y-y d x=\sqrt{x^2+y^2} d x, x>0, y(1)=0$, be $y=y(x)$. Then...

JEE Main Mathematics Medium

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Q JEE MAIN_2026_

The number of solutions of $\tan ^{-1} 4 x+\tan ^{-1} 6 x=\frac{\pi}{6}$, where $-\frac{1}{2 \sqrt{6}}

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Q JEE MAIN_2026_

If the chord joining the points $\mathrm{P}_1\left(x_1, y_1\right)$ and $\mathrm{P}_2\left(x_2, y_2\right)$ on the parabola $y^2=12 x$ subtends a right angle...

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Q JEE MAIN_2026_

The coefficient of $x^{48}$ in $(1+x)+2(1+x)^2+3(1+x)^3+\ldots+100(1+x)^{100}$ is equal to

JEE Main Mathematics Medium

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Q JEE MAIN_2026_

The value of $\int_{\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[x]+4}\right) d x$, where $[\cdot]$ denotes the greatest integer function, is

JEE Main Mathematics Medium

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Q JEE MAIN_2026_

If the domain of the function $f(x)=\sin ^{-1}\left(\frac{5-x}{3+2 x}\right)+\frac{1}{\log _{\varepsilon}(10-x)}$ is $(-\infty, \alpha] \cup[\beta, \gamma)-\{\delta\}$, then $6(\alpha+\beta+\gamma+\delta)$ is equal to

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Q JEE MAIN_2026_

If $\mathrm{A}=\left[\begin{array}{ll}2 & 3 \\ 3 & 5\end{array}\right]$, then the determinant of the matrix $\left(\mathrm{A}^{2025}-3 \mathrm{~A}^{2024}+\mathrm{A}^{2023}\right)$ is

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Q JEE MAIN_2026_

Let the set of all values of $r$, for which the circles $(x+1)^2+(y+4)^2=r^2$ and $x^2+y^2-4 x-2 y-4=0$ intersect at two...

JEE Main Chemistry Medium

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Q JEE MAIN_2026_

If the image of the point $\mathrm{P}(1,2, a)$ in the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{7-\mathrm{z}}{2}$ is $\mathrm{Q}(5, b, \mathrm{c})$, then $a^2+b^2+c^2$ is equal...

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Q JEE MAIN_2026

If the line $\alpha x+2 y=1$, where $\alpha \in \mathbb{R}$, does not meet the hyperbola $x^2-9 y^2=9$, then a possible...

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Q JEE MAIN 2026

The number of elements in the relation $\mathrm{R}=\left\{(x, y): 4 x^2+y^2

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Q JEE MAIN_2026_

Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function. If $6 \int_1^x f(t) d t=3 x f(x)+x^3-4$ for all $...

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Q JEE MAIN 2026

Let $\mathrm{P}(10,2 \sqrt{15})$ be a point on the hyperbola $\frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$, whose foci are S and $\mathrm{S}^{\prime}$. If the length of...

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Q JEE MAIN_2026_

The number of distinct real solutions of the equation $x|x+4|+3|x+2|+10=0$ is

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Q JEE MAIN 2026

Let $\mathrm{C}_{\mathrm{r}}$ denote the coefficient of $x^{\mathrm{r}}$ in the binomial expansion of $...

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Q JEE MAIN 2026

Let $\mathrm{S}=\left\{z \in \mathbb{C}: 4 z^2+\bar{z}=0\right\}$. Then $\sum_{z \in \mathrm{~S}}|z|^2$ is equal to:

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Q JEE MAIN_2026

If a random variable X has the probability distribution then $\mathrm{P}(3

JEE Main Mathematics Medium

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Q JEE MAIN 2026

Let $\alpha, \beta$ be the roots of the quadratic equation $12 x^2-20 x+3 \lambda=0, \lambda \in Z$. If $...

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Q JEE MAIN_2026_

Let $\mathrm{P}(\alpha, \beta, \gamma)$ be the point on the line $\frac{x-1}{2}=\frac{y+1}{-3}=z$ at a distance $4 \sqrt{14}$ from the point $(1,-1,0)$...

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Q JEE MAIN_2026_

Let $f(x)=x^{2025}-x^{2000}, x \in[0,1]$ and the minimum value of the function $f(x)$ in the interval $[0,1]$ be $(80)^{80}(n)^{-81}$. Then $n$...

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Q JEE MAIN_2026_

Let the line $x=-1$ divide the area of the region $\left\{(x, y): 1+x^2 \leq y \leq 3-x\right\}$ in the ratio...

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Q JEE MAIN_2026_

Two distinct numbers $a$ and $b$ are selected at random from $1,2,3, \ldots, 50$. The probability, that their product ab...

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Q JEE MAIN_2026

Let the relation $R$ on the set $M=\{1,2,3, \ldots, 16\}$ be given by $$ \mathrm{R}=\{(x, y): 4 y=5 x-3, x,...

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Q JEE MAIN_2026_

Let $\overrightarrow{\mathrm{AB}}=2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\overrightarrow{\mathrm{AD}}=\hat{i}+2 \hat{j}+\lambda \hat{k}, \lambda \in \mathbb{R}$. Let the projection of the vector $\vec{v}=\hat{i}+\hat{j}+\hat{k}$ on...

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Q JEE MAIN 2026

If P is a point on the circle $x^2+y^2=4, \mathrm{Q}$ is a point on the straight line $5 x+y+2=0$ and...

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Q JEE MAIN 2026

If $...

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Q JEE MAIN 2026

Let the maximum value of $\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2$ for $x \in\left[-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}\right]$ be $\frac{m}{n} \pi^2$, where $\operatorname{gcd}(m, n)=1$....

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Q JEE MAIN 2026

If $\int_0^1 4 \cot ^{-1}\left(1-2 x+4 x^2\right) \mathrm{d} x=\operatorname{atan}^{-1}(2)-\operatorname{blog}_{\mathrm{e}}(5)$, where $\mathrm{a}, \mathrm{b} \in \mathbf{N}$, then $(2 \mathrm{a}+\mathrm{b})$ is equal to

JEE Main Mathematics Hard

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Q JEE MAIN 2026

Let $[\cdot]$ denote the greatest integer function and $f(x)=\lim _{\mathrm{n} \rightarrow \infty} \frac{1}{\mathrm{n}^3} \sum_{\mathrm{k}=1}^{\mathrm{n}}\left[\frac{\mathrm{k}^2}{3^x}\right]$. Then $12 \sum_{\mathrm{j}=1}^{\infty} f(\mathrm{j})$ is equal...

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Q JEE MAIN 2026

The largest $\mathrm{n} \in \mathbf{N}$, for which $7^{\mathrm{n}}$ divides $101!$, is :

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Q JEE MAIN 2026

Let z be the complex number satisfying $|z-5| \leq 3$ and having maximum positive principal argument. Then $...

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Q JEE MAIN 2026

If the system of equations $$ \begin{aligned} & 3 x+y+4 z=3 \\ & 2 x+\alpha y-z=-3 \\ & x+2 y+z=4...

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Q JEE MAIN 2026

Let $\mathrm{A}=\{2,3,5,7,9\}$. Let R be the relation on A defined by $x \mathrm{R} y$ if and only if $...

JEE Main Mathematics Medium

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Q JEE MAIN 2026

Let $y=y(x)$ be the solution of the differential equation $...

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Q JEE MAIN 2026

For a triangle A B C, let $\vec{p}=\overrightarrow{B C}, \vec{q}=\overrightarrow{C A}$ and $\vec{r}=\overrightarrow{B A}$. If $|\vec{p}|=2 \sqrt{3},|\vec{q}|=2$ and $\cos \theta=\frac{1}{\sqrt{3}}$,...

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Q JEE MAIN 2026

For the matrices $\mathrm{A}=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}-29 & 49 \\ -13 & 18\end{array}\right]$, if $...

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Q JEE MAIN 2026

Let $\mathrm{A}=\left\{x:\left|x^2-10\right| \leq 6\right\}$ and $\mathrm{B}=\{x:|x-2|>1\}$. Then

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Q JEE MAIN 2026

Let $a_1, \frac{a_2}{2}, \frac{a_3}{2^2}, \ldots, \frac{a_{10}}{2^9}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. If $a_1+a_2+\ldots+a_{10}=62$, then $a_1$ is equal to:

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Q JEE MAIN 2026

If the area of the region $\left\{(x, y): 1-2 x \leq y \leq 4-x^2, x \geq 0, y \geq 0\right\}$...

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Q JEE MAIN 2026

A random variable X takes values $0,1,2,3$ with probabilities $\frac{2 a+1}{30}, \frac{8 a-1}{30}, \frac{4 a+1}{30}, b$ respectively, where $...

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Q JEE MAIN 2026

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+2 a x+(3 a+10)=0$ such that $\alpha

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Q JEE MAIN 2026

Let the line $\mathrm{L}_1$ be parallel to the vector $-3 \hat{i}+2 \hat{j}+4 \hat{k}$ and pass through the point $(2,6,7)$, and...

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Q JEE MAIN 2026

Let the line L pass through the point $(-3,5,2)$ and make equal angles with the positive coordinate axes. If the...

JEE Main Mathematics Medium

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Q JEE MAIN 2026

Let one end of a focal chord of the parabola $y^2=16 x$ be $(16,16)$. If $\mathrm{P}(\alpha, \beta)$ divides this focal...

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Q JEE MAIN 2026

Let $y^2=12 x$ be the parabola with its vertex at O . Let P be a point on the parabola...

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Q JEE MAIN 2026

If the line $\alpha x+4 y=\sqrt{7}$, where $\alpha \in \mathbf{R}$, touches the ellipse $3 x^2+4 y^2=1$ at the point P...

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Q JEE MAIN 2026

Let $f(x)=x^3+x^2 f^{\prime}(1)+2 x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in \mathbf{R}$. Then the value of $f^{\prime}(5)$ is:

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Q JEE MAIN 2026

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $f^{\prime \prime}(x)>0$ for all $x \in \mathbf{R}$ and...

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Q JEE MAIN 2026

The positive integer $n$, for which the solutions of the equation $x(x+2)+(x+2)(x+4)+\cdots+(x+2 n-2)(x+2 n)=\frac{8 n}{3}$ are two consecutive even integers,...

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Q JEE MAIN 2026

$6 \int_0^\pi|(\sin 3 x+\sin 2 x+\sin x)| d x$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2026

For some $\quad \alpha, \beta \in \mathbf{R}, \quad$ let $\quad A=\left[\begin{array}{ll}\alpha & 2 \\ 1 & 2\end{array}\right] \quad$ and $...

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Q JEE MAIN 2026

Let $S=\{(m, n): m, n \in\{1,2,3, \ldots, 50\}\}$. If the number of elements ( $\mathrm{m}, \mathrm{n}$ ) in S such...

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Q JEE MAIN 2026

Let $a_1=1$ and for $n \geq 1, a_{n+1}=\frac{1}{2} a_n+\frac{n^2-2 n-1}{n^2(n+1)^2}$. Then $\left|\sum_{n=1}^{\infty}\left(a_n-\frac{2}{n^2}\right)\right|$ is equal to

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Q JEE MAIN 2026

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that the quadratic equation $f(x) \mathrm{m}^2-2 f^{\prime}(x) \mathrm{m}+f^{\prime \prime}(x)=0$...

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Q JEE MAIN 2026

Let O be the vertex of the parabola $x^2=4 y$ and Q be any point on it. Let the locus...

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Q JEE MAIN 2026

The sum of all the roots of the equation $(x-1)^2-5|x-1|+6=0$, is :

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Q JEE MAIN 2026

The value of $\operatorname{cosec} 10^{\circ}-\sqrt{3} \sec 10^{\circ}$ is equal to :

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Q JEE MAIN 2026

If $x^2+x+1=0$, then the value of $\left(x+\frac{1}{x}\right)^4+\left(x^2+\frac{1}{x^2}\right)^4+\left(x^3+\frac{1}{x^3}\right)^4+\ldots .+\left(x^{25}+\frac{1}{x^{25}}\right)^4$ is:

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Q JEE MAIN 2026

If the coefficient of x in the expansion of $\left(a x^2+b x+c\right)(1-2 x)^{26}$ is -56 and the coefficients of $x^2$...

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Q JEE MAIN 2026

Let PQ and MN be two straight lines touching the circle $x^2+y^2-4 x-6 y-3=0$ at the points A and B...

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Q JEE MAIN 2026

Let $(\alpha, \beta, \gamma)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5,4,2)$ on the...

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Q JEE MAIN 2026

Let the mean and variance of 7 observations $2,4,10, x, 12,14, y, x>y$, be 8 and 16 respectively. Two numbers...

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Q JEE MAIN 2026

The value of $\int_{-\pi / 6}^{\pi / 6}\left(\frac{\pi+4 x^{11}}{1-\sin (|x|+\pi / 6)}\right) d x$ is equal to:

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Q JEE MAIN 2026

Let the foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$. If the eccentricity of the hyperbola...

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Q JEE MAIN 2026

Let $f: \mathbf{R} \rightarrow(0, \infty)$ be a twice differentiable function such that $f(3)=18, f^{\prime}(3)=0$ and $f^{\prime \prime}(3)=4$. Then $...

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Q JEE MAIN 2026

The number of strictly increasing functions $f$ from the set $\{1,2,3,4,5,6\}$ to the set $\{1,2,3, \ldots, 9\}$ such that $...

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Q JEE MAIN 2026

Let $y=y(x)$ be the solution curve of the differential equation $\left(1+x^2\right) d y+\left(y-\tan ^{-1} x\right) d x=0, y(0)=1$. Then the...

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Q JEE MAIN 2026

Let $\vec{c}$ and $\vec{d}$ be vectors such that $|\vec{c}+\vec{d}|=\sqrt{29}$ and $\vec{c} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \vec{d}$....

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Q JEE MAIN 2026

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms such that $a_2 \cdot a_3 \cdot a_4=64$ and...

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Q JEE MAIN 2026

Let $\overrightarrow{\mathrm{a}}=-\hat{i}+2 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=8 \hat{i}+7 \hat{j}-3 \hat{k}$ and $\overrightarrow{\mathrm{c}}$ be a vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}}$. If $...

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Q JEE MAIN 2026

Let a point A lie between the parallel lines $L_1$ and $L_2$ such that its distances from $L_1$ and $L_2$...

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Q JEE MAIN 2026

The number of relations, defined on the set $\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\}$, which are both reflexive and symmetric, is equal...

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Q JEE MAIN 2026

The area of the region, inside the ellipse $x^2+4 y^2=4$ and outside the region bounded by the curves $y=|x|-1$ and...

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Q JEE MAIN 2026

If the domain of the function $f(x)=\cos ^{-1}\left(\frac{2 x-5}{11-3 x}\right)+\sin ^{-1}\left(2 x^2-3 x+1\right)$ is the interval $[\alpha, \beta]$, then $...

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Q JEE MAIN_2019

$$ \text { If } \int \frac{d x}{\left(x^2-2 x+10\right)^2}=A\left(\tan ^{-1}\left(\frac{x-1}{3}\right)+\frac{f(x)}{x^2-2 x+10}\right)+C $$ where C is a constant of integration, then...

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Q JEE MAIN_2019_

ABC is a triangular park with $\mathrm{AB}=\mathrm{AC}=100$ metres. A vertical tower is situated at the mid-point of $B C$. If...

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Q JEE MAIN_2019_

Let $f \sim R \rightarrow R$ be differentiable at $c \in R$ and $f(c)=$ 0. If $g(x)=|f(x)|$, then at $...

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Q JEE MAIN_2019_

If $y=y(x)$ is the solution of the differential equation $\frac{\mathrm{dt}}{\mathrm{dx}}=(\tan \mathrm{x}-\mathrm{y}) \sec ^2 \mathrm{x}, \mathrm{x} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, such that $...

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Q JEE MAIN_2019_

If $\alpha$ and $\beta$ are the roots of the quadratic equation, $x^2+x \sin \theta-2 \sin \theta=0, \quad \theta \in\left(0, \frac{\pi}{2}\right)$,...

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Q JEE MAIN_2019

Let $f(x)=e^x-x$ and $g(x)=x^2-x, \forall x \in R$. Then the set of all $x \in R$, where the function $...

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Q _ JEE MAIN_2019_

The sum $$ \frac{3 \times 1}{1^2}+\frac{5 \times\left(1^3+2^3\right)}{1^2+2^2}+\frac{7 \times\left(1^3+2^3+3^3\right)}{1^2+2^2+3^2}+\ldots $$ upto $10^{\text {th }}$ term, is.

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Q JEE MAIN_2019

If the coefficients of $x^2$ and $x^3$ are both zero, in the expansion of the expression $...

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Q JEE MAIN_2019_

Let $f(x)=x^2, x \in R$. For any $A \subseteq R$, define $g(A)=\{x \in R: f(x) \in A\}$. If $S=[0,4]$, then...

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Q JEE MAIN_2019_

If $Q(0,-1,-3)$ is the image of the point $P$ in the plane $3 x-y+4 z=2$ and $R$ is the point...

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Q JEE MAIN_2019_

If $a_1, a_2, a_3, \ldots \ldots .$, an are in A.P. and $a_1+a_4+a_7+ a_{16}+a_{16}=114$, then $a_1+a_6+a_{11}+a_{16}$ is equal to:

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Q JEE MAIN_2019_

The region represented by $|x-y| \leq 2$ and $|x+y| \leq 2$ is bounded by a:

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Q JEE MAIN_2019_

If the circles $x^2+y^2+5 K x+2 y+K=0$ and $2\left(x^2+\right. \left.y^2\right)+2 K x+3 y-1=0,(K \in R)$, intersect at the points $P$...

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Q Statistics_ JEE MAIN_2019_

If for some $x \in R$, the frequency distribution of the marks obtained by 20 students in a test is:

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Q JEE MAIN_2019_

Assume that each born child is equally likely to be a boy or a girl. If two families have two...

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Q JEE MAIN_2019_

If the system of linear equations $$ \begin{aligned} & x+y+z=5 \\ & x+2 y+2 z=6 \\ & x+3 y+\lambda z=\mu,(\lambda,...

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Q JEE MAIN_2019_

Let $A(3,0-1), B(2,10,6)$ and $C(1,2,1)$ be the vertices of a triangle and $M$ be the mid point of $A C$....

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Q JEE MAIN_2019_

If the line $x-2 y=12$ is tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the point $\left(3, \frac{-9}{2}\right)$, then the length of...

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Q JEE MAIN_2019_

If a directrix of a hyperbola centred at the origin and passing through the point $(4,-2 \sqrt{3})$ is $...

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Q JEE MAIN_2019

If $\lim _{x \rightarrow 1} \frac{x^4-1}{x-1}=\lim _{x \rightarrow *} \frac{x^3-k^3}{x^2-k^2}$, then $k$ is

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Q JEE MAIN_2019

Which one of the following Boolean expressions is a tautology ?

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Q JEE MAIN_2019_

The line x = y touches a circle at the point (1, 1). If the circle also passes through the...

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Q JEE MAIN_2019_

$\lim _{n \rightarrow \infty}\left(\frac{(n+1)^{1 / 3}}{n^{4 / 3}}+\frac{(n+2)^{1 / 3}}{n^{1 / 3}}+\ldots . .+\frac{(2 n)^{1 / 3}}{n^{4 / 3}}\right)$ is...

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Q JEE MAIN 2019

Four persons can hit a target correctly with probabilities $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and $\frac{1}{8}$ respectively. If all hit at the...

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Q JEE MAIN 2019

If the function $f: R-\{1,-1\} \rightarrow A$ defined by $f(x)=\frac{x^2}{1-x^2}$, is surjective, then $A$ is equal to :

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Q JEE MAIN 2019

Let the sum of the first $n$ terms of a nonconstant A.P., $a_1, a_2, a_3, \ldots$. be $50 n+\frac{n(n-7)}{2} A$,...

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Q JEE MAIN 2019

The solution of the differential equation $x \frac{d y}{d x}+2 y=x^2(x \neq 0)$ with $y(1)=1$, is :

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Q JEE MAIN 2019

The value of $\int_0^{\pi / 2} \frac{\sin ^3 x}{\sin x+\cos x} d x$ is :

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Q JEE MAIN 2019

Slope of a line passing through P(2, 3) and intersecting the line, x + y = 7 at a distance...

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Q JEE MAIN 2019

A committee of 11 members is to be formed from 8 males and 5 females. If m is the number...

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Q JEE MAIN 2019

The value of $\cos ^2 10^{\circ}-\cos 10^{\circ} \cos 50^{\circ}+\cos ^2 50^{\circ}$ is:

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Q JEE MAIN 2019

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+x+1=0$. Then for $y \neq 0$ in $...

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Q JEE MAIN 2019

If the line $y=m x+7 \sqrt{3}$ is normal to the hyperbola $\frac{x^2}{24}-\frac{y^2}{18}=1$, then a value of $m$ is :

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Q JEE MAIN_2019_

If the length of the perpendicular from the point ( $\beta$, $0, \beta)(\beta \neq 0)$ to the line, $\frac{x}{1}=\frac{y-1}{0}=\frac{z+1}{-1}$ is...

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Q _ JEE MAIN_2019_

$$ \text { If } f(x)=\left\{\begin{array}{cc} \frac{\sin (p+1) x+\sin x}{x} & , x0 \end{array}\right. $$ is continuous at $x=0$, then...

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Q JEE MAIN 2019

The area (in sq. units) of the region $A=\left\{(x, y): x^2 \leq y \leq x+2\right\}$ is :

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Q JEE MAIN_2019_

If $...

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Q JEE MAIN 2019

Let f(x) = 15 – |x – 10|; x $\in$ R. Then the set of all values of x, at...

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Q JEE MAIN 2019

Let $S=\left\{\theta \in[-2 \pi, 2 \pi] ; 2 \cos ^2 \theta+3 \sin \theta=0\right\}$ . Then the sum of the elements...

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Q JEE MAIN_2019_

The value of $\int_0^{2 \pi}[\sin 2 x(1+\cos 3 x)] d x$, where $[t]$ denotes the greatest integer function, is:

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Q JEE MAIN_2019_

The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9...

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Q JEE MAIN 2019

$...

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Q JEE MAIN_2019

If $a>0$ and $z=\frac{(1+i)^2}{a-i}$, has magnitude $\sqrt{\frac{2}{5}}$, then $\bar{z}$ is equal to:

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Q JEE MAIN 2019

Let $\sum_{k=1}^{10} f(a+k)=16\left(2^{10}-1\right)$ , where the function f satisfies f(x + y) = f(x) f(y) for all natural numbers x,...

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Q JEE MAIN_2019_

All the pairs ( $x, y$ ) that satisfy the inequality $...

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Q JEE MAIN 2019

Let $\vec{\alpha}=3 \hat{i}+\hat{j}$ and $\vec{\beta}=2 \hat{i}-\hat{j}+3 \hat{k}$. If $\vec{\beta}=\vec{\beta}_1-\vec{\beta}_2$ , where $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular...

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Q JEE MAIN 2019

All the points in the set $\mathrm{S}=\left\{\frac{\alpha+\mathrm{i}}{\alpha-\mathrm{i}}: \alpha \in \mathrm{R}\right\}(\mathrm{i}=\sqrt{-1})$ lie on a :

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Q JEE MAIN 2019

If the function f defined on $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ by $...

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Q JEE MAIN 2019

If one end of a focal chord of the parabola, $y^2$ = 16x is at (1, 4), then the length...

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Q JEE MAIN 2019

The integral $\int \sec ^{2 / 3} x \operatorname{cosec}^{4 / 3}$ is equal to : (Here C is a constant...

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Q JEE MAIN 2019

If the fourth term in the Binomial expansion of $\left(\frac{2}{x}+x^{\log _8 x}\right)^6$ (x > 0) is 20 $\times 8^7$ ,...

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Q JEE MAIN 2019

If the standard deviation of the numbers –1, 0, 1, k is $\sqrt{5}$ where k > 0, then k is...

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Q JEE MAIN 2019

If f(x) is a non-zero polynomial of degree four, having local extreme points at x = –1, 0, 1; then...

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Q JEE MAIN 2019

If a tangent to the circle $x^2+y^2=1$ intersects the coordinate axes at distinct points P and Q, then the locus...

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Q JEE MAIN 2019

If the line, $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ meets the plane, x + 2y + 3z = 15 at a point P, then the...

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Q JEE MAIN 2019

Let p, q$\in$ R. If $2-\sqrt{3}$ is a root of the quadratic equation, $x^2+p x+q=0$, then :

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Q JEE MAIN 2019

If the line $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}$ intersects the plane $2 x+3 y-z+13=0$ at a point $P$ and the plane $3 x+y+4 z=16$...

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Q JEE MAIN 2019

If $m$ is the minimum value of $k$ for which the function $f(x)=x \sqrt{k x-x^2}$ is increasing in the interval...

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Q JEE MAIN 2019

For $x \in R$, let $[x]$ denote the greatest integer $\leq x$, then the sum of the series $...

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Q JEE MAIN 2019

If $B=\left[\begin{array}{ccc}5 & 2 \alpha & 1 \\ 0 & 2 & 1\end{array}\right]$ is the inverse of a $...

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Q JEE MAIN 2019

If $\alpha$ and $\beta$ are the roots of the equation $375 x^2-25 x-2=0$, then $...

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Q JEE MAIN 2019

If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix such that $...

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Q JEE MAIN 2019

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Q JEE MAIN 2019

Let $z \in C$ with $\operatorname{lm}(z)=10$ and it satisfies $\frac{2 z-n}{2 z+n}=2 i-1$ for some natural number $n$. Then :

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Q JEE MAIN 2019

A value of $\alpha$ such that $$ \int_\alpha^{\alpha+1} \frac{\mathrm{dx}}{(\mathrm{x}+\alpha)(\mathrm{x}+\alpha+1)}=\log _{\mathrm{e}}\left(\frac{9}{8}\right) \text { is: } $$

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Q JEE MAIN 2019

Let A, B and C be sets such that φ ≠ A ∩ B ⊆ C. Then which of the...

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Q JEE MAIN 2019

A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice),...

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Q JEE MAIN 2019

A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through...

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Q JEE MAIN 2019

The term independent of $x$ in the expansion of $\left(\frac{1}{60}-\frac{x^8}{81}\right) \cdot\left(2 x^2-\frac{3}{x^2}\right)^6$ is equal to :

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Q JEE MAIN 2019

$\lim _{x \rightarrow 0} \frac{x+2 \sin x}{\sqrt{x^2+2 \sin x+1}-\sqrt{\sin ^2 x-x+1}}$ is :

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Q JEE MAIN 2019

If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. such that the equations $...

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Q JEE MAIN 2020


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Q JEE MAIN 2020

If the variance of the terms in an increasing A.P., $b_1, b_2, b_3, \ldots, b_{11}$ is 90 , then the...

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Q JEE MAIN 2020

For a positive integer $n,\left(1+\frac{1}{x}\right)^n$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are...

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Q JEE MAIN 2020

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of $...

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Q JEE MAIN 2019

A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (–1,...

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Q JEE MAIN 2020

If $y=\sum_{k=1}^6 k \cos ^{-1}\left\{\frac{3}{5} \cos k x-\frac{4}{5} \sin k x\right\}$, then $\frac{d y}{d x}$ at $x=0$ is

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Q JEE MAIN 2019

If ${ }^{20} \mathrm{C}_1+\left(2^2\right)^{20} \mathrm{C}_2+\left(3^2\right)^{20} \mathrm{C}_3+\ldots \ldots . .+\left(20^2\right)^{20} \mathrm{C}_{20}=\mathrm{A}\left(2^\beta\right)$, then the ordered pair $(\mathrm{A}, \beta)$ is equal to:

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Q JEE MAIN 2020

The area (in sq. units) of an equilateral triangle inscribed in the parabola $y^2=8 x$, with one of its vertices...

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Q JEE MAIN 2020

If a curve $\mathrm{y}=f(\mathrm{x})$, passing through the point $(1,2)$, is the solution of the differential equation, $...

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Q JEE MAIN 2019

Let $\alpha \in(0, \pi / 2)$ be fixed. If the integral $...

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Q JEE MAIN 2020

If the equation $\cos ^4 \theta+\sin ^4 \theta+\lambda=0$ has real solutions for $\theta$, then $\lambda$ lies in the interval

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Q JEE MAIN 2019

The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines

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Q JEE Main 2020

The integral $\int_0^2 \| x-1|-x| d x$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2020

Let $f:(-1, \infty) \longrightarrow R$ be defined by $f(0)=1$ and $f(x)=\frac{1}{x} \log _e(1+x), x \neq 0$. Then the function $f$...

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Q JEE MAIN 2019

Let α ∈ R and the three vectors

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Q JEE Main 2020

If $\lim _{x \rightarrow 1} \frac{x+x^2+x^3+\ldots .+x^n-n}{x-1}=820,(n \in N)$ then the value of $n$ is equal to $\_\_\_\_$

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Q JEE MAIN 2020

$\lim _{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+x\right)\right)^{\frac{1}{x}}$ is equal to :

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Q JEE MAIN 2019

Let S be the set of all α ∈ R such that the equation, cos2x + αsinx = 2α –...

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Q JEE MAIN 2020

Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function which satisfies $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x})+f(\mathrm{y}) \forall \mathrm{x}, \mathrm{y} \in \mathrm{R}$. If $f(1)=2$ and $...

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Q JEE MAIN 2019

A plane which bisects the angle between the two given planes 2x – y + 2z – 4 = 0...

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Q JEE MAIN 2020

A plane passing through the point $(3,1,1)$ contains two lines whose direction ratios are $1,-2,2$ and $2,3,-1$ respectively. If this...

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Q JEE MAIN 2019

The general solution of the differential equation $\left(y^2-x^3\right) d x-x y d y=0(x \neq 0)$ is : (where c is...

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Q JEE MAIN 2020

Let $E^C$ denote the complement of an event $E$. Let $E_1, E_2$ and $E_3$ be any pairwise independent events with...

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Q JEE MAIN 2020

Let $S$ be the sum of the first 9 terms of the series : $\{x+k a\}+\left\{x^2+(k+2) a\right\}+\left\{x^3+(k+4) a\right\}+\left\{x^4+(k+6) a\right\}+\ldots$ where...

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Q JEE Main 2020

If the letters of the word ‘MOTHER’ be permuted and all the words so formed (with or without meaning) be...

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Q JEE Main 2020

The number of integral values of $k$ for which the line, $3 x+4 y=k$ intersects the circle, $...

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Q JEE MAIN 2020

Let $n>2$ be an integer. Suppose that there are $n$ Metro stations in a city located along a circular path....

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Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $|\vec{a}-\vec{b}|^2+|\vec{a}-\vec{c}|^2=8$. Then $|\vec{a}+2 \vec{b}|^2+|\vec{a}+2 \vec{c}|^2$ is equal to

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Q JEE MAIN 2020

Consider a region $\mathrm{R}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathrm{R}^2 \cdot \mathrm{x}^2 \leq \mathrm{y} \leq 2 \mathrm{x}\right\}$. if a line $\mathrm{y}=\alpha$ divides the...

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Q JEE MAIN 2019

The Boolean expression ~ (p ⇒ (~ q)) is equivalent to

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Q JEE-MAIN 2019

The Boolean expression $((p \wedge q) \vee(p \vee \sim q)) \wedge(\sim p \wedge \sim q)$ is equivalent to

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Q JEE Main 2020

Let $\alpha>0, \beta>0$ be such that $\alpha^3+\beta^2=4$. If the maximum value of the term independent of $x$ in the binomial...

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Q JEE Main 2020

Area (in sq. units) of the region outside $\frac{|x|}{2}+\frac{|y|}{3}=1$ and inside the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is :

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Q JEE-MAIN 2019

Let $S_k=\frac{1+2+3+\ldots .+k}{k}$. If $\mathrm{S}_1^2+\mathrm{S}_2^2+\ldots \ldots+\mathrm{S}_{10}^2=\frac{5}{12} \mathrm{~A}$, then A is equal to

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Q JEE MAIN 2019

A value of θ ∈ (0, π/3), for which

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Q JEE MAIN 2020

Let f(x) be a quadratic polynomial such that f(–1) + f(2) = 0. If one of the roots of f(x)...

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Q JEE-MAIN 2019

The integral $\int \cos \left(\log _e x\right) d x$ is equal to (where $C$ is a constant of integration)

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Q JEE MAIN 2020

If the sum of first 11 terms of an A.P., $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3, \ldots$ is $0\left(\mathbf{a}_1 \neq 0\right)$, then the...

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Q JEE Main 2020

The sum of the first three terms of a G.P. is S and their product is 27. Then all such...

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Q JEE MAIN 2019

If [x] denotes the greatest integer ≤ x, then the system of linear equations [sinθ]x + [– cosθ]y = 0...

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Q JEE-MAIN 2019

Let $S$ be the set of all points in $(-\pi, \pi)$ at which the function, $f(x)=\min \{\sin x, \cos x\}$...

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Q JEE Main 2020

The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y,...

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Q JEE Main 2020

Let $A$ be a $2 \times 2$ real matrix with entries from $\{0,1\}$ and $|A| \neq 0$. Consider the following...

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Q JEE MAIN 2019

The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be...

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Q JEE MAIN 2020

The set of all possible values of  in the interval (0, ) for which the points (1, 2) and...

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Q JEE Main 2020

If $R=\left\{(x, y) ; x, y \in Z, x^2+3 y^2 \leq 8\right\}$ is a relation on the set of integers...

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Q JEE MAIN 2022

Let $\ell$ be a line which is normal to the curve $y=2 x^2+x+2$ at a point $P$ on the curve....

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Q JEE MAINN 2019

If $a_1, a_2, a_3, \ldots .$. are in A.P. such that $a_1+a_7+a_{16}=40$, then the sum of the first 15 terms...

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Q JEE MAIN 2020

For some $\theta \in\left(0, \frac{\pi}{2}\right)$, if the eccentricity of the hyperbola, $x^2-y^2 \sec ^2 \theta=10$ is $\sqrt{5}$ times the eccentricity...

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Q JEE MAIN 2020

Let $A=\left\{X=(x, y, z)^{\top}: P X=0\right.$ and $\left.x^2+y^2+z^2=1\right\}$, where $...

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Q JEE Main 2020

Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A...

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Q JEE MAIN 2019

A group of students comprises of 5 boys and n girls. If the number of ways, in which a team...

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Q JEE MAIN 2020

The imaginary part of $(3+2 \sqrt{-54})^{1 / 2}-(3-2 \sqrt{-54})^{1 / 2}$ can be :

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Q JEE MAIN 2019

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Q JEE MAIN 2020

Let $a, b, c \in R$ be all non-zero and satisfy $a^3+b^3+c^3=2$. If the matrix $$ A=\left(\begin{array}{lll} a & b...

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Q JEE Main 2020

A line parallel to the straight line $2 x-y=0$ is tangent to the hyperbola $\frac{x^2}{4}-\frac{y^2}{2}=1$ at the point $\left(x_1, y_1\right)$....

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Q JEE MAIN 2019

If the vertices of a hyperbola be at (–2, 0) and (2, 0) and one of its foci be at...

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Q JEE MAIN 2019

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Q JEE MAIN 2019

The maximum area (in sq. units) of a rectangle having its base on the $x$-axis and its other two vertices...

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Q JEE MAIN 2019

If $\frac{z-\alpha}{z+\alpha}(a \in R)$ is a purely imaginary number and $|z|=2$, then a value of $\alpha$ is

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Q JEE-MAIN 2019

If the straight line, $2 x-3 y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15, \beta)$,...

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Q JEE-MAIN 2019

Considering only the principal values of inverse functions, the set $A=\left\{x \geq 0 ; \tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\right\}$

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Q JEE-MAIN 2019

If $\lambda$ be the ratio of the roots of the quadratic equation in $x, 3 m^2 x^2+m(m-4) x+2=0$, then the...

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Q JEE-MAIN 2019

The sum of the distinct real values of $\mu$, for which the vectors, $\mu \hat{j}+\hat{j}+\hat{k}, \hat{i}+\mu \hat{j}+\hat{k}, \hat{i}+\hat{j}+\mu \hat{k}$ are...

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Q JEE-MAIN 2019

Let $C 1$ and $C 2$ be the centres of the circles $x^2+y^2-2 x-2 y-2=0$ and $x^2+y^2-6 x-6 y+14=0$ respectively....

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Q JEE-MAIN 2019

The area (in sq. units) of the region bounded by the parabola, $y=x^2+2$ and the lines, $y=x+1, x=0$ and $x=3$,...

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Q JEE MAIN 2019

Let $y=y(x)$ be the solution of the differential equation, $x \frac{d y}{d x}+y=x \log _e x x,(x>1)$. If $...

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Q JEE-MAIN 2019

If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is

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Q JEE-MAIN 2019

A tetrahedron has vertices $P(1,2,1), Q(2,1,3), R(-1,1,2)$ and $O(0,0,0)$. The angle between the faces $O P Q$ and $...

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Q JEE-MAIN 2019

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first...

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Q JEE MAIN 2019

The number of natural numbers less than 7,000 which can be formed by using the digits 0, 1, 3, 7,9...

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Q JEE-MAIN 2019

If a variable line, $3 x+4 y-\lambda=0$ is such that the two circles $x^2+y^2-2 x-2 y+1=0$ and $x^2+y^2-18 x-2 y+78=0$...

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Q JEE Main 2020

If a function f(x) defined by be continuous for some $a, b, c \in R$ and $f^{\prime}(0)+f^{\prime}(2)=e$, then the value...

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Q JEE-MAIN 2019

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the...

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Q JEE MAIN 2019

Let the equations of two sides of a triangle be 3x – 2y + 6 = 0 and 4x +...

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Q JEE Main 2020

Let $y=y(x)$ be the solution of the differential equation, $\frac{2+\sin x}{y+1} \cdot \frac{d y}{d x}-\cos x, y>0, y(0)=1$. If $...

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Q JEE-MAIN 2019

Consider three boxes, each containing 10 balls labelled 1, 2, ...,10. Suppose one ball is randomly drawn from each of...

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Q JEE MAIN 2019

The perpendicular distance from the origin to the plane containing the two lines, $\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7}$ and $\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7}$ is

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Q JEE MAIN 2019

An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the...

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Q JEE Main 2020

The value of $\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)$ is :

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Q JEE MAIN 2019

$$ \text { The derivative of } \tan ^{-1}\left(\frac{\sin x-\cos x}{\sin x+\cos x}\right) \text {, with respect to } \frac{x}{2}...

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Q JEE-MAIN 2019

Let $f$ and $g$ be continuous functions on $[0$, a $]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$, then $...

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Q JEE MAIN 2019

The area of the region A = {(x, y) : 0 ≤ y ≤ x|x| + 1 and –1 ≤...

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Q JEE Main 2020

The domain of the function $f(x)=\sin ^{-1}\left(\frac{|x|+5}{x^2+1}\right)$ is $(-\infty,-a] \cup[a, \infty)$. Then a is equal to:

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Q JEE MAIN 2019

For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that...

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Q JEE-MAIN 2019

A ratio of the 5 th term from the beginning to the $5^{\text {th }}$ term from the end in...

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Q JEE MAIN 2019

A data consists of $n$ observations $x_1, x_2 \ldots, x_n$. If $\sum_{i=1}^n\left(x_i+1\right)^2=9 n$ and $\sum_{i=1}^n\left(x_i-1\right)^2=5 n$, then the standard deviation...

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Q JEE Main 2020

Let S be the set of all $\lambda \in \mathrm{R}$ for which the system of linear equations has no solution....

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Q JEE MAIN 2019

An ellipse, with foci at (0, 2) and (0, –2) and minor axis of length 4, passes through which of...

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Q JEE MAIN 2019

The equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$...

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Q JEE-MAIN 2019

The maximum value of $3 \cos \theta+5 \sin \left(\theta-\frac{\pi}{6}\right)$ for any real value of $\theta$ is

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Q JEE Main 2020

If $|x|

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Q JEE MAIN 2019

A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length...

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Q JEE Main 2020

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and...

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Q JEE MAIN 2019

If the circles $x^2+y^2-16 x-20 y+164=r^2$ and $(x-4)^2+(y-7)^2=36$ intersect at two distinct points, then

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Q JEE MAIN 2019

Let $\mathrm{a}, \mathrm{b}$ and c be the $7^{\text {th }}, 11^{\text {th }}$ and $13^{\text {th }}$ terms respectively of...

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Q JEE Main 2020

Let $X=\{x \in N: 1 \leq x \leq 17\}$ and $Y\{a x+b: x \in X$ and $...

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Q JEE MAIN 2019

The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation, $...

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Q JEE MAIN 2019

If $x=3$ tan $t$ and $y=3$ sect, then the value of $\frac{d^2 y}{d x^2}$ at $t=\frac{\pi}{4}$, is

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Q JEE Main 2020

Let $\alpha$ and $\beta$ be the roots of the equation, $5 x^2+6 x- 2=0$. If $S_n=\alpha^n+\beta^n, n=1,2,3, \ldots$, then :

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Q JEE MAIN 2019

$ \int_0^{\pi / 3} \frac{\tan \theta}{\sqrt{2 \mathrm{ksec} \theta}} \mathrm{~d} \theta=1-\frac{1}{\sqrt{2}},(\mathrm{k}>0) $ then the value of $k$ is

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Q JEE MAIN 2019

Let $\mathrm{A}(4,-4)$ and $\mathrm{B}(9,6)$ be points on the parabola, $\mathrm{y}^2=4 \mathrm{x}$. Let C be chosen on the arc AOB of...

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Q JEE MAIN 2019

For each $x \in R$, let $[x]$ be the greatest integer less than or equal to $x$. Then $...

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Q JEE MAIN 2019

The coefficient of $t^4$ in the expansion of $\left(\frac{1-t^6}{1-t}\right)^3$ is

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Q JEE MAIN 2019

Let $A=\{x \in R: x$ is not a positive integer $\}$. Define a function $f: A \rightarrow R$ as $...

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Q JEE MAIN 2019

If $...

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Q JEE MAIN 2020

If for $x \geq 0, y=y(x)$ is the solution of the differential equation, $(x+1) d y=\left((x+1)^2+y-3\right) d x, y(2)=0$, then...

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Q JEE MAIN 2020

The number of distinct solutions of the equation, $\log _{\frac{1}{2}}|\sin x|=2-\log _{\frac{1}{2}}|\cos x|$ in the interval $[0,2 \pi]$, is $\_\_\_\_$...

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Q JEE MAIN 2019

Let $z_0$ be a root of the quadratic equation, $x^2+x+1=0$. If $z=3+6 i z_0^{81}-3 i z_0^{93}$, then arg $z$ is...

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Q JEE MAIN 2020

If the vectors, $\vec{p}=(a+1) \hat{i}+a \hat{j}+a \hat{k}$, $\overrightarrow{\mathrm{q}}=\mathrm{a} \hat{\mathrm{i}}+(\mathrm{a}+1) \hat{\mathrm{j}}+\mathrm{a} \hat{\mathrm{k}}$, and $\vec{r}=a \hat{i}+a \hat{j}+(a+1) \hat{k}(a \in R)$ are coplanar...

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Q JEE MAIN 2020

The coefficient of $x^4$ in the expansion of $\left(1+x+x^2\right)^{10}$ is. $\_\_\_\_$

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Q JEE MAIN 2019

If $x=\sin ^{-1}(\sin 10)$ and $y=\cos ^{-1}(\cos 10)$, then $y-x$ is equal to

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Q JEE MAIN 2020

The projection of the line segment joining the point (1, –1, 3) and (2, –4, 11) on the line joining...

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Q JEE MAIN 2020

A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts...

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Q JEE MAIN 2019

Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x)-f(y)| \leq 2|x-y|^{\frac{3}{2}}$, for all $...

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Q JEE MAIN 2020

If $f(x)= \begin{cases}\frac{\sin (a+2) x+\sin x}{x} & ; x0\end{cases}$ is continuous at $x=0$, then $a+2 b$ is equal to

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Q JEE MAIN 2020

If the number of five digit numbers with distinct digits and 2 at the $10^{\text {th }}$ place is 336...

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Q JEE MAIN 2020

The value of $\cos ^3\left(\frac{\pi}{8}\right) \cdot \cos \left(\frac{3 \pi}{8}\right)+\sin ^3\left(\frac{\pi}{8}\right) \cdot \sin \left(\frac{3 \pi}{8}\right)$ is

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Q JEE MAIN 2019

Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the...

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Q JEE MAIN 2019

The logical statement $[\sim(\sim p \vee q) \vee(p \wedge r)] \wedge(\sim q \wedge r)$ is equivalent to

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Q JEE MAIN 2020

If for all real triplets $(a, b, c), f(x)=a+b x+c x^2$; then $\int_0^1 f(x) d x$ is equal to

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Q JEE MAIN 2019

If the system of linear equations x – 4y + 7z = g 3y – 5z = h – 2x...

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Q JEE MAIN 2020

Let the observations $x_i(1 \leq i \leq 10)$ satisfy the equations, $\sum_{\mathrm{i}=1}^{10}\left(\mathrm{x}_{\mathrm{i}}-5\right)=10$ and $\sum_{\mathrm{i}=1}^{10}\left(\mathrm{x}_{\mathrm{i}}-5\right)^2=40$. If $\mu$ and $\lambda$ are the...

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Q JEE MAIN 2019

If $0 \leq x

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Q JEE MAIN 2020

Negation of the statement : ' $\sqrt{5}$ is an integer of 5 is irrational is

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Q JEE MAIN 2020

The product $2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}}$. to $\infty$ is equal to

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Q JEE MAIN 2020

The value of $\int_0^{2 \pi} \frac{x \sin ^8 x}{\sin ^8 x+\cos ^8 x} d x$ is equal to

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Q JEE MAIN 2019

If the lines x = ay + b, z = cy + d and x = a′z + b′, y...

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Q JEE MAIN 2019

Let $f:[0,1] \rightarrow R$ be such that $f(x y)=f(x) \cdot f(y)$, for all $x, y \in[0,1]$, and $f(0) \neq 0$....

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Q JEE MAIN 2020

The integral $\int \frac{d x}{(x+4)^{\frac{8}{7}}(x-3)^{\frac{6}{7}}}$ is equal to (where C is a constant of integration)

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Q JEE MAIN 2020

Let C be the centroid of the triangle with vertices (3, –1), (1, 3) and (2, 4). Let P be...

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Q JEE MAIN 2020

If $e_1$ and $e_2$ are the eccentricitis of the ellipse, $\frac{x^2}{18}+\frac{y^2}{4}=1 \quad$ and the hyperbola, $\quad \frac{x^2}{9}-\frac{y^2}{4}=1$ respectively and (...

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Q JEE MAIN 2020

The number of real roots of the equation, $e^{4 x}+e^{3 x}-4 e^{2 x}+e^x+1=0$ is

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Q JEE MAIN 2022

The number of real solutions of $x^7+5 x^3+3 x+1=0$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2020

Let $f(x)=x \cdot\left[\frac{x}{2}\right]$ for $-10

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Q JEE MAIN 2020

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from...

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Q JEE MAIN 2020

The natural number m , for which the coefficient of $x$ in the binomial expansion of $\left(x^m+\frac{1}{x^2}\right)^{22}$ is 1540 ,...

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Q JEE MAIN 2020

If the line, $2 x-y+3=0$ is at a distance $\frac{1}{\sqrt{5}}$ and $\frac{2}{\sqrt{5}}$ from the lines $4 x-2 y+\alpha=0$ and $...

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Q JEE MAIN 2020

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up...

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Q JEE MAIN 2020

If the volume of a parallelopiped, whose coterminus edges are given by the vectors $\vec{a}=\hat{i}+\hat{j}+n \hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-n \hat{k}$...

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Q JEE MAIN 2020

If the point $P$ on the curve, $4 x^2+5 y^2=20$ is farthest from the point $\mathrm{Q}(0,-4)$, then $\mathrm{PQ}^2$ is equal...

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Q JEE MAIN 2020

The product of the roots of the equation $9 x^2-18 |x|+5=0$, is :

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Q JEE MAIN 2020

If $2^{10}+2^9 \cdot 3^1+2^8 \cdot 3^2+\ldots .+2 \cdot 3^9+3^{10}=S-2^{11}$, then $S$ is equal to:

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Q JEE MAIN 2020

A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x...

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If the minimum and the maximum values of the function $f: f:\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow R$, defined by $$ f(\theta)=\left|\begin{array}{ccc} -\sin...

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Q JEE MAIN 2020

If $\alpha$ is the positive root of the equation, $p(x)=x^2-x -2=0$, then $\lim _{x \rightarrow \alpha^{+}} \frac{\sqrt{1-\cos (p(x))}}{x+\alpha-4}$ is equal...

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Q JEE MAIN 2020

If the four complex numbers $z, \bar{z}, \bar{z}-2 \operatorname{Re}(\bar{z})$ and $z-2 \operatorname{Re}(z)$ represent the vertices of a square of side...

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Q JEE MAIN 2020

The mean and variance of 7 observations are 8 and 16, respectively. If five observations are 2, 4, 10, 12,...

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Q JEE MAIN 2020

If $3^{2 \sin 2 \alpha-1}, 14$ and $3^{4-2 \sin 2 \alpha}$ are the first three terms of an A.P. for...

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Q JEE MAIN 2020

If $(a, b, c)$ is the image of the point $(1,2,-3)$ in the line, $\frac{x+1}{2}=\frac{y-2}{-2}=\frac{z}{-1}$, then $a+b+c$ is equal to...

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Q JEE MAIN 2020

If S is the sum of the first 10 terms of the series $\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{13}\right)+\tan ^{-1}\left(\frac{1}{21}\right)+\ldots \ldots$ then...

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Q JEE MAIN 2020

The value of $\int_{-\pi / 2}^{\pi / 2} \frac{1}{1+\mathrm{e}^{\sin x}} \mathrm{dx}$ is:

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Q JEE MAIN 2020

Let $A D$ and $B C$ be two vertical poles at $A$ and $B$ respectively on a horizontal ground. If...

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Q JEE MAIN 2020

If $y=y(x)$ is the solution of the differential equation $\frac{5+\mathrm{e}^{\mathrm{x}}}{2+\mathrm{y}} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{e}^{\mathrm{x}}=0$ satisfying $\mathrm{y}(0)=1$, then a value of $...

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Q JEE MAIN 2020

If $\vec{a}$ and $\vec{b}$ are unit vectors, then the greatest value of $\sqrt{3}|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|$ is

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Q JEE MAIN 2019

The set of all values of $\lambda$ for which the system of linear equations $$ \begin{aligned} & x-2 y-2 z=\lambda...

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Q JEE MAIN 2019

The total number of irrational terms in the binomial expansion of $\left(7^{1 / 5}-3^{1 / 10}\right)^{60}$ is :

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Q JEE MAIN 2020

The angle of elevation of the top of a hill from a point on the horizontal plane passing through the...

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Q JEE MAIN 2019

if the sum of the first 15 terms of the series $\left(\frac{3}{4}\right)^3+\left(1 \frac{1}{2}\right)^3+\left(2 \frac{1}{4}\right)^3+3^3+\left(3 \frac{3}{4}\right)^3+\ldots$. is equal to $225 k$,...

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Q JEE MAIN 2019

$\lim _{x \rightarrow 1} \frac{\sqrt{\pi}-\sqrt{2 \sin ^{-1} x}}{\sqrt{1-x}}$ is equal to :

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Q JEE MAIN 2020

Set A has m elements and Set B has n elements. If the total number of subsets of A is...

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If a straight line passing through the point P(–3, 4) is such that its intercepted portion between the coordinate axes...

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Q JEE MAIN 2020

If $\alpha$ and $\beta$ be two roots of the equation $x^2-64 x+256=0$. Then the value of $...

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Q JEE MAIN 2019

The equation of a tangent to the parabola, $x^2=8 y$, which makes an angle $\theta$ with the positive direction of...

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Q JEE MAIN 2020

Let $a, b, c, d$ and $p$ be any non zero distinct real numbers such that $...

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Q JEE MAIN 2019

The integral $\int \frac{3 x+2 x}{\left(2 x^4+3 x^2+1\right)^4} d x$ is equal to (where C is a constant of integration)

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Q JEE MAIN 2020

If $\sum_{i=1}^n\left(x_i-a\right)=n$ and $\sum_{i=1}^n\left(x_i-a\right)^2=n a,(n, a>1)$ then the standard deviation of $n$ observations $x_1, x_2, \ldots, x_n$ is :

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Q JEE MAIN 2019

$\lim _{n \rightarrow \infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\ldots \ldots+\frac{1}{5 n}\right)$ is equal to :

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Q JEE MAIN 2019

If a circle of radius R pases through the origin O and intersects the coordinate axes at A and B,...

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If ${ }^{\mathrm{n}} \mathrm{C}_4,{ }^{\mathrm{n}} \mathrm{C}_5$ and ${ }^{\mathrm{n}} \mathrm{C}_6$ are in A.P., then n can be :

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Q JEE MAIN 2020

The position of a moving car at time $t$ is given by $f(t)=a t^2+b t+c, t>0$, where $a, b$ and...

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Q JEE MAIN 2019

The number of integral values of $m$ for which the quadratic expression, $(1+2 m) x^2-2(1+3 m) x+4(1+m)$, $x \in R$,...

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Q JEE MAIN 2020

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the...

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Q JEE MAIN 2020

Consider the data on $x$ taking the values $0,2,4,8$, $\ldots, 2^n$ with frequencies $...

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Q JEE MAIN 2019

Let $z_1$ and $z_2$ be two complex numbers satisfying $\left|z_1\right|=9$ and $\left|z_2-3-4 i\right|=4$. Then the minimum value of $...

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Q JEE MAIN 2020

If the common tangent to the parabolas, $y^2=4 x$ and $x^2=4 y$ also touches the circle, $x^2+y^2=c^2$, then c is...

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Q JEE MAIN 2020

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Q JEE MAIN 2019

If an angle between the line, $\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}$ and the plane, $x-2 y-k z=3$ is $\cos ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)$

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Q JEE MAIN 2019

Let $S$ be the set of all real values of $\lambda$ such that a plane passing through the points $...

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Q JEE MAIN 2020

The number of words (with or without meaning) that can be formed from all the letters of the word “LETTER”...

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Q JEE MAIN 2020

If the co-ordinates of two points $A$ and $B$ are $(\sqrt{7}, 0)$ and $(-\sqrt{7}, 0)$ respectively and $P$ is any...

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Q JEE MAIN 2019

The mean and the variance of five observations are 4 and 5.20, respectively. If three of the observations are 3,...

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Q JEE MAIN 2020

The negation of the Boolean expression $x \leftrightarrow \sim y$ is equivalent to:

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Q JEE MAIN 2020

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Q JEE MAIN 2019

The expression $\sim(\sim p \rightarrow q)$ is logically equivalent to

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The sum of distinct values of λ for which the system of equations (λ – 1)x + (3λ + 1)y...

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Q JEE MAIN 2019

Let $Z$ be the set of integers. If $A=\left\{x \in Z: 2^{(x+2)\left(x^2-5 x+9\right)}=1\right\}$ and $B=\{x \in Z:-3

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Q JEE MAIN 2020

The area (in sq. units) of the region $A=\left\{(x, y)\right.$ : $\left.|x|+|y| \leq 1,2 y^2 \geq|x|\right\}$ is :

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Q JEE MAIN 2020

If the function $f(x)\left\{\begin{array}{ll}k_1(x-\pi)^2-1, & x \leq \pi \\ k_2 \cos x, & x>\pi\end{array}\right.$ is twice differentiable, then the ordered...

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Q JEE MAIN 2019

There are m men and two women participating in a chess tournament. Each participant plays two games with every other...

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The values of $\lambda$ and $\mu$ for which the system of linear equations $$ \begin{aligned} & x+y+z=2 \\ & x+2...

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Q JEE MAIN 2020

Let $\lambda \in \mathrm{R}$. The system of linear equations $$ \begin{aligned} & 2 x_1-4 x_2+\lambda x_3=1 \\ & x_1-6 x_2+x_3=2...

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Q JEE MAIN 2020

Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the...

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Q JEE MAIN 2019

If the function f given by $\mathrm{f}(\mathrm{x})=\mathrm{x}^3-3(\mathrm{a}-2) \mathrm{x}^2+3 \mathrm{ax}+7$, for some $\mathrm{a} \in \mathrm{R}$ is increasing in $(0,1]$ and decreasing...

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Q JEE MAIN 2020

If $\quad \int\left(e^{2 x}+2 e^x-e^{-x}-1\right) e^{\left(e^x+e^{-x}\right)} d x=g(x) e^{\left(e^x+e^{-x}\right)}+c$ where c is a constant of integration, then $\mathrm{g}(0)$ is equal...

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Q JEE MAIN 2019

Let f be a differentiable function such that $f(1)=2$ and $f^{\prime}(x)=f(x)$ for all $x \in R$. If $h(x)=f(f(x))$, then $h^{\prime}(1)$...

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Q JEE MAIN 2020

A plane P meets the coordinate axes at A, B and C respectively. The centroid of ΔABC is given to...

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Q JEE MAIN 2020

Two families with three members each and one family with four members are to be seated in a row. In...

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Let $S$ and $S^{\prime}$ be the foci of an ellipse and $B$ be any one of the extremities of its...

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Q JEE MAIN 2020

The area (in sq. units) of the region enclosed by the curves $y=x^2-1$ and $y=1-x^2$ is equal to :

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Q JEE MAIN 2019

The integral $\int_1^e\left\{\left(\frac{x}{e}\right)^{2 x}-\left(\frac{e}{x}\right)^x\right\} \log _e x d x$ is equal to

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Q JEE MAIN 2019

If $\sin ^4 \alpha+4 \cos ^4 \beta+2=4 \sqrt{2} \sin \alpha \cos \beta ; \alpha, \beta \in[0, \pi]$, then $...

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Q JEE MAIN 2019

If a curve passes through the point $(1,-2)$ and has slope of the tangent at any point $(x, y)$ on...

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Q JEE MAIN 2020

If $I_1=\int_0^1\left(1-x^{50}\right)^{100} d x$ and $I_2=\int_0^1\left(1-x^{50}\right)^{101} d x$ such that $I_2=\alpha I_1$ then $\alpha$ equals to:

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Q JEE MAIN 2019

In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair...

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Q JEE MAIN 2019

If $...

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Let $z=x+$ iy be a non-zero complex number such that $z^2=i|z|^2$, where $i=\sqrt{-1}$, then $z$ lies on the

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Q JEE MAIN 2020

The region represented by $\{z=x+i y \in C:|z|-\operatorname{Re}(z) \leq 1\}$ is also given by the inequality :

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Q JEE MAIN 2019

If the angle of elevation of a cloud from a point P which is 25 m above a lake be...

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For a suitably chosen real constant a , let a function, $\mathrm{f}: \mathrm{R}-\{-\mathrm{a}\} \rightarrow \mathrm{R}$ be defined by $f(x)=\frac{a-x}{a+x}$. Further...

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Q JEE MAIN 2019

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors, out of which vectors $\vec{b}$ and $\vec{c}$ are non-parallel. If $\alpha$...

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Q JEE MAIN 2019

In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC...

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Q JEE MAIN 2020

$$ \text { The integral } \int_1^2 e^x \cdot x^x\left(2+\log _e x\right) d x \text { equals : } $$

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Q JEE MAIN 2020

If $\{p\}$ denotes the fractional part of the number $p$, then $\left\{\frac{3^{200}}{8}\right\}$, is equal to:

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Q JEE MAIN 2020

For all twice differentiable functions f : R → R, with f(0) = f(1) = f′(0) = 0,

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Q JEE MAIN 2020

A ray of light coming from the point $(2,2 \sqrt{3})$ is incident at an angle $30^{\circ}$ on the line $x=1$...

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Q JEE MAIN 2020

Consider the statement: "For an integer n , if $\mathrm{n}^3-1$ is even, then n is odd." The contrapositive statement of...

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Q JEE MAIN 2020

The general solution of the differential equation $\sqrt{1+x^2+y^2+x^2 y^2}+x y \frac{d y}{d x}=0$ is : (where C is a constant...

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Q JEE MAIN 2020

JEE Main Mathematics Hard

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Q JEE MAIN 2020

The shortest distance between the lines $\frac{x-1}{0}=\frac{y+1}{-1}=\frac{z}{1}$ and $x+y+z+1=0,2 x-y+z+3=0$ is :

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Q JEE MAIN 2020

Let $f: R \rightarrow R$ be a function defined by $f(x)=\max \left\{x, x^2\right\}$. Let $S$ denote the set of all...

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Q JEE MAIN 2020

The negation of the Boolean expression $p \vee(\sim p \wedge q)$ is equivalent to

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Q JEE MAIN 2020

If the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{k}{x^2}\right)^{10}$ is 405 , then $|k|$ equals :

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If $f(x+y)=f(x) f(y)$ and $\sum_{x=1}^{\infty} f(x)=2 x, y \in N$, where $N$ is the set of all natural numbers, then...

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Q JEE MAIN 2020

The angle of elevation of the summit of a mountain from a point on the ground is 45°. After climbing...

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Let $L_1$ be a tangent to the parabola $y^2=4(x+1)$ and $L_2$ be a tangent to the parabola $y^2=8(x+2)$ such that...

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The common difference of the A.P. $b_1, b_2, \ldots, b_m$ is 2 more than the common difference of A.P. $...

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Q JEE MAIN 2020

Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they...

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Let $\theta=\frac{\pi}{5}$ and $\mathrm{A}=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]$. If $\mathrm{B}=\mathrm{A}+\mathrm{A}^4$, then $\operatorname{det}(\mathrm{B})$ :

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If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor...

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If α and β are the roots of the equation 2x(2x + 1) = 1, then β is equal to...

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The probabilities of three events $\mathrm{A}, \mathrm{B}$ and C are given by $\mathrm{P}(\mathrm{A})=0.6, \mathrm{P}(\mathrm{B})=0.4$ and $\mathrm{P}(\mathrm{C})=0.5$. If $...

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Q JEE MAIN 2019

The number of natural numbers less than 7,000 which can be formed by using the digits 0, 1, 3, 7,9...

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The centre of the circle passing through the point $(0,1)$ and touching the parabola $\mathrm{y}=\mathrm{x}^2$ at the point $(2,4)$ is...

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Let the equations of two sides of a triangle be $3 x-2 y+6=0$ and $4 x+5 y-20=0$. If the orthocentre...

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An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the...

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The area of the region $A=\{(x, y): 0 \leq y \leq x|x|+1$ and $-1 \leq x \leq 1\}$ in sq....

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A data consists of $n$ observations $x_1, x_2, \ldots ., x n$. If $\sum_{i=1}^n\left(x_i+1\right)^2=9 n$ and $\sum_{i=1}^n\left(x_i-1\right)^2=5 n$, then the...

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Q JEE MAIN 2019

The equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$...

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Q JEE MAIN 2019

A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length...

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Q JEE MAIN 2019

If the circles $x^2+y^2-16 x-20 y+164=r^2$ and $(x-4)^2+(y-7)^2=36$ intersect at two distinct points, then

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Q JEE MAIN 2019

Let $\mathrm{a}, \mathrm{b}$ and c be the $7^{\text {th }}, 11^{\text {th }}$ and $13^{\text {th }}$ terms respectively of...

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Q JEE MAIN 2019

The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation, $...

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Q JEE MAIN 2019

Let $\ddot{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}}, \ddot{\mathrm{b}}=\mathrm{b}_1 \hat{\mathrm{i}}+\mathrm{b}_2 \hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}}$ and $\ddot{\mathrm{c}}=5 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}}$ be three vectors such that the projection vector of...

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Q JEE MAIN 2019

If $x=3 \tan t$ and $y=3$ sect, then the value of $\frac{d^2 y}{d x^2}$ at $t=\frac{\pi}{4}$, is

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Q JEE MAIN 2019

The sum of the following series $$ 1+6+\frac{9\left(1^2+2^2+3^2\right)}{7}+\frac{12\left(1^2+2^2+3^2+4^2\right)}{9}+\frac{15\left(1^2+2^2+\ldots \ldots+5^2\right.}{11}+\ldots . $$ up to 15 terms, is

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$\int_0^{\pi / 3} \frac{\tan \theta}{\sqrt{2 k \sec \theta}} d \theta=1-\frac{1}{\sqrt{2}},(k>0)$, then the value of $k$ is

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Let $\mathrm{A}(4,-4)$ and $\mathrm{B}(9,6)$ be points on the parabola, $\mathrm{y}^2=4 \mathrm{x}$. Let C be chosen on the arc AOB of...

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Q JEE MAIN 2019

For each $x \in R$, let $[x]$ be the greatest integer less than or equal to $x$. Then $...

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Q JEE MAIN 2019

The coefficient of $t^4$ in the expansion of $\left(\frac{1-t^6}{1-t}\right)^3$ is

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Q JEE MAIN 2019

Let $A=\{x \in R: x$ is not a positive integer $\}$. Define a function $f: A \rightarrow R$ as $...

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Q JEE MAIN 2019

lf $$ A=\left[\begin{array}{ccc} e^t & e^{-t} & e^{-t} \sin t \\ e^t & -e^t \cos t-e^{-t} \sin t & -e^{-t}...

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Q JEE MAIN 2020

The coef ficient of $x^4$ in the expansion of $\left(1+x+x^2+x^3\right)^6$ in powers of $x$, is $\_\_\_\_$ ...

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Q JEE MAIN 2019

Let $z_0$ be a root of the quadratic equation, $x^2+x+1=0$. If $z=3+6 i z_0^{81}-3 i z_0^{93}$, then arg $z$ is...

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Q JEE MAIN 2020

Let the vectors $\vec{a}, \vec{b}, \vec{c}$, be such that $|\vec{a}|=2,|\vec{b}|=4$ and $|\vec{c}|=4$ and $|\vec{c}|=4$. If the projection of $\vec{b}$ on...

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Q JEE MAIN 2020

In a bombing attack, there is 50% chance that a bomb will hit the target. Atleast two independent hits are...

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If the lines $x+y=a$ and $x-y=b$ touch the curve $y=x^2-3 x+2$ at the points where the curve intersects the $x$-axis,...

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Q JEE MAIN 2019

If both the roots of the quadratic equation $x^2-m x+4=0$ are real and distinct and they lie in the interval...

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Q JEE MAIN 2020

Let $A=\{a, b, c\}$ and $B=\{1,2,3,4\}$. Then the number of elements in the set $...

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Q JEE MAIN 2020

The area (in sq. units) of the region $...

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The derivative of $\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to $\tan ^{-1}\left(\frac{2 x \sqrt{1-x^2}}{1-2 x^2}\right)$ at $x=\frac{1}{2}$ is :

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Q JEE MAIN 2020

If $\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos ^2 \theta} d \theta=A \log _e|B(\theta)|+C$, where $C$ is a constant of integration,...

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Q JEE MAIN 2020

If the line $y=m x+c$ is a common tangent to the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ and the circle $x^2+y^2=36$, then which one...

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Q JEE MAIN 2020

If $L=\sin ^2\left(\frac{\pi}{16}\right)-\sin ^2\left(\frac{\pi}{8}\right)$ and $M=\cos ^2\left(\frac{\pi}{16}\right)-\sin ^2\left(\frac{\pi}{8}\right)$, then

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Q JEE MAIN 2020

If the system of linear equations $...

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Q JEE MAIN 2019

A galvanometer, whose resistance is 50 ohm , has 25 divisions in it. When a current of $...

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The numer of all $3 \times 3$ matrices A , with enteries from the set $\{-1,0,1)$ such that the sum...

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Q JEE MAIN 2020

The least positive value of ' $a$ ' for which the equation, $2 x^2+(a-10) x+\frac{33}{2}=2 a$ has real roots is...

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Q JEE MAIN 2020

The sum $\sum_{k=1}^{20}(1+2+3+\ldots . .+k)$ is $\_\_\_\_$ .

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Q JEE MAIN 2019

If one real root of the quadratic equation $81 x^2+k x+256=0$ is cube of the other root, then a value...

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Let the normal at a point $P$ on the curve $y^2-3 x^2+y+10=0$ intersect the $y$-axis at $\left(0, \frac{3}{2}\right)$. If $m$...

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The value of the $\int_{-2}^2 \frac{\sin ^2 x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}}$ (where $[x]$ denotes the greatest integer less than or equal to $x$...

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Q JEE MAIN 2019

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its...

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Q JEE MAIN 2020

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which...

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Let the line $y=m x$ and the ellipse $2 x^2+y^2=1$ intersect at a point $P$ in the first quadrant. If...

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Q JEE MAIN 2019

The sum of the real values of $x$ for which the middle term in the binomial expansion of $\left(\frac{x^3}{3}+\frac{3}{x}\right)^8$ equals...

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Q JEE MAIN 2019

Let $f_x(x)=\frac{1}{k}\left(\sin ^k x+\cos ^k x\right)$ for $k=1,2,3, \ldots$. Then for all $x \in R$, the value of $f_4(x)-f_6(x)$ is...

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Q JEE MAIN 2020

If $\int \frac{\cos x d x}{\sin ^3 x\left(1+\sin ^6 x\right)^{2 / 3}}=f(x)\left(1+\sin ^6 x\right)^{1 / 2}+c$ where $c$ is a...

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Q JEE MAIN 2019

The maximum value of the function $f(x)=3 x^3-18 x^2+27 x-40$ on the set $S=\left\{x \in R: x^2+30 \leq 11 x\right\}$...

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Q JEE MAIN 2020

For $a>0$, let the curves $C_1: y^2=a x$ and $C_2: x^2=$ ay intersect at origin $O$ and a point $P$....

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Q JEE MAIN 2019

The direction ratios of normal to the plane through the points $(0,-1,0)$ and $(0,0,1)$ and making an angle $\frac{\pi}{4}$ with...

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Q JEE MAIN 2020

The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola,...

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Q JEE MAIN 2020

If the equation, $x^2+b x+45=0(b \in R)$ has conjuate complex roots and they satisfy $|z+1|=2 \sqrt{10}$, then

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Q JEE MAIN 2019

If $y(x)$ is the solution of the differential equation $\frac{d y}{d x}+\left(\frac{2 x+1}{x}\right) y=e^{-2 x}, x>0$ where $y(1)=\frac{1}{2} e^{-2}$ then

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Q JEE MAIN 2020

The shortest distance between the lines $\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{2}$ is

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Q JEE MAIN 2019

In a triangle, the sum of lengths of two sides is $x$ and the product of the lengths of the...

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Q JEE MAIN 2020

The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations...

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Q JEE MAIN 2019

A square is inscribed in the circle $x^2+y^2-6 x+8 y-103=0$ with its sides parallel to the coordinate axes. Then the...

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Q JEE MAIN 2020

If $a, b$ and $c$ are the greatest values of ${ }^{19} \mathrm{C}_{\mathrm{p}},{ }^{20} \mathrm{C}_{\mathrm{q}}$ and ${ }^{21} \mathrm{C}_{\mathrm{r}}$ respectively,...

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Q JEE MAIN 2020

Which one of the following is a tautology?

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Q JEE MAIN 2019

If tangents are drawn to the ellipse $x^2+2 y^2=2$ at all points on the ellipse other than its four vertices...

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Q JEE MAIN 2019

Equation of a common tangent to the parabola $y^2=4 x$ and the hyperbola $x y=2$ is:

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Q JEE MAIN 2020

For which of the following ordered pairs $(\mu, \delta)$, the system of linear equations $...

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Q JEE MAIN 2020

Let the volume of a parallelopiped whose coterminous edges are given by $\overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{v}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{w}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$...

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Q JEE MAIN 2019

Two integers are selected at random from the set {1, 2, ..., 11}. Given that the sum of selected numbers...

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Q JEE MAIN 2020

Let $f: R \rightarrow R$ be such that for all $x \in R\left(2^{1+x}+2^{1-x}\right), f(x)$ and $\left(3^x+3^{-x}\right)$ are in A.P., then...

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Q JEE MAIN 2019

If $\int \frac{\sqrt{1-x^2}}{x^4} d x=A(x)\left(\sqrt{1-x^2}\right)^m+C$, for a suitable chosen integer $m$ and a function $A(x)$, where $C$ is a constant...

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Q JEE MAIN 2019

If $x \log _e\left(\log _e x\right)-x^2+y^2=4(y>0)$, then $\frac{d y}{d y}$ at $x=e$ is equal to:

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Q JEE MAIN 2019

The value of r for which $...

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Q JEE MAIN 2019

Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point...

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Q JEE MAIN 2019

Let $A=\left(\begin{array}{ccc}0 & 2 q & r \\ p & q & -r \\ p & -q & r\end{array}\right)$. If...

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Q JEE MAIN 2019

Let $[\mathrm{x}]$ denote the greatest integer less than or equal to x . Then $...

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Q JEE MAIN 2019

The straight line $x+2 y=1$ meets the coordinate axes at A and B. A circle is drawn through A, B...

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Q JEE MAIN 2019

If the system of linear equations $$ \begin{aligned} & 2 x+2 y+3 z=a \\ & 3 x-y+5 z=b \\ &...

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Q JEE MAIN 2019

Let $a_1, a_2, \ldots, a_{10}$ be a G.P. If $\frac{a_3}{a_1}=25$, then $\frac{a_9}{a_5}$ equals

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Q JEE MAIN 2020

Let $f$ be any function continuous on $[a, b]$ and twice differentiable on $(a, b)$. If for all $...

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Q JEE MAIN 2020

In a box, there are 20 cards, out of which 10 are labelled as A and the remaining 10 are...

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Q JEE MAIN 2019

The area (in sq. units) of the region bounded by the curve $x^2=4 y$ and the straight line $x=4 y-2$...

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Q JEE MAIN 2020

A circle touches the y-axis at the point (0, 4) and passes through the point (2, 0). Which of the...

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Q JEE MAIN 2019

Let $\vec{a}=\hat{i}+2 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}+\lambda \hat{j}+4 \hat{k}$ and $\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\left(\lambda^2-1\right) \hat{\mathrm{k}}$ be coplanar vectors. Then the non-zero vector $...

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Q JEE MAIN 2020

If for some $\alpha$ and $\beta$ in R , the intersection of the following three planes $...

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Q JEE MAIN 2020

Let $f(x)=x \cos ^{-1}(-\sin |x|), x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, then which of the following is true?

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Q JEE MAIN 2020

Let $z$ be a complex number such that $\left|\frac{z-i}{z+2 i}\right|=1$ and $|z|=\frac{5}{2}$. Then the value of $|z+3 i|$ is

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Q JEE MAIN 2019

The outcome of each of 30 items was observed; 10 items gave an outcome $\frac{1}{2}$-d each, 10 items gave outcome...

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Q JEE MAIN 2019

Let $f: R \rightarrow R$ be defined by $f(x)=\frac{x}{1+x^2} x \in R$. Then the range of $r$ is

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Q JEE MAIN 2020

$\lim _{x \rightarrow 0}\left(\frac{3 x^2+2}{7 x^2+2}\right)^{\frac{1}{x^2}}$ is equal to

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Q JEE MAIN 2019

The plane containing the line $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z-1}{3}$ and also containing its projection on the plane $2 x+3 y-z=$ 5 , contains...

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Q JEE MAIN 2019

The negation of the Boolean expression $\sim s \vee(\sim r \wedge s)$ is equivalent to:

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Q JEE MAIN 2019

Let $f(x)=\left\{\begin{array}{cc}-1, & -2 \leq x

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Q JEE MAIN 2019

If $\cos ^{-1} x-\cos ^{-1} \frac{y}{2}=\alpha$, where $1 \leq x \leq 1,-2 \leq y \leq 2$, $x \leq \frac{y}{2}$, then...

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Q JEE MAIN 2019

The distance of the point having position vector $-\hat{i}+2 \hat{j}+6 \hat{k}$ from the straight line passing through the point (...

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Q JEE MAIN 2019

The tangent and normal to the ellipse $3 x^2+5 y^2=$ 32 at the point $P(2,2)$ meet the $x$-axis at $Q$...

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Q JEE MAIN 2019

If the line $a x+y=c$, touches both the curves $x^2+y^2=1$ and $y^2=4 \sqrt{2} x$, then $|c|$ is equal to :

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Q JEE MAIN 2019

The integral $\int_{\pi / 6}^{\pi / 3} \sec ^{2 / 3} x \operatorname{cosec}^{4 / 3} x d x$ is equal...

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Q JEE MAIN 2019

Let $\lambda$ be a real number for which the system of linear equations $$ \begin{aligned} & x+y+z=6 \\ & 4...

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Q JEE MAIN 2019

If 5 x+9=0 is the directrix of the hyperbola $16 x^2- 9 y^2=144$, then its corresponding focus is :

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Q JEE MAIN 2019

Minimum number of times a fair coin must be tossed so that the probability of getting at least one head...

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A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts...

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Q JEE MAIN 2019

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the...

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Q JEE MAIN 2020

Let two points be $\mathrm{A}(1,-1)$ and $\mathrm{B}(0,2)$. If a point $\mathrm{P}\left(\mathrm{x}^{\prime}, \mathrm{y}^{\prime}\right)$ be such that the area of $\Delta \mathrm{PAB}=5$...

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Q JEE MAIN 2019

Lines are drawn parallel to the line $4 x-3 y+2=0$, at a distance $\frac{3}{5}$ from the origin. Then which one...

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Q JEE MAIN 2019

The number of real roots of the equation $5+\left|2^x-1\right|=2^x\left(2^x-2\right)$ is :

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Q JEE MAIN 2020

The inverse function of $f(x)=\frac{8^{2 x}-8^{-2 x}}{8^{2 x}+8^{-2 x}}, x \in(-1,1)$, is $\_\_\_\_$ .

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Q JEE MAIN 2019

If $\lim _{x \rightarrow 1} \frac{x^2-a x+b}{x-1}=5$, then $a+b$ is equal to :

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Q JEE MAIN 2019

If the plane $2 x-y+2 z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4 x-2 y+4 z+\lambda=0$...

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Q JEE MAIN 2020

Let $A$ and $B$ be two independent events such that $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{6}$. Then, which of the following is TRUE...

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Q JEE MAIN 2019

Let $a_1, a_2, a_3, \ldots .$. be an A.P. with $a_6=2$. Then the common difference of this A.P., which maximises...

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Q JEE MAIN 2019

Let $y=y(x)$ be the solution of the differential equation, $\frac{d y}{d x}+y \tan 2 x=x^2+x^2 \tan x$, $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$,...

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Q JEE MAIN_2021_

If $\lim _{x \rightarrow \infty}\left(\sqrt{x^2-x+1}-a x\right)=b$, then the ordered pair $(a b)$ is :

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Q JEE MAIN_2021_

The value of the inkegral $\int_0^1 \frac{\sqrt{x} d x}{(1+x)(1+3 x)(3+x)}$ is :

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Q JEE-MAIN 2020

If the length of the chord of the circle, $x^2+y^2=r^2(r>0)$ along the line, $y-2 x=3$ is $r$, then $r^2$ is...

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Q JEE MAIN_2021_

If $0

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Q JEE MAIN 2019

A perpendicular is drawn from a point on the line $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{1}$ to the plane $x+y+z=3$ such that the foot of...

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Q JEE-MAIN 2020

If $\alpha$ and $\beta$ are the roots of the equation, $7 x^2-3 x-2=0$, then the value of $\frac{\alpha}{1-\alpha^2}+\frac{\beta}{1-\beta^2}$ is equal...

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Q JEE MAIN 2020

If $c$ is a point at which Rolle's theorem holds for the function, $f(x)=\log _e\left(\frac{x^2+\alpha}{7 x}\right)$ in interval $[3,4]$, where...

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Q JEE MAIN_2021_

Two poles, $A B$ of length a metres and $C D$ of length $a+b(b \neq a)$ metres are erected at...

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Q JEE MAIN 2019

The area (in sq. units) of the region bounded by the curves $y=2^x$ and $y=|x+1|$, in the first quadrant is...

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Q JEE-MAIN 2020

If for some $\alpha \in \mathrm{R}$, the lines $L_1: \frac{x+1}{2}=\frac{y-2}{-1}=\frac{z-1}{1}$ and $\mathrm{L}_2: \frac{\mathrm{x}+2}{\alpha}=\frac{\mathrm{y}+1}{5-\alpha}=\frac{\mathrm{z}+1}{1}$ are coplanar, then the line $L_2$ passes...

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Q JEE-MAIN 2020

If $x=1$ is a critical point of the function $f(x)=\left(3 x^2+a x-2-a\right) e^x$, then :

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Q JEE MAIN 2019

Let $\mathrm{a}, \mathrm{b}$ and c be in G.P. with common ratio r , where $\mathrm{a} \neq 0$ and $0

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Q JEE MAIN_2021_

If $y(x)=\cot ^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right), x \in\left(\frac{\pi}{2}, \pi\right)$, then $\frac{d y}{d x}$ at $x=\frac{5 \pi}{6}$ is :

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Q JEE MAIN_2021_

The area of the region bounded by the parabola $(y-2)^2=(x-1)$, the tangent to it at the point whose ordinate is...

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Q JEE MAIN 2019

If $\int x^5 e^{-x^2} d x=g(x) e^{-x^2}+c$, where $c$ is a constant of integration, then $\mathrm{g}(-1)$ is equal to :

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Q JEE-MAIN 2020

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a...

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Q JEE MAIN_2021_

Let $\mathbb{N}$ be the set of all integers, $$ \begin{aligned} & A=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}:(x-2)^2+y^2 \leq 4\right\} \\...

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Q JEE MAIN_2021_

The Boolean expression $(p \wedge q) \Rightarrow((r \wedge q) \wedge p)$ is equivalent to $i$

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Q JEE-MAIN 2020

If the mean and the standard deviation of the data 3, 5, 7, a, b are 5 and 2 respectively,...

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Q JEE MAIN 2019

If both the mean and the standard deviation of 50 observations $x_1, x_2, \ldots x_{50}$ are equal to 16 ,...

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Q JEE MAIN 2019

The smallest natural number n , such that the coefficient of $x$ in the expansion of $\left(x^2+\frac{1}{x^3}\right)^n$ is $...

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Q JEE-MAIN 2020

$\lim _{x \rightarrow 0} \frac{x\left(e^{\left(\sqrt{1+x^2+x^4}-1\right) / x}-1\right)}{\sqrt{1+x^2+x^4}-1}$

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Q JEE MAIN 2020

When a car is at rest, its driver sees rain drops falling on it vertically. When driving the car with...

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Q JEE MAIN 2019

The sum $1+\frac{1^3+2^3}{1+2}+\frac{1^3+2^3+3^3}{1+2+3}+\ldots \ldots$ $$ +\frac{1^3+2^3+3^3+\ldots \ldots+15^3}{1+2+3+\ldots \ldots+15}-\frac{1}{2}(1+2+3 \ldots .+15) $$ is equal to :

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Let $y=y(x)$ be the solution of the differential equation $...

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Q JEE MAIN_2021_

A box open from top is made from a rectangular sheet of dimension a × b by cutting squares each...

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Q JEE-MAIN 2020

Which of the following points lies on the tangent to the curve $x^4 e^y+2 \sqrt{y+1}=3$ at the point (1, 0)?

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Q JEE MAIN 2019

If $z$ and $w$ are two complex numbers such that $|z w|=1$ and $\arg (z)-\arg (w)=\frac{\pi}{2}$, then :

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Let $\{x\}$ and $[x]$ denote the fractional part of $x$ and the greatest integer $\leq x$ respectively of a real...

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Q JEE-MAIN 2020

The value of $\left(\frac{-1+i \sqrt{3}}{1-i}\right)^{30}$ is :

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Q JEE MAIN 2019

The sum of the real roots of the equation $...

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Q JEE MAIN 2020

A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The...

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Q JEE-MAIN 2020

If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum...

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Q JEE MAIN 2019

The angles $A, B$ and $C$ of a triangle $A B C$ are in A.P. and $a: b=1: \sqrt{3}$. If...

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Q JEE MAIN 2020

If $\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k}$, then the value of $|\hat{i} \times(\vec{a} \times \hat{i})|^2+|\hat{j} \times(\vec{a} \times \hat{j})|^2+|\hat{k} \times(\vec{a} \times \hat{k})|^2$ is equal...

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Q JEE MAIN 2019

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Let $P Q$ be a diameter of the circle $x^2+y^2=9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars...

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Q JEE MAIN 2020

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Let $A$ be a point on the line $\vec{r}=(1-3 \mu) \hat{i}+(\mu-1) \hat{j}+(2+5 \mu) \hat{k}$ and $B(3,2,6)$ be a point in...

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Q JEE MAIN 2020

The integral $$ \int_{\pi / 6}^{\pi / 3} \tan ^3 x \cdot \sin ^2 3 x\left(2 \sec ^2 x \cdot...

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Q JEE Main 2019

If the area enclosed between the curves $y=k x^2$ and $x=k y^2,(k>0)$, is 1 square unit. Then $k$ is :

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Q JEE Main 2019

Let $\vec{a}=2 \hat{i}+\lambda_1 \hat{j}+3 \hat{k}, \vec{b}=4 \hat{i}+\left(3-\lambda_2\right) \hat{j}+6 \hat{k}$ and $\vec{c}=3 \hat{i}+6 \hat{j}+\left(\lambda_3-1\right) \hat{k}$ be three vectors such that $...

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Q JEE MAIN 2020

In a game two players A and B take turns in throwing a pair of fair dice starting with player...

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Q JEE Main 2019

For each tR, let [t] be the greatest integer less than or equal to t. Then,

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Q JEE Main 2019

If $5,5 r, 5 r^2$ are the lengths of the sides of a triangle, then $r$ cannot be equal to...

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Q JEE MAIN 2020

The minimum value of $2^{\sin x}+2^{\cos x}$ is :

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Q JEE MAIN 2020

If $\mathrm{C}_{\mathrm{r}} \equiv{ }^{25} \mathrm{C}_{\mathrm{r}}$ and $\mathrm{C}_0+5 . \mathrm{C}_1+9 . \mathrm{C}_2+\ldots . .+(101) . \mathrm{C}_{25}= 2^{25} \cdot \mathrm{k}$, then k...

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Q JEE Main 2019

The plane passing through the point $(4,-1,2)$ and parallel to the lines $\frac{x+2}{3}=\frac{y-2}{-1}=\frac{z+1}{2}$ and $\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-4}{3}$ also passes through the point:

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Q JEE MAIN 2020

The circle passing through the intersection of the circles, $x^2+y^2-6 x=0$ and $x^2+y^2-4 y=0$, having its centre on the line,...

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Q JEE Main 2019

Let $\mathrm{n} \geq 2$ be a nutural number and $0

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Q JEE MAIN 2020

If the curves, $x^2-6 x+y^2+8=0$ and $x^2-8 y+y^2+ 16-\mathrm{k}=0,(\mathrm{k}>0)$ touch each other at a point, then the largest value of...

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Q JEE Main 2019

In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number...

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Q JEE MAIN 2020

Let $a_1, a_2, \ldots a_n$ be a given A.P. whose common difference is an integer and $S_n=a_1+a_2+\ldots+$ an. If $...

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Q JEE MAIN 2020

If the distance between the plane, $23 x-10 y-2 z+ 48=0$ and the plane containing the lines $$ \frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3} $$...

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Q JEE MAIN 2020

If $a$ and $b$ are real numbers such that $(2+\alpha)^4=a+b \alpha$, where $\alpha=\frac{-1+i \sqrt{3}}{2}$, then $a+b$ is equal to

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Q JEE Main 2019

The shortest distance between the point $\left(\frac{3}{2}, 0\right)$ and the curve $y=\sqrt{x},(x>0)$ is

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Q JEE Main 2019

https://competishun.com/let-f-r-rightarrow-r-be-a-function-such-that-fxx3x2-fprime1x-fprime-prime2fprime-prime-prime3-x-in-r-then-f2-equals/ has infinitely many solutions, then $\beta-\alpha$ equals

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Q JEE MAIN 2020

The number of terms common to the two A.P.’s 3, 7, 11,.....,407 and 2, 9, 16,.....,709 is______.

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Q JEE MAIN 2020

Let $x=4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac{1}{2}$. If...

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Q JEE Main 2019

Let $f: R \rightarrow R$ be a function such that $f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in R$. Then...

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Q JEE MAIN 2020

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Q JEE MAIN 2020

Let $\bigcup_{i=1}^{50} X_i=\bigcup_{i=1}^n Y_i=T$ where each $X_i$ contains 10 elements and each $Y_i$ contains 5 elements. If each element of...

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Q JEE Main 2019

The equation of a tangent to the hyperbola $4 x^2-5 y^2=20$ parallel to the line $x-y=2$ is

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Q JEE MAIN 2020

If $\int \frac{d \theta}{\cos ^2 \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _e|f(\theta)|+C$ where C is a constant of...

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Q JEE MAIN 2020

Contrapositive of the statement : ‘If a function f is differentiable at a, then it is also continuous at a’,...

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Q JEE Main 2019

The sum of all values of $\theta \in\left(0, \frac{\pi}{2}\right)$ satisfying $\sin ^2 2 \theta+\cos ^4 2 \theta=\frac{3}{4}$ is

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Q JEE Main 2019

If the parabolas $y^2=4 b(x-c)$ and $y \quad l=8 a x$ have a common normal, then which one of the...

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Q JEE MAIN 2020

If $A=\{x \in R:|x|

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Q JEE MAIN 2020

The angle of elevation of a cloud $C$ from a point $P, 200 \mathrm{~m}$ above a still lake is $30^{\circ}$....

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Q JEE Main 2019

A point $P$ moves on the line $2 x-3 y+4=0$. If $Q(1,4)$ and $R(3,-2)$ are fixed points, then the locus...

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Q JEE Main 2019

Consider the statement: " $P(n): n^2-n+41$ " is prime." Then which one of the following is true?

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Q JEE MAIN 2020

Let $\lambda \neq 0$ be in R . If $\alpha$ and $\beta$ are the roots of the equation, $\mathrm{x}^2-\mathrm{x}+2 \lambda=0$...

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Q JEE MAIN 2020

Let $[t]$ denote the greatest integer $\leq t$ and $\lim _{x \rightarrow 0} x\left[\frac{4}{x}\right]=A$. Then the function, $...

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Q JEE Main 2019

Let $f(x)=\left\{\begin{array}{ll}\max \left\{|x|, x^2\right\}, & |x| \leq 2 \\ 8-2|x|, & 2

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Q JEE Main 2019

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are...

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Q JEE MAIN 2020

Suppose the vectors $x_1, x_2$ and $x_3$ are the solutions of the system of linear equations, $A x=b$ when the...

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Q JEE MAIN 2020

Let a function $f:[0,5] \rightarrow R$ be continuous, $f(1)=3$ and $F$ be defined as : $$ F(x)=\int_1^x t^2 g(t) d...

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Q JEE Main 2019

If the third term in the binomial expansion of $\left(1+x^{\log _2 x}\right)^5$ equals 2560 , then a possible value of...

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Q JEE Main 2019

If the line $3 x+4 y-24=0$ intersects the $x$-axis at the point $A$ and the $y$-axis at the point $B$,...

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Q JEE MAIN_2021

The set of all values of $k>-1$, for which the equation $...

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Q JEE MAIN_2021_

Let [ $\lambda$ ] be the greatest integer less than or equal to $\lambda$. The set of all values of...

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Q JEE MAIN_2021_

Let $A(a, 0), B(b, 2 b+1)$ and $C(0, b), b \neq 0,|b| \neq 1$, be points such that the area...

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Q JEE MAIN_2021_

If the solution curve of the differential equation $\left(2 x-10 y^3\right) d y+y d x=0$, passes through the points $(0,1)$...

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Q JEE Main 2019

Consider the quadratic equation $(c-5) x^2-2 c x+(c-4)=0, c \neq 5$. Let $S$ be the set of all integral values...

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Q JEE Main 2019

An unbiased coin is tossed. If the outcome is a head, then a pair of unbiased dice is rolled and...

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The equation of the plane passing through the line of intersection of the planes $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=1$ and $...

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Q JEE Main 2019

Let $I=\int_a^b\left(x^4-2 x^2\right) d x$ If $I$ is minimum then the ordered pair $(a, b)$ is

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Q JEE MAIN 2020

If the equation of a plane P , passing through the intersection of the planes $x+4 y-z+7=0$ and $...

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Q JEE MAIN_2021

If two tangents drawn from a point $P$ to the parabola $y^2=16(x-3)$ are at right angles, then the locus of...

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Q JEE MAIN 2020

$\left(2 x^2+3 x+4\right)^{10}=\sum_{r=10}^{20} a_r x^r$. Then $\frac{a_7}{a_{13}}$ is equal to $\_\_\_\_$ .

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Q JEE Main 2019

Consider a triangular plot ABC with sides AB = 7 m, BC = 5 m and CA = 6 m....

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Q JEE MAIN 2020

Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y)=f(x)+f(y)+x y^2+x^2 y$, for all real $x$ and $...

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Q JEE MAIN_2021_

A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is...

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Q JEE MAIN 2020

If the system of equations $$ \begin{aligned} & x-2 y+3 z=9 \\ & 2 x+y+z=b \end{aligned} $$ $x-7 y+a z=24$,...

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Q JEE Main 2019

If $\frac{d y}{d x}+\frac{3}{\cos ^2 x} y=\frac{1}{\cos ^2 x}, x \in\left(\frac{-\pi}{3}, \frac{\pi}{3}\right)$ and $y\left(\frac{\pi}{4}\right)=\frac{4}{3}$, they $y\left(-\frac{\pi}{4}\right)$ equals

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Q JEE MAIN 2020

The probability of a man hitting a target is $\frac{1}{10}$. The least number of shots required, so that the probability...

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Q JEE MAIN 2020

Let $[\mathrm{t}]$ denote the greatest integer $\leq \mathrm{t}$. Then the equation in $x,[x]^2+2[x+2]-7=0$ has :

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Q JEE Main 2019

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is

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Q JEE Main 2019

If $\sum_{\mathrm{i}=1}^{20}\left(\frac{{ }^{20} \mathrm{C}_{\mathrm{i}-1}}{{ }^{20} \mathrm{C}_{\mathrm{i}}+{ }^{20} \mathrm{C}_{\mathrm{i}-1}}\right)^3=\frac{\mathrm{k}}{21}$, then k equals

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Q JEE MAIN 2020

If $(a+\sqrt{2} b \cos x)(a-\sqrt{2 b} \cos y)=a^2-b^2$, where $a >\mathrm{b}>0$, then $\frac{\mathrm{dx}}{\mathrm{dy}} \mathrm{at}\left(\frac{\pi}{4}, \frac{\pi}{4}\right)$ is :

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Q JEE Main 2019

If a circle $C$ passing through the point $(4,0)$ touches the circle $x^2+y^2+4 x-6 y=12$ externally at the point $(1$,...

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Q JEE MAIN_2021_

Each of the persons A and B independently tosses three fair coins. The probability that both of them get the...

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Q JEE MAIN 2020

Let $u=\frac{2 z+i}{z-k i}, z=x+i y$ and $k>0$. If the curve represented by $\operatorname{Re}(\mathrm{u})+\operatorname{Im}(\mathrm{u})=1$ intersects the y axis at the...

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Q JEE MAIN 2020

Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8, f^{\prime}(2)=5, f^{\prime}(x) \geq 1$ and $f^{\prime \prime}(x) \geq 4$,...

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Q JEE MAIN 2020

Let $a-2 b+c=1$. $f(x)=\left|\begin{array}{lll}x+a & x+2 & x+1 \\ x+b & x+3 & x+2\end{array}\right|$, then

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Q JEE MAIN 2020

Let $y=y(x)$ be the solution of the differential equation, $x y^{\prime}-y=x^2(x \cos x+\sin x), x>0$. If $y(\pi)=\pi$, then $y^{\prime \prime}\left(\frac{\pi}{2}\right)+y\left(\frac{\pi}{2}\right)$...

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Q JEE Main 2019

Let $d \in R$, and $$ A\left[\begin{array}{ccc} -2 & 4+d & (\sin \theta)-2 \\ 1 & (\sin \theta)+2 & d...

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Q JEE MAIN 2020

If one end of a focal chord AB of the parabola $\mathrm{y}^2= 8 x$ is at $A\left(\frac{1}{2},-2\right)$, then the equation...

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Q JEE MAIN 2020

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ be a given ellipse, length of whose latus rectum is 10 . If its eccentricity is the maximum...

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Q JEE MAIN 2020

Let $f(x)=\int \frac{\sqrt{x}}{(1+x)^2} d x(x \geq 0)$. Then $f(3)-f(1)$ is equal to:

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Q JEE MAIN 2020

The length of the minor axis (along $y$-axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse...

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Q JEE MAIN 2020

If $1+\left(1-2^2 \cdot 1\right)+\left(1-4^2 \cdot 3\right)+\left(1-6^2 \cdot 5\right)+\ldots \ldots+ \left(1-20^2 \cdot 19\right)=\alpha-220 \beta$, then an ordered pair $(\alpha, \beta)$ is...

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Q JEE MAIN 2020

Let $x_0$ be the point of local maxima of $f(x)=\vec{a} \cdot(\vec{b} \times \vec{c})$, where $...

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Q JEE MAIN 2020

If z be a complex number satisfying |Re(z)| + |km(z)| = 4, then |z| cannot be

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Q JEE MAIN 2020

The value of $\sum_{\mathrm{r}=0}^{20}{ }^{50-\mathrm{r}} \mathrm{C}_6$ is equal to:

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Q JEE MAIN 2020

The function $f(x)=\left\{\begin{array}{l}\frac{\pi}{4}+\tan ^{-1} x,|x| \leq 1 \\ \frac{1}{2}(|x|-1),|x|>1\end{array}\right.$ is :

JEE Main Mathematics Hard

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Q JEE MAIN 2020

Let $\mathrm{P}(3,3)$ be a point on the hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal to it at P intersects the x axis...

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Q JEE MAIN 2020

A random variable X has the following probability distribution X : 1 2 3 4 5 $...

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Q JEE MAIN 2020

A triangle $A B C$ lying in the first quadrant has two vertices as $\mathrm{A}(1,2)$ and $\mathrm{B}(3,1)$. If $\angle \mathrm{BAC}=90^{\circ}$,...

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Q JEE MAIN 2020

The area (in sq. units) of the largest rectangle $A B C D$ whose vertices $A$ and $B$ lie on...

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Q JEE MAIN 2020

Given the following two statements: $\left(\mathrm{S}_1\right):(\mathrm{q} \vee \mathrm{p}) \rightarrow(\mathrm{p} \leftrightarrow \sim \mathrm{q})$ is a tautology: $...

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Q JEE MAIN 2020

The following system of linear equations 7x + 6y – 2z = 0 3x + 4y + 2z = 0...

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Q JEE MAIN 2020

The distance of the point $(1,-2,3)$ from the plane $x-y+z=5$ measured parallel to the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}$ is :

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Q JEE MAIN 2020

$$ \text { If } A=\left[\begin{array}{cc} \cos \theta & i \sin \theta \\ i \sin \theta & \cos \theta \end{array}\right],\left(\theta=\frac{\pi}{24}\right)...

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Q JEE MAIN 2020

Let $f:(0, \infty) \rightarrow(0, \infty)$ be a dif ferentiable function such that $f(1)=e$ and $...

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Q JEE MAIN 2020

The integral $\int\left(\frac{x}{x \sin x+\cos x}\right)^2 d x$ is equal to (where C is a constant of integration):

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Q JEE MAIN 2020

Let $f(\mathrm{x})=|\mathrm{x}-2|$ and $\mathrm{g}(\mathrm{x})=f(f(\mathrm{x})), \mathrm{x} \in[0,4]$. Then $\int^3(g(x)-f(x)) d x$ is equal to:

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Q JEE MAIN 2020

The solution of the dif ferential equation $\frac{d y}{d x}-\frac{y+3 x}{\log _e(y+3 x)}+3=0$ is (where C is a constant of...

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Q JEE MAIN 2019

The area (in sq. units) in the first quadrant bounded by the parabola, $y=x^2+1$, the tangent to it at the...

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Q JEE MAIN 2020

Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with...

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Q JEE MAIN 2020

$...

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Q JEE MAIN 2020

The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are 5, 7,...

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Q JEE MAIN 2019

Let $K$ be the set of all real values of $x$ where the function $f(x)=\sin |x|-|x|+2(x-\pi)$ is not differentiable. Then...

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Q JEE MAIN 2020

If for some positive integer $n$, the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in...

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Q JEE MAIN 2020

Let $\alpha$ and $\beta$ be the roots of $x^2-3 x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^2-6 x+q=0$....

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Q JEE MAIN 2019

The solution of the differential equation, $\frac{d y}{d x}=(x-y)^2$, when $y(1)=1$, is

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Q JEE MAIN 2020

If the perpendicular bisector of the line segment joining the points P(1, 4) and Q(k, 3) has y-intercept equal to...

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Q JEE MAIN 2020

Let $f$ and $g$ be differentiabel functions on $R$ such that fog is the identity function. If for some $...

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Q JEE MAIN 2019

Let $\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $\beta \hat{i}+(1-\beta) \hat{j}$ respectively be the position vectors of the points $A, B$ and...

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Q JEE MAIN 2019

All $x$ satisfying the inequality $\left(\cot ^{-1} x\right)^2-7\left(\cot ^{-1} x\right)+10>0$, lie in the interval

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Q JEE MAIN 2020

If the system of equations $$ \begin{aligned} & x+y+z=2 \\ & 2 x+4 y-z=6 \\ & 3 x+2 y+\lambda z=\mu...

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Q JEE MAIN 2020

If $p \rightarrow(p \wedge \sim q)$ is false, then the truth values of $p$ and $q$ are respectively :

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Q JEE MAIN 2019

If the point $(2, \alpha, \beta)$ lies on the plane which passes through the points $(3,4,2)$ and $(7,0,6)$ and is...

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Q JEE MAIN 2020

If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of...

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Q JEE MAIN 2019

Let S = {1, 2, ...., 20}. A subset B of S is said to be “nice”, if the sum...

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Q JEE MAIN 2019

Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the...

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Q JEE MAIN 2020

If $\frac{d y}{d x}=\frac{x y}{x^2+y^2} ; y(1)=1 ; \quad$ then a value of $x$ satisfying $y(x)=e$ is :

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Q JEE MAIN 2020

A particle of mass m is fixed to one end of a light spring having force constant k and unstretched...

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Q JEE MAIN 2019

Let $S_n=1+q+q^2+\ldots+q^n$ and $T_n=1+\left(\frac{q+1}{2}\right)+\left(\frac{q+1}{2}\right)^2+\ldots .+\left(\frac{q+1}{2}\right)^n$ where $q$ is a real number and $q \neq 1$. If $...

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Q JEE MAIN 2020

If the tangent to the curve, $y=e^x$ at a point $\left(c, e^c\right)$ and the normal to the parabola, $y^2=4 x$...

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Q JEE MAIN 2020

Let $a, b \in R, a \neq 0$ be such that the equation, $a x^2- 2 \mathrm{bx}+5=0$ has a repeated...

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Q JEE MAIN 2020

https://competishun.com/let-s-be-the-set-of-all-integer-solutions-x-y-z-of-the-system-of-equationsbr-beginaligned-x-2-y5-z0-2-x4-yz0-7-x14-y9-z0-endaligned-such-that-15-leq-x2y2z2-leq-150-then-the-number-of-elements-in-the-s/

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Q JEE MAIN 2019

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13,...

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Q JEE-MAIN 2020

Let S be the set of all integer solutions, (x, y, z), of the system of equations $$ \begin{aligned} &...

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Q JEE MAIN 2020

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Q JEE-MAIN 2020

The total number of 3-digit numbers, whose sum of digits is 10, is ______.

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Q JEE MAIN 2019

A bag contains 30 white ball and 10 red balls. 16 balls are drawn one by one randomly from the...

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Q JEE-MAIN 2020

Let a plane P contain two lines $$ \overrightarrow{\mathrm{r}}=\hat{\mathrm{i}}+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}), \lambda \in \mathrm{R} \text { and } \overrightarrow{\mathrm{r}}=-\hat{\mathrm{i}}+\mu(\hat{\mathrm{j}}-\hat{\mathrm{k}}), \mu \in \mathrm{R}...

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Q JEE-MAIN 2020

The probability that a randomly chosen 5-digit number is made from exactly two digits is :

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Q JEE-MAIN 2020

If $x^3 d y+x y d x=x^2 d y+2 y d x ; x(2)=e$ and $x>1$, then $x(4)$ is equal...

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Q JEE MAIN 2019

Let $\alpha$ and $\beta$ the roots of the quadratic equation $x^2 \sin \theta-x(\sin \theta \cos \theta+1)+\cos \theta=0\left(0

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Q JEE MAIN 2020

The sum, $\sum_{n=1}^7 \frac{n(n+1)(2 n+1)}{4}$ is equal toThe sum, $\sum_{n=1}^7 \frac{n(n+1)(2 n+1)}{4}$ is equal to

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Q JEE-MAIN 2020

$\lim _{x \rightarrow a} \frac{(a+2 x)^{\frac{1}{3}}-(3 x)^{\frac{1}{3}}}{(3 a+x)^{\frac{1}{3}}-(4 x)^{\frac{1}{3}}}(a \neq 0)$ is equal to : $(3 a+x)^{\frac{1}{3}}-(4 x)^{\frac{1}{3}}$

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Q JEE MAIN 2019

The integral $\int_{\pi / 6}^{\pi / 4} \frac{d x}{\sin 2 x\left(\tan ^5 \times \cot ^5 x\right)}$ equals :

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Q JEE MAIN 2020

Let a line y=m x(m>0) intersect the parabola, $y^2=x$ at a point $P$, other than the origin. Let the tangent...

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Q JEE MAIN 2020

In the expansion of $\left(\frac{x}{\cos \theta}+\frac{1}{x \sin \theta}\right)^{16}$, if $l_1$ is the least value of the term independent of x...

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Q JEE MAIN 2020

Let f(x) be a polynomial of degree 3 such that f(–1) = 10, f(1) = –6, f(x) has a critical...

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Q JEE-MAIN 2020

If the surface area of a cube is increasing at a rate of $3.6 \mathrm{~cm}^2 / \mathrm{sec}$, retaining its shape;...

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Q JEE MAIN 2019

The number of function f from {1, 2, 3, ...,20} onto {1, 2, 3, ..., 20} such that f(k) is...

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Q JEE MAIN 2020

The plane which bisects the line joining, the points (4, –2, 3) and (2, 4, –1) at right angles also...

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Q JEE MAIN 2020

The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the...

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Q JEE MAIN 2020

If a $\Delta$ABC has vertices A(–1, 7), B(–7, 1) and C(5, –5), then its orthocentre has coordinates

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Q JEE MAIN 2020

If $\frac{\sqrt{2} \sin \alpha}{\sqrt{1+\cos 2 \alpha}}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2 \beta}{2}}=\frac{1}{\sqrt{10}}, \alpha, \beta, \in\left(0, \frac{\pi}{2}\right)$, then $\tan (\alpha+2 \beta)$ is equal...

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Q JEE MAIN 2020

Let $f:(1,3) \rightarrow R$ be a function defined by $f(x)=\frac{x[x]}{1+x^2}$, where $[x]$ denotes the greatest integer $\leq x$. Then the...

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Q JEE MAIN 2019

If 19 th term of a non-zero A.P. is zero, then its $\left(49^{\text {th }}\right.$ term) $(29$ th term $)$...

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Q JEE-MAIN 2020

Let $x i(1 \leq i \leq 10)$ be ten observations of a random variable $X$. If $\sum_{i=1}^{10}\left(x_i-p\right)=3$ and $\sum_{i=1}^{10}\left(x_i-p\right)^2=9$ where...

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Q JEE MAIN 2020

Let $\mathrm{a}_{\mathrm{n}}$ be the $\mathrm{n}^{\text {th }}$ term of a G.P. of positive terms. If $\sum_{n=1}^{100} a_{2 n+1}=200$ and $...

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Q JEE MAIN 2020

Let A and B be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and...

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Q JEE MAIN 2019

Given $\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$ for $\triangle A B C$ with usual natation. If $\frac{\cos A}{\alpha}=\frac{\cos B}{\beta}=\frac{\cos C}{12}$, then the ordered triplet $(\alpha$,...

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Q JEE MAIN 2020

Let $\alpha=\frac{-1+\mathrm{i} \sqrt{3}}{2}$. If $\mathrm{a}=(1+\alpha) \sum_{\mathrm{k}=0}^{100} \alpha^{2 \mathrm{k}}$ and $\mathrm{b}=\sum_{\mathrm{k}=0}^{100} \alpha^{3 \mathrm{k}}$, then a and b are the roots of...

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If a line, y=m x+c is a tangent to the circle $\ldots(x-3)^2+y^2=1$ and it is perpendicular to a line $L_1$,...

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Q JEE MAIN 2019

Let $(x+10)^{50}+(x-10)^{50} =a_0+a_1 x+a_2 x^2+\ldots+a_{50} x^{50}$, for all $x \in R$ : then $\frac{a_2}{a_0}$ is equal to :

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Q JEE MAIN 2020

If $\alpha$ and $\beta$ be the coefficients of $x^4$ and $x^2$ respectively in the expansion of $\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6$, then

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Q JEE MAIN 2019

Two lines $\frac{x-3}{1}=\frac{y+1}{3}=\frac{z-3}{4}$ and $\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4}$ intersect at the point $R$. The reflection of $R$ in the $x y$-plane has coordinates...

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Q JEE MAIN 2019

$\lim _{x \rightarrow 0} \frac{x \cot (4 x)}{\sin ^2 x \cot ^2(2 x)}$ is equal to

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Q JEE MAIN 2019

If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2,...

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Q JEE MAIN 2019

Contrapositive of the statement “If two numbers are not equal, then their squares are not equal.” is :

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Q JEE MAIN 2019

Let $f(x)=\frac{x}{\sqrt{a^2+x^2}}-\frac{(d-x)}{\sqrt{b^2+(d-x)^2}}, x \in R$ where $a, b$ and $d$ are non-zero real constants. Then :

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Q JEE MAIN 2019

Let $z$ be a complex number such that $|z|+z=3+i($ where $i=\sqrt{-1})$. Then $|z|$ is equal to :

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Q JEE MAIN 2019

If the area of the triangle whose one vertex is at the vertex of the parabola, $y^2+4\left(x-a^2\right)=0$ and the other...

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Q JEE MAIN 2019

If $\int \frac{x+1}{\sqrt{2 x-1}} d x=f(x) \sqrt{2 x-1}+C$, where $C$ is a constant of integration, then $f(x)$ is equal to...

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Q JEE MAIN 2020

The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was found...

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Q JEE MAIN 2020

Let S be the set of all real roots of the equation, $3^x\left(3^x-1\right)+2=\left|3^x-1\right|+\left|3^x-2\right|$. Then $S$

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Q JEE MAIN 2020

If the $10^{\text {th }}$ term of an A.P. is $\frac{1}{20}$ and its $20^{\text {th }}$ term is $\frac{1}{10}$, then...

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Q JEE MAIN 2019

If $...

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Q JEE MAIN 2020

$\lim _{x \rightarrow 0} \frac{\int_0^x t \sin (10 t) d t}{x}$ is equal to

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Q JEE MAIN 2020

If a hyperbola passes through the point $\mathrm{P}(10,16)$ and it has vertices at $( \pm 6,0)$, then the equation of...

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Q JEE-MAIN 2020

If $z_1, z_2$ are complex numbers such that $\operatorname{Re}\left(z_1\right)=\left|z_1-1\right|, \operatorname{Re}\left(z_2\right)=\left|z_2-1\right|$, and $\arg \left(z_1-z_2\right)=\frac{\pi}{6}$, then $\operatorname{Im}\left(z_1+z_2\right)$ is equal to

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Q JEE MAIN 2019

Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of the expression $...

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Q JEE-MAIN 2020

If the sum of the series $20+19 \frac{3}{5}+19 \frac{1}{5}+18 \frac{4}{5}+\ldots .$. upto $n$th term is 488 and the $n$th term...

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Q JEE MAIN 2020

The length of the perpendicular from the origin, on the normal to the curve, $x^2+2 x y-3 y^2=0$ at the...

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Q JEE MAIN 2020

The set of all real values of $\lambda$ for which the quadratic equations, $\left(\lambda^2+1\right) x^2-4 \lambda x+2=0$ always have exactly...

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A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant...

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The system of linear equations $...

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If the term independent of $x$ in the expansion of $\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9$ is $k$, then $18 k$ is equal...

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Let $e_1$ and $e_2$ be the eccentricities of the ellipse, $\frac{x^2}{25}+\frac{y^2}{b^2}=1(b

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Which of the following statements is a tautology?

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Let $A$ and $B$ be two invertible matrices of order $3 \times 3$. If $\operatorname{det}\left(A B A^{\top}\right)=8$ and $\operatorname{det}\left(A B^{-1}\right)=8$,...

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The mirror image of the point $(1,2,3)$ in a plane is $\left(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3}\right)$. Which of the following points lies on this...

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Suppose $f(x)$ is a polynomial of degree four, having critical points at $-1,0,1$. If $T=\{x \in R \mid f(x)=f(0)\}$, then...

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Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x)-f(y)| \leqslant 2|x-y|^{\frac{3}{2}}$, for all $...

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Let $S$ be the set of points where the function, $f(x)=|2-|x-3| \|, x \in R$, is not differentiable. Then $...

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Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the...

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If the sum of the coefficients of all even powers of x in the product $\left(1+x+x^2+\ldots \ldots+x^{2 n}\right)\left(1-x+x^2-x^3+\ldots .+x^{2 n}\right)$...

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If the variance of the first n natural numbers is 10 and the variance of the first m even natural...

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The logical statement (MF ST-1) $[\sim(\sim p \vee q) \vee(p \wedge r)] \wedge(\sim q \wedge r)$ is equivalent to

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$\lim _{x \rightarrow 2} \frac{3^x+3^{3-x}-12}{3^{-x / 2}-3^{1-x}}$ is equal to $\_\_\_\_$ .

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Let $\mathrm{A}(1,0), \mathrm{B}(6,2)$ and $\mathrm{C}\left(\frac{3}{2}, 6\right)$ be the vertices of a triangle $A B C$. If $P$ is a point...

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If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then...

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If $y(\alpha)=\sqrt{2\left(\frac{\tan \alpha+\cot \alpha}{1+\tan ^2 \alpha}\right)+\frac{1}{\sin ^2 \alpha}}, \alpha \in\left(\frac{3 \pi}{4}, \pi\right)$ then $\frac{\mathrm{dy}}{\mathrm{d} \alpha}$ at $\alpha=\frac{5 \pi}{6}$ is

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If the system of linear equations $$ \begin{gathered} x-4 y+7 z=g \\ 3 y-5 z=h \\ -2 x+5 y-9 z=k...

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Let the function, $\mathrm{f}:[-7,0] \rightarrow \mathrm{R}$ be continuous on $[- 7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $...

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If $0 \leq x

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$f$ the lines $x=a y+b, z=c y+d$ and $x=a^{\prime} z+b^{\prime}, y=c^{\prime} z+d^{\prime}$ are perpendicular, then

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If the sum and product of the first three terms in an A.P. are 33 and 1155, respectively, then a...

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Let $f:[0,1] \rightarrow R$ be such that $f(x y)=f(x) \cdot f(y)$, for all $x, y \in[0,1]$, and $f(0) \neq 0$....

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If $f(x)=\int \frac{5 x^8+7 x^6}{\left(x^2+1+2 x\right)^2} d x,(x \geq 0)$, and $f(0)=0$, then the value of $f(1)$ is

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If $f: R \rightarrow R$ is a differentiable function and $f(2)=6$, then $...

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Let S be the set of all functions f : [0, 1]  R, which are continuous on [0, 1]...

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If the tangent to the parabola $\mathrm{y}^2=\mathrm{x}$ at a point $(\alpha, \beta),(\beta>0)$ is also a tangent to the ellipse, $...

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Let $\vec{a}=\hat{i}-2 \hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$ be two vectors. If $\vec{c}$ is a vector such that $\vec{b} \times \vec{c}=\vec{b} \times \vec{a}$...

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The value of sin10° sin30° sin50° sin70° is

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If the two lines $x+(a-1) y=1$ and $2 x+a^2 y=1(a \in R-\{0,1\})$ are perpendicular, then the distance of their point...

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The area (in sq. units) of the region $A=\left\{(x, y): \frac{y^2}{2} \leq x \leq y+4\right\}$ is

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The area (in sq. units) of the region $\left\{(x, y) \in \mathbb{R}^2: x^2 \leq y \leq 3-2 x\right\}$, is

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The value of the integral $\int_0^1 x \cot ^{-1}\left(1-x^2+x^4\right) d x$ is

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Let $a, b c \in R$ be such that $a^2+b^2+c^2=1$. If $a \cos \theta=b \cos \left(\theta+\frac{2 \pi}{3}\right)=c \cos \left(\theta+\frac{4 \pi}{3}\right)$,...

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The differential equation of the family of curves, $x^2=4 b(y+b), b \in R$, is

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Let the latus ractum of the parabola $y^2=4 x$ be the common chord to the circles $C_1$ and $C_2$ each...

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A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $\tan ^{-1}\left(\frac{1}{2}\right)$. Water is...

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Let $p, q, r$ be three statements such that the truth value of $(p \wedge q) \rightarrow(\sim q \vee r)$...

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If $\mathrm{A}=\left(\begin{array}{ll}2 & 2 \\ 9 & 4\end{array}\right)$ and $\mathrm{I}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$, then $10 \mathrm{~A}^{-1}$ is...

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Two newspapers A and B are published in a city. It is known that 25% of the city population reads...

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Let $R_1$ and $R_2$ be two relation defined as follows : $\mathrm{R}_1=\left\{(\mathrm{a}, \mathrm{b}) \in \mathrm{R}^2: \mathrm{a}^2+\mathrm{b}^2 \in \mathrm{Q}\right\}$ and $...

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Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the...

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If the value of the integral $\int_0^{1 / 2} \frac{x^2}{\left(1-x^2\right)^{3 / 2}} d x$ is $\frac{k}{6}$, then $k$ is equal...

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The mean and the median of the following ten numbers in increasing order $10,22,26,29,34, x, 42,67,70, y$ are 42 and...

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If $p \Rightarrow(q \vee r)$ is false, then truth values of $p, q, r$ are respectively:

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The area (in sq. units) of the smaller of the two circles that touch the parabola, $y^2=4 x$ at the...

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The sum of the series 1 + 2 × 3 + 3 × 5 + 4 × 7 + ......

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The domain of the definition of the function $f(x)=\frac{1}{4-x^2}+\log _{10}\left(x^3-x\right)$ is :

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The common tangent to the circles $x^2+y^2=4$ and $x^2+y^2+6 x+8 y-24=0$ also passes through the point :

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If $f(x)=[x]-\left[\frac{x}{4}\right], x \in R$, where $[x]$ denotes the greatest integer function, then :

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Let P be the plane, which contains the line of intersection of the planes, x + y + z –...

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Let $z \in C$ be such that $|z|

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If the function $f(x)=\left\{\begin{array}{ll}a|\pi-x|+1, & x \leq 5 \\ b|x-\pi|+3, & x>5\end{array}\right.$ is continuous at $x=5$, then the value of...

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Let $f(x)$ be a cubic polynomial with $f(1)=-10, f(-1)=6$, and has a local_minima at $x=1$, and $f^{\prime}(x)$ has a local...

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If a unit vector $\vec{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}, \frac{\pi}{4}$ with $\hat{j}$ and $\theta \in(0, \pi)$ with $\hat{k}$, then...

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Let $B$ be the centre of the circle $x^2+y^2-2 x+4 y+1=0$. Let the tangents at two point $P$ and $Q$...

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A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If...

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If the line $y=m x$ bisects the area enclosed by the lines $x=0, y=0, x=3 / 2$ and the curve...

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If some three consecutive coefficients in the binomial expansion of $(x+1)^n$ in powers of $x$ are in the ratio $...

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The number of elements in the set $\left\{A=\left(\begin{array}{ll}a & b \\ 0 & d\end{array}\right): a, b, d \in\{-1,0,1\}\right.$ and $\left.(I-A)^3=I-A^3\right\}$...

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If $S=\frac{7}{5}+\frac{9}{5^2}+\frac{13}{5^3}+\frac{19}{5^4}+\ldots$, then 160 S is equal to $\_\_\_\_$ .

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If $\int e^{\sec x}\left(\sec x \tan x f(x)+\sec x \tan x+\sec ^2 x\right) d x \quad=e^{\sec x} f(x)+C$, then a...

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A tangent line L is drawn at the point $(2,-4)$ on the parabola $\mathrm{y}^2=8 \mathrm{x}$. If the line L is...

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If $...

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An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that...

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If $\left(\frac{3^6}{4^4}\right) k$ is the term, independent of x , in the binomial expansion of $\left(\frac{x}{4}-\frac{12}{x^2}\right)^{12}$, then k is equal...

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The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is ______.

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If $x \phi(\mathrm{x})=\int_5^x\left(3 t^2-2 \phi^{\prime}(\mathrm{t})\right) d t, x>-2$, and $\phi(0)=4$, then $\phi(2)$ is

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The vertices $B$ and $C$ of a $\triangle A B C$ lie on the line, $\frac{x+2}{3}=\frac{y-1}{0}=\frac{z}{4}$ such that $B C=5$...

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The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so...

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If the variable line $3 x+4 y=\alpha$ lies between the two circles $(x-1)^2+(y-1)^2=1$ and $(x-9)^2+(y-1)^2=4$, without intercepting a chord on...

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If $m$ is chosen in the quadratic equation $\left(m^2+1\right) x^2-3 x+\left(m^2+1\right)^2=0$ such that the sum of its roots is greatest,...

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If the coefficient of $a^7 b^8$ in the expansion of $(a+2 b+4 a b)^{10}$ is $K .2^{16}$, then $K$ is...

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The mean of 10 numbers $$ 7 \times 8,10 \times 10,13 \times 12,16 \times 14, \ldots . \text { is...

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If the system of equations 2x+3y-z=0, x+ky-2z=0 and $2 x-y+z=0$ has a non-trivial solution $(x, y, z)$, then $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k$ is...

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The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then...

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If ' $R$ ' is the least value of ' $a$ ' such that the function $f(x)=x^2+a x+1$ is increasing...

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Two poles standing on a horizontal ground are of heights 5 m and 10 m respectively. The line joining their...

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The square of the distance of the point of intersection of the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+1}{6}$ and the plane $2 x-y+z=6$ from...

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A point $Z$ moves in the complex plane such that arg $\left(\frac{z-2}{z+2}\right)-\frac{\pi}{4}$, then the minimum value of $|Z-9 \sqrt{2}-2 \mathrm{i}|^2$...

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Let $[\mathrm{t}]$ denote the greatest integer $\leq 1$. Then the value of 8 . $\int_{-\frac{1}{2}}^1([2 x]|\mathrm{x}|) d x$ is

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Let $f$ be any continuous function on $[0,2]$ and twice differentiable on $(0,2)$. If $f(0)=0, f(1)=1$ and $f(2)=2$, then

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The line $12 x \cos \theta+5 y \sin \theta=60$ is tangent to which of the following curves?

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Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear,...

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If $\operatorname{Re}\left(\frac{z-1}{2 z+i}\right)=1$, where $z=x+i y$, then the point $(x, y)$ lies on a

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If $y=y(x)$ is the solution of the differential equation, $e^y\left(\frac{d y}{d x}-1\right)=e^x$ such that $y(0)=0$, then $y(1)$ is equal to

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If $a_r=\cos \frac{2 r \pi}{9}+i \sin \frac{2 r \pi}{9}, r=1,2,3, \ldots, \mathrm{i}=\sqrt{-1}$ then the determinant $...

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A vertical pole fixed to the horizontal ground is divided in the ratio 3 : 7 by a mark on...

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Q JEE MAIN 2020

An unbiased coin is tossed 5 times. Suppose that a variable X is assigned the value k when k consecutive...

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If $y=m x+4$ is a tangent to both the parabolas, $y^2 =4 x$ and $x^2=2 b y$, then $b$ is...

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Five number are in A.P., whose sum is 25 and product is 2520 . If one of these five numbers...

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A vector $\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}(\alpha, \beta \in R)$ lies in the plane of the vectors, $\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}+4 \hat{k}$....

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$\lim _{x \rightarrow 0} \frac{\sin ^2\left(\pi \cos ^4 x\right)}{x^4}$ is equal to:

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Let $\alpha$ and $\beta$ be two real roots of the equation $(\mathrm{k}+1) \tan ^2 \mathrm{x}-\sqrt{2} \cdot \lambda \tan \mathrm{x}=(1-\mathrm{k})$, where...

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The value of $\lambda$ such that sum of the squares of the roots of the quadratic equation, $x^2+(3-\lambda) x+2=\lambda$ has...

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If $\frac{d y}{d x}=\frac{2^{x+y}-2^x}{2 y}, \mathrm{y}(0)=1$, then $\mathrm{y}(1)$ is equal to :

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Let $x^k+y^k=a^k,(a, k>0)$ and $\frac{d y}{d x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0$, then k is :

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If $g(x)=x^2+x-1$ and $(g \circ f)(x)=4 x^2-10 x+5$, then $\mathrm{f}\left(\frac{5}{4}\right)$ is equal to:

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The value of $\cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \ldots \ldots \cos \frac{\pi}{2^{10}} \cdot \sin \frac{\pi}{2^{10}}$ is

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Let $\alpha$ be a root of the equation $x^2+x+1=0$ and the matrix $...

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Let P be a plane through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R...

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If mean and standard deviation of 5 observations $x_1, x_2, x_3, x_4, x_5$ are 10 and 3 , respectively, then...

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If the function $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}\frac{1}{x} \log _e\left(\frac{1+\frac{a}{x}}{1-\frac{x}{b}}\right) & , x0\end{array}\right.$ Is continuous at $\mathrm{x}=0$, then $\frac{1}{a}+\frac{1}{b}+\frac{4}{k}$ is equal to :

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The area of the region, enclosed by the circle $x^2+ y^2=2$ which is not common to the region bounded by...

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If $\int_0^x f(t) d t=x^2+\int t^2 f(t) d t$ then $f(1 / 2)$ is :

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If the system of linear equations 2x + 2ay + az = 0 2x + 3by + bz = 0...

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The greatest positive integer k , for which $49^{\mathrm{k}}+1$ is a factor of the sum $49^{125}+49^{124}+\ldots \ldots+49^2+49+1$, is

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Q JEE MAIN 2020

The logical statement $(p \Rightarrow q) \wedge(q \Rightarrow \sim p)$ is equivalent to

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The length of the latus rectum of a parabola, whose vertex and focus are on the positive $x$-axis at a...

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The number of solutions of the equation $32^{\tan ^2 x}+32^{x c^2 x}=81,0 \leq x \leq \frac{\pi}{4}$ is :

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Let N be the set of natural numbers and two functions f and g be defined as $...

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If the following system of linear equations $...

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Let A be the set of all points $(\alpha, \beta)$ such that the area of triangle formed by the points...

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cosec18º is a root of the equation :

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Two vertices of a triangle are (0, 2) and (4, 3). If its orthocentre is at the origin, then its...

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If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines, $...

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Let $a_1, a_2, a 3, \ldots$ be an A.P. If $\frac{a_1+a_2+\ldots+a_{10}}{a_1+a_2+\ldots+a_p}=\frac{100}{p^2}, p \neq 10$, then $\frac{a_{11}}{a_{10}}$ is equal to :

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If $\sum_{\mathrm{r}=0}^{25}\left\{{ }^{50} \mathrm{C}_{\mathrm{r}} \cdot{ }^{50-\mathrm{r}} \mathrm{C}_{25-\mathrm{r}}\right\}=\mathrm{K}\left({ }^{50} \mathrm{C}_{25}\right)$, then K is equal to:

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The integral $\int \frac{1}{\sqrt[4]{(x-1)^3(x+2)^5}} d x$ is equal to :

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If $z$ is a complex number such that $\frac{z-1}{z-1}$ is purely imaginary, then the minimum value of $|z-(3+3 i)|$ is...

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Let $f$ be a differentiable function such tht $f^{\prime}(x)=7-\frac{3 f(x)}{4 x},(x>0)$ and $f(1) \neq 4$. Then $...

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\text { The sum of the roots of the equation } x+1-2 \log _2\left(3+2^x\right)+2 \log _4\left(10-2^{-x}\right)=0 \text {, is: }

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The value of $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{d x}{[x]+[\sin x]+4}$ where $[t]$ denotes the greatest integer less than or equal to $t$, is...

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Which of the following is not correct for relation R on the set of real numbers ?

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Let $A=\left[\begin{array}{ccc}2 & b & 1 \\ b & b^2+1 & b \\ 1 & b & 2\end{array}\right]$ where $b>0$....

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If $y \frac{d y}{d x}=x\left|\frac{y^2}{x^2}+\frac{\phi\left(\frac{y^2}{x^2}\right)}{\phi^{\prime}\left(\frac{y^2}{x^2}\right)}\right|, x>0, \phi>0$, and $y(1)=-1$, then $\phi\left(\frac{y^2}{4}\right)$ is equal to :

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With the usual notation, in $\triangle A B C$, if $\angle A+\angle B=120^{\circ}, a=\sqrt{3}+1$ and $b=\sqrt{3}-1$, then the ratio $...

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Q JEE MAIN 2019

Let $a_1, a_2, a_3 \ldots a_{10}$ in G.P with $a_i>0$ for $i=1,2, \ldots \ldots, 10$ and $S$ be the set...

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Q JEE MAIN 2019

Two sides of a parallelogram are along the lines, $x+y=3$ and $x-y+3=0$. If its diagonals intersect at (2,4) then one...

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Q JEE MAIN 2019

Let $S=\left\{(x, y) \in R^2: \frac{y^2}{1+r}-\frac{x^2}{1-r}=1\right\}$, where where $r \neq \pm 1$. Then $S$ represents

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Q JEE MAIN 2019

On which of the following lines lies the point of intersection of the line, $\frac{x-4}{2}=\frac{y-5}{2}=\frac{z-3}{1}$ and the plane, $...

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Q JEE MAIN 2019

If $\int x^5 e^{-4 x^3} d x=\frac{1}{48} e^{-4 x^3} f(x)+C$ where $C$ is a constant of integration, then $f(x)$ is...

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Q JEE MAIN 2019

The length of the chord of the parabola $x^2=4 y$ having equation $x-\sqrt{2} y+4 \sqrt{2}=0$ is

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Q JEE MAIN 2019

The curve amongst the family of curves represented by the differential equation, $\left(x^2-y^2\right) d x+2 x y d y=0$ which...

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Q JEE MAIN 2019

The value of $\cot \left(\sum_{n=1}^{19} \cot ^{-1}\left(1+\sum_{p=1}^n 2 p\right)\right)$ is

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Q JEE MAIN 2019

Let $f:(-1,1) \rightarrow R$ be a function defined by $f(x)=\max \left\{-|x|,-\sqrt{1-x^2}\right\}$. If $K$ be the set of all points at...

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Q JEE MAIN 2019

If the probability of hitting a target by a shooter, in any shot, is $\frac{1}{3}$, then the minimum number of...

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Q JEE MAIN 2019

Let $\mathrm{z}=\left(\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}\right)^5+\left(\frac{\sqrt{3}}{2}-\frac{\mathrm{i}}{2}\right)^5$. If $\mathrm{R}(\mathrm{z})$ and $\mathrm{I}(\mathrm{z})$ respectively denote the real and imaginary parts of z , then

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Q JEE MAIN 2019

The plane which bisects the line segment joining the points (–3, –3, 4) and (3, 7, 6) at right angles,...

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Q JEE MAIN 2019

The positive value of $\lambda$ for which the co-efficient of $x^2$ in the expression $x^2\left(\sqrt{x}+\frac{\pi}{x^2}\right)^{10}$ is 720 , is

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Q JEE MAIN 2019

The number of values of $\theta \in(0, \pi)$ for which the system of linear equations $$ \begin{aligned} & x+3 y+7...

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Q JEE MAIN 2019

\text { If the area of an equilateral triangle inscribed in the circle, } x^2+y^2+10 x+12 y+c=0 \text { is...

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Q JEE MAIN 2019

Let $\alpha=(\lambda-2) \mathrm{a}+\mathrm{b}$ and $\beta=(4 \lambda-2) \mathrm{a}+3 \mathrm{~b}$ be two given vectors where vectors a and b are non-collinear. The...

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Q JEE Main 2021

If $\frac{d y}{d x}=\frac{2^x y+2^y \cdot 2^x}{2^x+2^{x+y} \log _e 2}, y(0)$, then for $y=1$, the value of $x$ lies in...

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Q JEE Main 2021

The locus of mid-points of the line segments joining $(-3,-5)$ and the points on the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is :

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Q JEE MAIN 2021

Three numbers are in an increasing geometric progression with common ratio $r$. If the middle number is doubled, then the...

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Q JEE Main 2021

If $\alpha=\lim _{x \rightarrow \pi / 4} \frac{\tan ^3 x-\tan x}{\cos \left(x+\frac{\pi}{4}\right)}$ and $\beta=\lim _{x \rightarrow 0}(\cos x)^{\cot x}$ are...

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Q JEE MAIN 2021

The function $f(x)=\left|x^2-2 x-3\right| . e^{\left|9 x^2-12 x+4\right|}$ is not differentiable at exactly:

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Q JEE MAIN 2021

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|2 \vec{a}+3 \vec{b}|=|3 \vec{a}+\vec{b}|$ and the angle between $\vec{a}$ and $\vec{b}$...

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Q JEE Main 2021

Negation of the statement $(p \vee r) \Rightarrow(q \vee r)$ is :

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Q JEE MAIN 2021

Let f be a non - negative function in $[0,1]$ and twice differentiable in $(0,1)$. If $...

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Q JEE Main 2021

The distance of the point $(-1,2,-2)$ from the line of intersection of the planes $2 x+3 y+2 z=0$ and $...

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Q JEE MAIN 2021

Let the equation of the plane, that passes through the point $(1,4,-3)$ and contains the line of intersection of the...

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Q JEE MAIN 2021

The sum of 10 terms of the series $\frac{3}{1^2 \times 2^2}+\frac{5}{2^2 \times 3^2}+\frac{7}{3^2 \times 4^2}+\ldots$ is :

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Q JEE Main 2021

Let $f: N \rightarrow N$ be a function such that $f(m+n)=f(m)+f(n)$ for every $m, n \in N$. If $f(6)=18$, then...

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Q JEE MAIN 2021

The number of real roots of the equation $\mathrm{e}^{4 \mathrm{x}}+2 \mathrm{e}^{3 \mathrm{x}}-\mathrm{e}^{\mathrm{x}}-6=0$ is :

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Q JEE Main 2021

Let $S=\{1,2,3,4,5,6\}$. Then the probability that a randomly chosen onto function $g$ from $S$ to $S$ satisfies $g(3) =2 g(1)$...

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Q JEE MAIN 2021

Let *, □ $\in\{\wedge, \vee\}$ be such that the Boolean expression $\left(\mathrm{p}^* \sim q\right) \Rightarrow(\mathrm{p}$ □ q) is a tautology....

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Q JEE Main 2021

The domain of the function $f(x)=\sin ^{-1}\left(\frac{3 x^2+x-1}{(x-1)^2}\right)+\cos ^{-1}\left(\frac{x-1}{x+1}\right)$ is :

JEE Main Mathematics Medium

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Q JEE MAIN 2021

A wire of length 20 m is to be cut into two pieces. One of the pieces is to be...

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Q JEE MAIN 2021

If $\int_6^{16} \frac{\log _e x^2}{\log _e x^2+\log _e\left(x^2-44 x+484\right)} d x$ is equal to :

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Q JEE Main 2021

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\vec{r}$...

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Q JEE Main 2021

If $\alpha+\beta+\gamma=2 \pi$, then the system of equations $$ \begin{aligned} & x+(\cos \gamma) y+(\cos \beta) z=0 \\ & (\cos \gamma)...

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Q JEE MAIN 2021

If $x^2+9 y^2-4 x+3=0, x, y \in \mathbb{R}$, then $x$ and $y$ respectively lie in the intervals:

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Q JEE MAIN 2021

When a certain biased die is rolled, a particular face occurs with probability $\frac{1}{6}-x$ and its opposite face occurs with...

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Q JEE MAIN 2021

If $\alpha, \beta$ are the distinct roots of $x^2+b x+c=0$, then $\lim _{x \rightarrow \beta} \frac{e^{2\left(x^2+b x+c\right)}-1-2\left(x^2+b x+c\right)}{(x-\beta)^2}$ is equal...

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Q JEE MAIN 2021

Let $\frac{\sin A}{\sin B}=\frac{\sin (A-C)}{\sin (C-B)}$, where $A, B, C$ are angles of a triangle $A B C$. If the...

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Q JEE MAIN 2021

A tangent and a normal are drawn at the point $\mathrm{P}(2,-4)$ on the parabola $\mathrm{y}^2=8 \mathrm{x}$, which meet the directrix...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

$\sum_{\mathrm{k}=0}^{20}{ }^{20} \mathrm{C}_{\mathrm{k}} \cdot{ }^{20} \mathrm{C}_{20-\mathrm{k}}$ is equal to :

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let us consider a curve, $y=f(x)$ passing through the point $(-2,2)$ and the slope of the tangent to the curve...

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Q JEE MAIN 2021

The statement $(p \wedge(p \rightarrow q) \wedge(q \rightarrow r)) \rightarrow r$ is :

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Q JEE MAIN 2021

If $U_n=\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right)^2 \cdots\left(1+\frac{n^2}{n^2}\right)^n$, then $\lim _{n \rightarrow \infty}\left(U_n\right)^{n^{\frac{-4}{2}}}$ is equal to :

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Q JEE MAIN 2021

Equation of a plane at a distance $\sqrt{\frac{2}{21}}$ from the origin, which contains the line of intersection of the planes...

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Q JEE Main 2021

Let $a, b \in R, b \neq 0$, Define a function $...

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Q JEE Main 2021

The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of...

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Q JEE Main 2021

If $y=y(x)$ is an implicit function of $x$ such that $\log _e(x+y)=4 x y$, then $\frac{d^2 y}{d x^2}$ at $x=0$...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let y=y(x) be the solution of the differential equation $\frac{d y}{d x}=2(y+2 \sin x-5) x-2 \cos x$ such that $y(0)=7$....

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Q JEE Main 2021

The locus of a point, which moves such that the sum of squares of its distances from the points $(0,0),(1,0)$,...

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Q JEE MAIN 2021

If $S=\left\{z \in \mathbb{C}: \frac{z-i}{z+2 i} \in \mathbb{R}\right\}$, then:

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Q JEE Main 2021

The area of the region $S=\left\{(x, y): 3 x^2 \leq 4 y \leq 6 x+24\right\}$ is $\_\_\_\_$ .

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Q JEE MAIN 2021

The distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel...

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Q JEE Main 2021

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

If the matrix $A=\left[\begin{array}{cc}0 & 2 \\ k & -1\end{array}\right]$ satisfies $A\left(A^3+3 I\right)=2 I$, then the value of $K$ is...

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Q JEE Main 2021

If ${ }^1 \mathrm{P}_1+2 \cdot{ }^2 \mathrm{P}_2+3 \cdot{ }^3 \mathrm{P}_3+\ldots+15 \cdot{ }^{15} \mathrm{P}_{15}={ }^9 \mathrm{P}_{\mathrm{r}}-\mathrm{s}, 0 \leq \mathrm{s} \leq 1$,...

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Q JEE MAIN 2021

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Q JEE Main 2021

Let the line $L$ be the projection of the line $\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}$ in the plane $x-2 y-z=3$. If $d$ is the...

JEE Main Mathematics Medium

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Q JEE Main 2021

The sum of all integral values of $k(k \neq 0)$ for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real...

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Q JEE MAIN 2021

Let A be a fixed point (0, 6) and B be a moving point (2t, 0), Let M be the...

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Q JEE MAIN 2021

If for $x, y \in R, x>0$, $y=\log _{10} x+\log _{10} x^{1 / 3}+\log _{10} x^{1 / 9}+\ldots \ldots$ upto...

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Q JEE Main 2021

Let $z=\frac{1-i \sqrt{3}}{2}, i=\sqrt{-1}$. Then the value of $21+\left(z+\frac{1}{z}\right)^3+\left(z^2+\frac{1}{z^2}\right)^3+\left(z^3+\frac{1}{z^3}\right)^3+\ldots+\left(z^{21}+\frac{1}{z^{21}}\right)^3$ is $\_\_\_\_$ .

JEE Main Mathematics Medium

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Q JEE Main 2021

If the truth value of the Boolean expression $((p \vee q) \wedge(q \rightarrow r) \wedge(\sim r)) \rightarrow(p \wedge q)$ is...

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Q JEE MAIN 2021

JEE Main Mathematics Medium

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Q JEE Main 2021

Let $A B C$ be a triangle with $A(-3,1)$ and $\angle A C B=\theta, 0

JEE Main Mathematics Medium

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Q JEE Main 2021

The value of $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{2 n-1} \frac{n^2}{n^2 4 r^2}$ is:

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Q JEE MAIN 2021

If $y^{1 / 4}+y^{-1 / 4}=2 x$, and $\left(x^2-1\right) \frac{d^2 y}{d x^2}+\alpha x \frac{d y}{d t}+\beta y=0$, then $|\alpha-\beta|$ is...

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Q JEE Main 2021

If the sum of an infinite GP a, ar, $a r^2, a r^3, \ldots$ is 15 and the sum of...

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Q JEE MAIN 2021

A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is...

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Q JEE Main 2021

If $A=\left(\begin{array}{cc}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}}\end{array}\right), B=\left(\begin{array}{ll}1 & 0 \\ i & 1\end{array}\right), i=\sqrt{-1}$, and $Q=A^{\top} B A$,...

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Q JEE MAIN 2021

If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x^2}{b^2}+\frac{y^2}{4 a^2}=1$ and the co-ordinate axis...

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Q JEE Main 2021

The value of $\int_{\frac{-1}{\sqrt{2}}}^{\frac{1}{\sqrt{2}}}\left(\left(\frac{x+1}{x-1}\right)^2+\left(\frac{x-1}{x+1}\right)^2-2\right)^{\frac{1}{2}} d x$ is

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Q JEE MAIN 2021

Let n be an odd natural number such that the variance of 1, 2, 3, 4, ...., n is 14....

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Q JEE MAIN 2021

If the system of linear equations $$ \begin{aligned} & 2 x+y-z=3 \\ & x-y-z=\alpha \\ & 3 x+3 y+\beta z=3...

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Q JEE Main 2021

If a line along a chord of the circle $4 x^2+4 y^2+120 x+675=0$, passes through the point $(-30,0)$ and is...

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Q JEE MAIN 2021

If $\int \frac{d x}{\left(x^2+x+1\right)^2}=a \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+b\left(\frac{2 x+1}{x^2+x+1}\right)+C, x>0$ where $C$ is the constant of integration, then the value of...

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Q JEE Main 2021

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{j}-\hat{k}$. If $\vec{c}$ is a vector-such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$, then $...

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Q JEE MAIN_2021_

Let $M$ and $m$ respectively be the maximum and minimum values of the function $f(x)=\tan ^{-1}(\sin x+\cos x)$ in $...

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Q JEE MAIN_2021_

Let $A=\left(\begin{array}{ccc}{[x+1]} & {[x+2]} & {[x+3]} \\ {[x]} & {[x+3]} & {[x+3]} \\ {[x]} & {[x+2]} & {[x+4]}\end{array}\right)$, where $[t]$...

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Q JEE MAIN 2021

If $A=\{x \in \mathbf{R}:|x-2|>1\}, B=\left\{x \in \mathbf{R}: \sqrt{x^2-3}>1\right\}, C=\{x \in \mathbf{R}:|x-4| \geq 2\}$ and $\mathbf{Z}$ is the set of all...

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Q JEE MAIN_2021_

The angle between the straight lines, whose direction cosines are given by the equations $2 \ell+2 \mathrm{~m}-\mathrm{n}=0$ and $...

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Q JEE Main 2021

The equation arg $\left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}$ represents a circle with:

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Q JEE MAIN 2021

Let the equation $x^2+y^2+p x+(1-p) y+5=0$ represent circles of varying radius $r \in(0,5]$. Then the number of elements in the...

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Q JEE MAIN_2021_

Let $S$ be the sum of all solutions (in radians) of the equation $...

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Q JEE MAIN 2021

The number of distinct real roots of the equation $3 x^4+4 x^3-12 x^2+4=0$ is $\_\_\_\_$ .

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Q JEE Main 2021

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are...

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Q JEE MAIN_2021_

Let $S$ be the mirror image of the point $Q(1,3,4)$ with respect to the plane $2 x-y+z+3=0$ and let $...

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Q JEE MAIN 2021

Let $\vec{a}=\hat{i}+5 \hat{j}+\alpha \hat{k}, \vec{b}=\hat{i}+3 \hat{j}+\beta \hat{k}$ and $\vec{c}=-\hat{i}+2 \hat{j}+3 \hat{k}$ be three vectors such that, $|\vec{b} \times \vec{c}|=5 \sqrt{3}$...

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Q JEE Main 2021

If ${ }^{20} \mathrm{C}_{\mathrm{r}}$ is the co-efficient of $\mathrm{x}^{\mathrm{r}}$ in the expansion of $(1+\mathrm{x})^{20}$, then the value of $...

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Q JEE MAIN_2021_

The probability distribution of random variable X is given by: Let $p=P(1

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The sum of the series $\frac{1}{x+1}+\frac{2}{x^2+1}+\frac{2^2}{x^4+1}+\ldots .+\frac{2^{100}}{x^{2^{100}}+1}$ when $x=2$ is

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Q JEE MAIN 2021

The least positive integer n such that $\frac{(2 \mathrm{i})^{\mathrm{n}}}{(1-\mathrm{i})^{\mathrm{n}-2}}, \mathrm{i}=\sqrt{-1}$ is a positive integer, is

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Q JEE MAIN 2021

Let A be a $3 \times 3$ real matrix. If $\operatorname{det}(2 \operatorname{Adj}(2 \operatorname{Adj}(\operatorname{Adj}(2 A))))=2^{41}$, then the value of $\operatorname{det}\left(A^2\right)$ equal...

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Q JEE MAIN_2021_

Let $z_1$ and $z_2$ be two complex numbers such that arg $\left(z_1-z_2\right)=\frac{\pi}{4}$ and $z_1, z_2$ satisfy the equation $|z-3|=$ $\operatorname{Re}(z)$....

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Q JEE Main 2021

Let $f(x)=\cos \left(2 \tan ^{-1} \sin \left(\cos ^{-1} \sqrt{\frac{1-x}{x}}\right)\right) .0

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let the mean and variance of four numbers $3,7, x$ and $y(x>y)$ be 5 and 10 respectively. Then the mean...

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Let $S=\{1,2,3,4,5,6,9\}$. Then the number of elements in the set $T=\{A \subseteq S ; A \neq f$ and the sum...

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Q JEE MAIN 2021

Let $\lambda \neq 0$ be in R . If $\alpha$ and $\beta$ are the roots of the equation $\mathrm{x}^2-\mathrm{x}+2 \lambda=0$,...

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Q JEE Main 2021

Let $\theta \in\left(0, \frac{\pi}{2}\right)$. If the system of linear equations $$ \begin{aligned} & \left(1+\cos ^2 \theta\right) x+\sin ^2 \theta y+4...

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Q JEE MAIN 2021

Let $\binom{n}{k}$ denotes ${ }^n C_k$ and $...

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Q JEE MAIN_2021_

Let $\mathrm{A}(\sec \theta, 2 \tan \theta)$ and $\mathrm{B}(\sec \phi, 2 \tan \phi)$, where $\theta+\phi=\pi / 2$, be two points on...

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Q JEE MAIN_2021_

Two circles each of radius 5 units touch each other at the point $(1,2)$. If the equation of their common...

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Q JEE MAIN_2021_

$3 \times 7^{22}+2 \times 10^{22}-44$ when divided by 18 leaves the remainder $\_\_\_\_$

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Q JEE MAIN 2019

The parallel combination of two air filled parallel plate capacitors of capacitance C and nC is connected to a battery...

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Q JEE MAIN_2021_

An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys...

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Q JEE MAIN_2021_

If $\int \frac{2 e^x+3 e^{-x}}{4 e^x+7 e^{-x}} d x=\frac{1}{14}\left(u x+v \log _e\left(4 e^x+7 e^{-x}\right)\right)+C$, where $C$ is a constant of...

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Q JEE Main 2021

Let $A$ and $B$ be independent events such that $P(A)=p, P(B)=2 p$. The largest value of $p$, for which $P$...

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Q JEE Main 2021

Let $y=y(x)$ be a solution curve of the differential equation $...

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Q JEE Main 2021

On the ellipse $\frac{x^2}{8}+\frac{y^2}{4}=1$ let $P$ be a point in the second quadrant such that the tangent at $P$ to...

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Q JEE Main 2021

The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by...

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Q JEE Main 2021

The sum of solutions of the equation $\frac{\cos x}{1+\sin x}=|\tan 2 x|, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)-\left\{\frac{\pi}{4}-\frac{\pi}{4}\right\}$ is :

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Q JEE Main 2019

The equation of the line passing through $(-4,3,1)$, parallel to the plane $x+2 y-z-5=0$ and intersecting the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}$ is...

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Q JEE MAIN 2019

The length of the perpendicular from the point ( $2,-$ 1,4) on the straight line, $\frac{x+3}{10}=\frac{y-2}{-7}=\frac{z}{1}$ is

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Q JEE MAIN 2021

Let $Q$ be the foot of the perpendicular from the point $P(7,-2,13)$ on the plane containing the lines $\frac{x+1}{6}=\frac{y-1}{7}=\frac{z-3}{8}$ and...

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Q JEE Main 2019

If a, b and c be three distinct real numbers in G.P. and a + b + c = xb,...

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Q JEE MAIN 2019

The sum of all natural numbers ' $n$ ' such that $100< n1$ is :

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Q JEE Main 2019

For $x^2 \neq n \pi+1, n \in N$ (the set of natural numbers), the integral $...

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Q JEE MAIN 2019

$\lim _{x \rightarrow 0} \frac{\sin ^2 x}{\sqrt{2}-\sqrt{1+\cos x}}$ equals :

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Q JEE Main 2019

If $\cos ^{-1}\left(\frac{2}{3 x}\right)+\cos ^{-1}\left(\frac{3}{4 x}\right)=\frac{\pi}{2}\left(x>\frac{3}{4}\right)$, then $x$ is equal to :

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Q JEE MAIN 2019

The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are 2, 4,...

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Q JEE MAIN 2019

The magnitude of the projection of the vector $2 \hat{i}+3 \hat{j}+\hat{k}$ on the vector perpendicular to the plane containing the...

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Q JEE MAIN 2021

Let $a_1, a_2 \ldots \ldots a_{10}$ be an AP with common difference -3 and $b_1, b_2 \ldots \ldots, b_{10}$ be...

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Q JEE Main 2019

If the Boolean expression $(p \oplus q) \wedge(\sim p \odot q)$ is equivalent to $\mathbf{p} \wedge \mathbf{q}$, where $...

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Q JEE MAIN 2021

If the projection of the vector $\hat{i}+2 \hat{j}+\hat{k}$ on the sum of the two vectors $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and...

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Q JEE MAIN 2019

The sum of the solutions of the equation $|\sqrt{x}-2|+\sqrt{x}(\sqrt{x}-4)+2=0,(x>0)$ is equal to:

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Q JEE MAIN 2021

Let $a$ and $b$ respectively be the points of local maximum and local minimum of the function $...

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Q JEE Main 2019

The maximum volume (in cu. m) of the right circular cone having slant height 3 m is :

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Q JEE MAIN 2021

The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit "1"...

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Q JEE MAIN 2019

The contrapositive of the statement “If you are born in India, then you are a citizen of India”, is:

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Q JEE Main 2019

If $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]$, then the matrix $A^{-50}$ when $\theta=\frac{\pi}{12}$, is equal...

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Q JEE MAIN 2019

Let $\quad \mathrm{A}=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right),(\alpha \in \mathrm{R}) \quad$ such that $...

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Q JEE Main 2019

Let $f: R \rightarrow R$ be a function defined as $$ f(x)=\left\{\begin{array}{rrr} 5, & \text { if } & x...

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Q JEE MAIN 2019

If $\cos (\alpha+\beta)=\frac{3}{5}, \sin (\alpha-\beta)=\frac{5}{13}$ and $0

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Q JEE MAIN 2019

The equation of a plane containing the line of intersection of the planes $2 x-y-4=0$ and $y+2 z -4=0$ and...

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Q JEE MAIN 2019

If $f(x)=\frac{2-x \cos x}{2+x \cos x}$ and $g(x)=\log _e x,(x>0)$ then the value of the integral $...

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Q JEE MAIN 2019

The area (in sq. units) of the region $...

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Q JEE Main 2019

$$ \lim _{y \rightarrow 0} \frac{\sqrt{1+\sqrt{1+y^2}}-\sqrt{2}}{y^4} $$

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Q JEE MAIN 2021

$\lim _{x \rightarrow 2}\left(\sum_{n=1}^9 \frac{x}{n(n+1) x^2+2(2 n+1) x+4}\right)$ is equal to :

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Q JEE MAIN 2019

If $\alpha$ and $\beta$ be the roots of the equation $x^2-2 x+2=$ 0 , then the least value of $n$...

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Q JEE MAIN 2021

A circle $C$ touches the line $x=2 y$ at the point $(2,1)$ and intersects the circle $...

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Q JEE MAIN 2019

Let $A$ and $B$ be two non-null events such that $A \subset$ B. Then, which of the following statements is...

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Q JEE Main 2019

If $y=y(x)$ is the solution of the differential equation, $x \frac{d y}{d x}+2 y=x^2$ satisfying $y(1)=1$, then $y\left(\frac{1}{2}\right)$ is equal...

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Q JEE MAIN 2021

The value of $\int_{-\frac{\pi}{2}}\left(\frac{\sin \pi^{\sin x}}{1+\pi^2}\right) \mathrm{dx}$

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Q JEE MAIN 2019

Let $\mathrm{O}(0,0)$ and $\mathrm{A}(0,1)$ be two fixed points. Then the locus of a point P such that the perimeter of...

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Q JEE Main 2019

Equation of a common tangent to the circle, $x^2+y^2-6 x=0$ and the parabola, $y^2=4 x$, is:

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Q JEE MAIN 2021

A 10 inches long pencil $A B$ with mid point $C$ and a small eraser $P$ are placed on the...

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Q JEE MAIN 2019

The greatest value of $c \in R$ for which the system of linear equations $$ \begin{aligned} & x-c y-c z=0...

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Q JEE Main 2019

The plane through the intersection of the planes x + y + z = 1 and 2x + 3y –...

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Q JEE MAIN 2019

All possible numbers are formed using the digits 1, 1, 2, 2, 2, 2, 3, 4, 4 taken all at...

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Q JEE MAIN 2021

Let $P$ be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes $...

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Q JEE Main 2019

The system of linear equations $$ \begin{aligned} & x+y+z=2 \\ & 2 x+3 y+2 z=5 \\ & 2 x+3 y+\left(a^2-1\right)...

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Q JEE MAIN 2021

A hall has a square floor of dimension $10 \mathrm{~m} \times 10 \mathrm{~m}$ (see the figure) and vertical walls. If...

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Q JEE MAIN 2019

The sum of the series $...

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Q JEE Main 2019

Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from...

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Q JEE MAIN 2021

If $(\sqrt{3}+i)^{100}=9^{99}(p+i q)$, then $p$ and $q$ are roots of the equation :

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Q JEE MAIN 2019

Let $f:[0,2] \rightarrow R$ be a twice differentiable function such that $f^{\prime \prime}(x)>0$, for all $x \in(0,2)$. If $\phi(x)=f(x)+ \mathrm{f}(2-x)$,...

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Q JEE MAIN 2019

If $\alpha=\cos ^{-1}\left(\frac{3}{5}\right), \beta=\tan ^{-1}\left(\frac{1}{3}\right)$, where $0

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Q JEE MAIN 2021

The value of $...

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Q JEE MAIN 2019

Let $y=y(x)$ be the solution of the differential equation, $\left(x^2+1\right)^2 \frac{d y}{d x}+2 x\left(x^2+1\right) y=1$ such that $y(0)=0$. If $...

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Q JEE MAIN 2021

The locus of the mid points of the chords of the hyperbola $x^2-y^2=4$, which touch the parabola $y^2=8 x$, is...

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Q JEE MAIN 2021

Two fair dice are thrown. The numbers on them are taken as l and m, and a system of linear...

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Q JEE MAIN 2019

If $f(x)=\log _e\left(\frac{1-x}{1+x}\right),|x|

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Q JEE MAIN 2019

The sum of the squares of the lengths of the chords intercepted on the circle, $x^2+y^2=16$, by the lines, $...

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Q JEE MAIN 2021

If $\sum_{r=1}^{50} \tan ^{-1} \frac{1}{2 r^2}=P$, then the value of $\tan p$ is :

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Q JEE MAIN 2019

If the tangents on the ellipse $4 x^2+y^2=8$ at the points $(1,2)$ and $(\mathrm{a}, \mathrm{b})$ are perpendicular to each other,...

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Q JEE MAIN 2021

A fair die is tossed until six is obtained on it. Let $X$ be the number of required tosses, then...

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Q JEE Main 2019

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable...

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Q JEE Main 2019

For any $\theta \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$, the expression $3(\sin \theta-\cos \theta)^4+6(\sin \theta+\cos \theta)^2+4 \sin ^6 \theta$ equals :

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Q JEE MAIN 2021

The domain of the function $\operatorname{cosec}^{-1}\left(\frac{1+x}{x}\right)$ is :

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Q JEE MAIN 2019

$\int \frac{\sin \frac{5 x}{2}}{\sin \frac{x}{2}} d x$ is equal to: (where c is a constant of integration)

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Q JEE Main 2019

Let $\vec{a}=\vec{i}-\vec{j}, \vec{b}=\vec{i}+\vec{j}+\vec{k}$ and $\vec{c}$ be a vector such that $\vec{a} \times \vec{c}+\vec{b}=\overrightarrow{0}$ and $\vec{a} \cdot \vec{c}=4$, then $|\vec{c}|^2$ is...

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Q JEE MAIN 2019

A point on the straight line, 3x + 5y = 15 which is equidistant from the coordinate axes will lie...

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Q JEE Main 2019

Let $a_1, a_2 \ldots . a_{30}$ be an A.P., $S=\sum_{i=1}^{30}$ ai and $T=\sum_{i=1}^{15} a_{(2 i-1)}$. If $a_5=27$ and $S-2 T=75$,...

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Q JEE MAIN 2021

Consider the two statements : $(\mathrm{S} 1):(\mathrm{p} \rightarrow \mathrm{q}) \vee(\sim \mathrm{q} \rightarrow \mathrm{p})$ is a tautology. $...

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Q JEE MAIN 2019

If $2 y=\left(\cot ^{-1}\left(\frac{\sqrt{3} \cos x+\sin x}{\cos x-\sqrt{3} \sin x}\right)\right)^2, x \in\left(0, \frac{\pi}{2}\right)$ then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to:

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Q JEE Main 2019

The value of $\int_0^\pi|\cos x|^3 d x$ is :

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Q JEE MAIN 2021

Let $y(x)$ be the solution of the differential equation $2 x^2 d y+\left(e^y-2 x\right) d x=0, x>0$. If $y(e)=1$, then...

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Q JEE Main 2019

Let $0

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Q JEE MAIN 2019

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $...

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Q JEE MAIN 2021

The point $\mathrm{P}(-2 \sqrt{6}, \sqrt{3})$ lies on the hyperbola $\frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$ having eccentricity $\frac{\sqrt{5}}{2}$. If the tangent and normal at $P$...

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Q JEE MAIN 2019

The sum of the co-efficients of all even degree terms in x in the expansion of $\left(x+\sqrt{x^3-1}\right)^6+\left(x-\sqrt{x^3-1}\right)^6,(x>1)$ is equal to...

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Q JEE MAIN 2019

The shortest distance between the line $\mathrm{y}=\mathrm{x}$ and the curve $y^2=x-2$ is :

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Q JEE Main 2019

The area (in sq. units) bounded by the parabola $y=x^2-1$, the tangent at the point $(2,3)$ to it and the...

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Q JEE MAIN 2021

If the value of the integral $\int_0^5 \frac{x+[x]}{e^{x-[x]}} d x=\alpha e^{-1}+\beta$, where $a, b \in R, 5 \alpha+6 \beta=0$, and...

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Q JEE Main 2019

Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3...

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Q JEE MAIN 2021

The local maximum value of the function $f(x)=\left(\frac{2}{x}\right)^{x^2}, x>0$, is

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Q JEE Main 2019

Let $\alpha$ and $\beta$ be two roots of the equation $x^2+2 x+2=0$, then $\alpha^{15}+\beta^{15}$ is equal to

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Q JEE Main 2019

Let $A=\left\{\theta \in\left(-\frac{\pi}{2}, \pi\right): \frac{3+2 i \sin \theta}{1-2 i \sin \theta}\right.$ is purely imaginary $\}$ Then the sum of the...

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Q JEE MAIN 2021

Let $\mathrm{A}=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0\end{array}\right)$. Then $\mathrm{A}^{2025}-\mathrm{A}^{2020}$...

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Q JEE Main 2019

Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis...

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Q JEE MAIN 2021

Let $[t]$ denote the greatest integer less than or equal to $t$. Let $f(x)=x-[x], g(x)=1-x+[x]$, and $...

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Q JEE Main 2019

If the fractional part of the number $\frac{2^{403}}{15}$ is $\frac{k}{15}$, then $k$ is equal to

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Q JEE MAIN 2021

A man starts walking from the point $P(-3,4)$, touches the $x$-axis at $R$, and then turns to reach at the...

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Q JEE Main 2019

5 students of a class have an average height 150 cm and variance $18 \mathrm{~cm}^2$. A new student, whose height...

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Q JEE MAIN 2021

Let $[t]$ denote the greatest integer $\leq t$. The number of points where the function $f(x)=[x]\left|x^2-1\right|+\sin \left(\frac{\pi}{[x]+3}\right)-[x+1], x \in(-2,2)$ is...

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Q JEE MAIN 2021

Let $\vec{a}-=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}--2 \hat{j}-\hat{k}$. Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}$. If...

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Q JEE MAIN 2021

If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096 , then the greatest coefficient in the...

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Q JEE MAIN 2021

All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R...

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Q JEE MAIN 2021

Let $f(x)$ be a polynomial of degree 3 such that $f(k)=-\frac{2}{k}$ for $k=2,3,4,5$. Then the value of $52-10 f(10)$ is...

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Q JEE MAIN 2021

Let the points of intersections of the lines x – y + 1 = 0, x – 2y + 3...

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Q JEE MAIN 2021

If for the complex numbers $z$ satisfying $|z-2-2 i| \leq 1$, the maximum value of $|3 i z+6|$ is attained...

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Q JEE MAIN 2021

Let $f(x)=x^6+2 x^4+x^3+2 x+3, x \in R$. Then the natural number $n$ for which $\lim _{x \rightarrow 1} \frac{x^n f(1)-f(x)}{x-1}=44$...

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Q JEE MAIN 2021

Let X be a random variable with distribution. If the mean of $X$ is 2.3 and variance of $X$ is...

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Q JEE MAIN 2021

Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{1}=1$ and the circle $x^2+y^2=3$ at their point...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

The function $f(x)$, that satisfies the condition $f(x)=x+\int_0^{\pi / 2} \sin x \cdot \cos y f(y) d y$, is :

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Q JEE MAIN 2021

Let $\mathrm{a}_1, \mathrm{a}_2 \ldots \ldots, \mathrm{a}_{21}$ be an AP such that $\sum_{\mathrm{n}=1}^{20} \frac{1}{\mathrm{a}_{\mathrm{n}} \mathrm{a}_{\mathrm{n}+1}}=\frac{4}{9}$. If the sum of this AP...

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Q JEE MAIN 2021

The range of the function, $f(x)=\log _{\sqrt{5}}\left(3+\cos \left(\frac{3 \pi}{4}+x\right)+\cos \left(\frac{\pi}{4}+x\right)+\cos \left(\frac{\pi}{4}-x\right)-\cos \left(\frac{3 \pi}{4}-x\right)\right)$ is :

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Q JEE MAIN 2021

Let $\mathrm{P}_1, \mathrm{P}_2 \ldots \ldots, \mathrm{P}_{15}$ be 15 points on a circle. The number of distinct triangles formed by points...

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Q JEE MAIN 2021

Let $S_n=1 \cdot(n-1)+2 \cdot(n-2)+3 \cdot(n-3)+\ldots \cdot+(n-1) \cdot n \geq 4$. The sum $\sum_{n=4}^{\infty}\left(\frac{2 S_n}{n!}-\frac{1}{(n-2)!}\right)$ is equal to :

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Q JEE MAIN 2021

The numbers of pairs $(a, b)$ of real numbers, such that whenever $a$ is a root of the equation $...

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Q JEE MAIN 2021

Consider the parabola with vertex $\left(\frac{1}{2}, \frac{3}{4}\right)$ and the directrix $y=\frac{1}{2}$. Let $P$ be the point where the parabola meets...

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Q JEE MAIN 2021

The distance of line 3y – 2z – 1 = 0 = 3x – z + 4 from the point...

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Q JEE MAIN 2021

The area, enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ and the lines $x=0, x=\frac{\pi}{2}$, is :

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Q JEE MAIN 2021

Let $J_{n, m}=\int_0^{\frac{1}{2}} \frac{x^n}{x^m-1} d x, \forall n>m$ and $m \in N$. Consider a matrix $A=\left[a_{i j}\right]_{3 \times 3}$ where...

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Q JEE MAIN_2021

Let $...

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Q JEE MAIN 2021

If $n$ is the number of solutions of the equation $2 \cos x\left(4 \sin \left(\frac{\pi}{4}+x\right) \sin \left(\frac{\pi}{4}-x\right)-1\right)=1, x \in[0, \pi]$...

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Q JEE MAIN_2021

The term independent of ' $x$ ' in the expansion of $\left(\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right)^{10}$, where $...

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Q _JEE MAIN_202

If $\alpha, \beta$ are roots of the equation $x^2+5(\sqrt{2}) x+10=0, \alpha>\beta$ and $P_n=\alpha^n-\beta^n$ for each positive integer $n$, then the...

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Q _JEE MAIN_2021_

There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the...

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Q JEE MAIN 2021

If $y=y(x)$ is the solution curve of the differential equation $x^2 d y+\left(y-\frac{1}{x}\right) d x=0 ; x>0$ and $y(1)=1$, then...

JEE Main Mathematics Hard

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Q JEE MAIN_2021

Let $M=\left\{A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right): a, b, c, d \in\{ \pm 3, \pm 2, \pm 1,0\}\right\}$ Difine...

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Q JEE MAIN 2021

Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common...

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Q JEE MAIN_2021

The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients...

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Q _JEE MAIN_2021

Let $\vec{p}=2 \hat{i}+3 \hat{j}+\hat{k}$ and $\vec{q}=\hat{i}+2 \hat{j}+\hat{k}$ be two vectors. If a vector $\vec{r}=(\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k})$ is perpendicular to...

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Q JEE MAIN 2021

Which of the following is equivalent to the Boolean expression $\mathrm{p} \wedge \sim \mathrm{q}$ ?

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Q JEE MAIN_2021

Consider the following frequency distribution

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Q JEE MAIN_2021

If the value of $\left(1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\ldots . \text { upto } \infty\right)^{\log _{1008}\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^2}+\ldots \text { uptoss }\right)}$ is $l$, then $l^2$...

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Q _JEE MAIN_2021

Let $y=y(x)$ be solution of the following differential equation $...

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Q _JEE MAIN_2021_

Let the foot of perpendicular from a point $P(1,2,-1)$ to the straight line $L: \frac{x}{1}=\frac{y}{0}=\frac{z}{-1}$ be $N$. Let a line...

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Q JEE MAIN 2021

Let the acute angle bisector of the two planes x – 2y – 2z + 1 = 0 and 2x...

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Q JEE MAIN 2021 S2

Consider the system of linear equations $$ \begin{aligned} & -x+y+2 z=0 \\ & 3 x-a y+5 z=1 \\ & 2...

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Q JEE MAIN 2021 S2

$\cos ^{-1}(\cos (-5))+\sin ^{-1}(\sin (6))-\tan ^{-1}(\tan (12))$ is equal to :

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Q JEE MAIN 2021 S2

Let $f: R \rightarrow R$ be a continuous function. Then $...

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Q JEE MAIN 2021 S2

If $A=\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right]$ and $...

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Q JEE MAIN 2021 S2

Let $A=\left\{n \in N \mid n^2 \leq n+10,000\right\}, B=\{3 k+1 \mid k \in N\}$ and $...

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Q JEE MAIN_2021_

Let an ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a^2>b^2$, passes through $\left(\sqrt{\frac{3}{2}}, 1\right)$ and has eccentricity $\frac{1}{\sqrt{3}}$. If a circle, centered at focus...

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Q JEE MAIN_2021_

The number of real roots of the equation $\mathrm{e}^{6 \mathrm{x}}-\mathrm{e}^{4 \mathrm{x}}-2 \mathrm{e}^{3 \mathrm{x}}-12 \mathrm{e}^{2 \mathrm{x}}+\mathrm{e}^{\mathrm{x}}+1=0$ is $i$

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Q _JEE MAIN_2021_

Let a parabola P be such that its vertex and focus lie on the positive x -axis at a distance...

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Q JEE MAIN_2021_

Let 9 distinct balls be distributed among 4 boxes, $B_1, B_2, B_3$ and $B_4$. If the probability than $B_3$ contains...

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Q JEE MAIN_2021_

The values of $a$ and $b$, for which the system of equations $$ \begin{aligned} & 2 x+3 y+6 z=8 \\...

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Q JEE MAIN 2021 S2

Let $n$ be a non-negative integer. Then the number of divisors of the form " $4 n+1$ " of the...

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Q JEE MAIN 2021 S2

Let $y=y(x)$ be the solution of the differential equation $d y=e^{\alpha x+y} d x ; \alpha \in N$. If $...

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Q JEE MAIN 2021 S2

The number of real roots of the equation $e^{4 x}-e^{3 x}-4 e^{2 x}-e^x+1=0$ is equal to $\_\_\_\_$

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Q JEE MAIN 2021 S2

If $\int_0^\pi\left(\sin ^3 x\right) e^{-\sin ^2 x} d x=\alpha-\frac{\beta}{e} \int_0^1 \sqrt{t} e^t d t$, then $\alpha+\beta$ is equal to $\_\_\_\_$...

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Q JEE MAIN 2021 S2

Let $E$ be an ellipse whose axes are parallel to the co-ordinates axes, having its center at $(3,-4)$, one focus...

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Q JEE MAIN 2021 S2

If the real part of the complex number $z=\frac{3+2 i \cos \theta}{1-3 i \cos \theta}, \theta \in\left(0, \frac{\pi}{2}\right)$ is zero,...

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Q JEE MAIN 2021 S2

The distance of the point $\mathrm{P}(3,4,4)$ from the point of intersection of the line joining the points. $\mathrm{Q}(3,-4,-5)$ and $R(2,-3,1)$...

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Q JEE MAIN 2021 S2

Let $\vec{a}=\hat{i}-\alpha \hat{j}+\beta \hat{k}, \vec{b}=3 \hat{i}+\beta \hat{j}+\alpha \hat{k}$ and $\vec{c}=-\alpha \hat{i}+2 \hat{j}+\hat{k}$, where $\alpha$ and $\beta$ are integers. If $...

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Q JEE MAIN 2021 S2

Let $f:(\mathrm{a}, \mathrm{b}) \rightarrow \mathrm{R}$ be twice differentiable function such that $\mathrm{f}(\mathrm{x})=\int_{\mathrm{a}}^{\mathrm{x}} \mathrm{g}(\mathrm{t}) \mathrm{dt}$ for a differentiable function $\mathrm{g}(\mathrm{x})$. If...

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Q JEE MAIN_2021

Let $f:[0, \infty) \rightarrow[0, \infty)$ be defined as $f:(x)=\int_0^x[y] d y$ where $[x]$ is the greatest integer less than or...

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Q JEE MAIN 2021 S2

Let $A$ and $B$ be two $3 \times 3$ real matrices such that $\left(A^2-B^2\right)$ is invertible matrix. If $A^5=B^5$ and...

JEE Main Mathematics Medium

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Q JEE MAIN 2021 S2

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}=\vec{b} \times(\vec{b} \times \vec{c})$. If magnitudes of the vectors $...

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Q _JEE MAIN_2021

Let $\mathrm{g}: \mathrm{N} \rightarrow \mathrm{N}$ be defined as $$ \begin{aligned} & g(3 n+1)=3 n+2 \\ & g(3 n+2)=3 n+3 \\...

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Q JEE MAIN 2021 S2

Consider a circle $C$ which touches the $y$-axis at $(0,6)$ and cuts off an intercept $6 \sqrt{5}$ on the $x$-axis....

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Q JEE MAIN_2021_

The sum of all values of $x$ in $[0,2 \pi]$, for which $\sin x+\sin 2 x+\sin 3 x+\sin 4 x=0$,...

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Q JEE MAIN 2019

A plane electromagnetic wave travels in free space along the x-direction. The electric field component of the wave at a...

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Q JEE MAIN_2021_

The area (in sq. units) of the region, given by the set $...

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Q JEE MAIN 2021 S2

Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $...

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Q JEE MAIN_2021_

The Boolean expression $(P \Rightarrow q) \wedge(q \Rightarrow \sim p)$ is equivalent to:

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Q JEE MAIN 2021 S2

Let $f:[0, \infty) \rightarrow[0,3]$ be a function defined by $...

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Q JEE MAIN_2021

Let $y=y(x)$ be the solution of the differential equation $\frac{d}{d x}=1+x e^{y-x},-\sqrt{2}

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Q JEE MAIN 2021 S2

Let $\alpha=\max _{x \in R}\left\{8^{2 \sin 3 x} \cdot 4^{4 \cos 3 x}\right\}$ and $...

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Q JEE MAIN 2021 S2

Two sides of a parallelogram are along the lines $4 x+5 y=0$ and $7 x+2 y=0$. If the equation of...

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Q JEE MAIN_2021

If $b$ is very small as compared to the value of $a$, so that the cube and other higher powers...

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Q JEE MAIN 2021 S2

The value of $\lim _{x \rightarrow 0}\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)$ is equal to:

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Q JEE MAIN_2021_

The value of the definite integral $\int_{\pi / 24}^{5 \pi / 24} \frac{d x}{1+\sqrt[3]{\tan 2 x}}$ is :

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Q JEE MAIN 2021 S2

Let $y=y(x)$ be the solution of the differential equation $\left(x-x^3\right) d y=\left(y+y x^2-3 x^4\right) d x, x>2$. If $y(3)=3$, then...

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Q JEE MAIN 2021 S2

The area of the region bounded by $y-x=2$ and $x^2=y$ is equal to :-

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Q JEE MAIN 2021 S2

If $\tan \left(\frac{\pi}{9}\right), x, \tan \left(\frac{7 \pi}{18}\right)$ are in arithmetic progression and $\tan \left(\frac{\pi}{9}\right), y, \tan \left(\frac{5 \pi}{18}\right)$ are also...

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Q JEE MAIN 2021 S2

A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability....

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Q JEE MAIN 2021 S2

Let be the set of all complex numbers. Let $$ \begin{aligned} & S_1=\{z \in \mathbb{C}:|z-2| \leq 1\} \text { and...

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Q JEE MAIN 2021 S2

Let $f: R-R$ be defined as $f(x+y)+f(x-y)=2 f(x) f(y), f\left(\frac{1}{2}\right)=-1$. Then, the value of $\sum_{k=1}^{20} \frac{1}{\sin (k) \sin (k+f(k))}$ is...

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Q JEE MAIN 2021 S2

For real numbers $\alpha$ and $\beta \neq 0$, if the point of intersection of the straight lines $\frac{x-\alpha}{1}=\frac{y-1}{2}=\frac{z-1}{3}$ and $\frac{x-4}{\beta}=\frac{y-6}{3}=\frac{z-7}{3}$,...

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Q JEE MAIN 2019

Let $f: R \rightarrow R$ be a differentiable function satifying $f^{\prime}(3)+f^{\prime}(2)=0$. Then $\lim _{x \rightarrow 0}\left(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)}\right)^{\frac{1}{x}}$ is equal to :

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Q JEE MAIN 2019

If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the...

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Q JEE MAIN 2019

If $\int \frac{d x}{x^3\left(1+x^6\right)^{2 / 3}}=x f(x)\left(1+x^6\right)^{\frac{1}{3}}+C$ where $C$ is a constant of integration, then the function $f(x)$ is equal...

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Q JEE MAIN 2019

Let $f(x)=a^x(a>0)$ be written as $f(x)=f_1(x)+f_2(x)$, where $f_1(x)$ is an even function and $f_2(x)$ is an odd function. Then $f_1(x+y)+f_1(x-y)$...

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Q JEE MAIN 2019

Let $f(x)=\int_0^x g(t) d t$, where $g$ is a non-zero even function. If $f(x+5)=g(x)$, then $\int_0^x f(t) d t$, equals:

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Q JEE MAIN 2019

If $f(1)=1, f^{\prime}(1)=3$, then the derivative of $f(f(f(x)))+(f(x))^2$ at $x=1$ is :

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Q JEE MAIN 2019

If the system of linear equations $$ \begin{aligned} & x-2 y+k z=1 \\ & 2 x+y+z=2 \\ & 3 x-y-k...

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Q JEE MAIN 2019

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is

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Q JEE MAIN 2019

If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest,...

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Q JEE MAIN 2019

If a point R(4, y, z) lies on the line segment joining the points P(2, –3, 4) and Q(8, 0,...

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Q JEE MAIN 2019

Let $\mathrm{S}(\alpha)=\left\{(\mathrm{x}, \mathrm{y}): \mathrm{y}^2 \leq \mathrm{x}, 0 \leq \mathrm{x} \leq \alpha\right\}$ and $\mathrm{A}(\alpha)$ is area of the region $\mathrm{S}(\alpha)$. If...

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Q JEE MAIN_2021_

Let $f: R \rightarrow R$ be defined as $...

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Q JEE MAIN 2019

Suppose that the points $(h, k),(1,2)$ and $(-3,4)$ lie on the line $L_1$. If a line $L_2$ passing through the...

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Q JEE MAIN 2021 S2

A possible value of ' $x$ ', for which the ninth term in the expansion of $...

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Q _JEE MAIN_2021

Let the vectors $(2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k},(1+b) \hat{i}+2 b \hat{j}-b \hat{k}$ and $...

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Q JEE MAIN 2021 S2

The point P (a,b) undergoes the following three transformations successively : (a) reflection about the line $\mathrm{y}=\mathrm{x}$.: (b) translation through...

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Q JEE MAIN 2019

Given that the slope of the tangent to a curve $y=y(x)$ at any point ( $x, y$ ) is $...

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Q JEE MAIN 2019

Let $f:[-1,3] \rightarrow R$ be defined as $$ f(x)=\left\{\begin{array}{cc} |x|+[x], & -1 \leq x

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Q _JEE MAIN_2021_

The locus of the centroid of the triangle formed by any point $P$ on the hyperbola $...

JEE Main Mathematics Medium

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Q JEE MAIN 2021 S2

If the lines $\frac{x-k}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x+1}{3}=\frac{y+2}{2}=\frac{z+3}{1}$ are co-planar, then the value of $k$ is

JEE Main Mathematics Easy

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Q JEE MAIN 2019

Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in meters)...

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Q JEE MAIN 2019

The sum $\sum_{k=1}^{20} k \frac{1}{2^k}$ is equal to :

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Q JEE MAIN_2021_

Let $S_n$ be the sum of the first $n$ terms of ah arithmetic progression. If $S_{3 n}=3 S_{2 n}$, then...

JEE Main Mathematics Medium

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Q JEE MAIN 2021 S2

If the co-efficient of $x^7$ and $x^8$ in the expansion of $\left(2+\frac{x}{3}\right)^n$ are equal, then the value of $n$ is...

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Q JEE MAIN 2021 S2

A fair coin is tossed $n$-times such that the probability of getting at least one head is at least 0.9...

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Q JEE MAIN_2021_

Let $f(x)=3 \sin ^4 x+10 \sin ^3 x+6 \sin ^2 x-3, x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right]$. Then, $f$ is -

JEE Main Mathematics Medium

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Q JEE MAIN 2021 S2

If $a+b+c=1, a b+b c+c a=2$ and $a b c=3$, then the value of $a 4+b 4+c 4$ is equal...

JEE Main Mathematics Medium

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Q JEE MAIN 2021 S2

Let a curve $y=f(x)$ pass through the point $\left(2,\left(\log _e 2\right)^2\right)$ and have slope $\frac{2 y}{x \log _e x}$ for...

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Q JEE MAIN 2019

The minimum number of times one has to toss a fair coin so that the probability of observing at least...

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Q JEE MAIN_2021_

A spherical gas balloon of radius 16 meter subtends an angle $60^{\circ}$ at the eye of the observer A while...

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Q JEE MAIN 2021 S2

If $(\vec{a}+3 \vec{b})$ is perpendicular to $(7 \vec{a}-5 \vec{b})$ and $(\vec{a}-4 \vec{b})$ is perpendicular to $(7 \vec{a}-2 \vec{b})$, then the...

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Q JEE MAIN 2019

Let the numbers 2, $\mathrm{b}, \mathrm{c}$ be in an A.P. and $...

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Q JEE MAIN 2021 S2

If a rectangle is inscribed in an equilateral triangle of side length $2 \sqrt{2}$ as shown in the figure, then...

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Q JEE MAIN 2021

The equation of a circle is $\operatorname{Re}\left(z^2\right)+2(\operatorname{lm}(z))^2+2 \operatorname{Re}(z)=0$, where $z=x+i y$. A line which passes through the center of the...

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Q JEE MAIN 2019

The number of integral values of $m$ for which the equation $\left(1+m^2\right) x^2-2(1+3 m) x+(1+8 m)=0$ has no real root...

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Q JEE MAIN 2021

If the matrix $\mathrm{A}=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1\end{array}\right]$...

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Q JEE MAIN 2019

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}(i=\sqrt{-1})$, then $\left(1+i z+z^5+i z^8\right)^9$ is equal to :

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Q JEE MAIN 2021

Let $X_1, X_2, \ldots \ldots X_{18}$ be eighteen observations such that $\sum_{i=1}^{18}\left(x_i-\alpha\right)=36$ and $\sum_{i=1}^{18}\left(x_i-\beta\right)^2=90$, where $\alpha$ and $\beta$ are distinct...

JEE Main Mathematics Easy

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Q JEE MAIN 2019

If three distinct numbers $a, b, c$ are in G.P. and the equations $a x^2+2 b x+c=0$ and $...

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Q JEE MAIN 2021

Consider the function $$ \begin{array}{ll} f(x)=\frac{P(x)}{\sin (x-2)}, & x \neq 2 \\ =7 & x=2 \end{array} $$ Where $\mathrm{P}(\mathrm{x})$ is...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let a be an integer such that all the real roots of the polynomial $2 x^5+5 x^4+10 x^3+10 x^2+10 x+10$...

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Q JEE MAIN 2019

The tangent to the parabola $\mathrm{y}^2=4 \mathrm{x}$ at the point where it intersects the circle $\mathrm{x}^2+\mathrm{y}^2=5$ in the first quadrant,...

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Q JEE MAIN 2021

Let $n \in N$ and $[x]$ denote the greatest integer less than or equal to $x$. If the sum of...

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Q JEE MAIN 2019

If the fourth term in the binomial expansion of $\left(\sqrt{\frac{1}{x^{1+\log _{10} x}}}+x^{\frac{1}{12}}\right)^6$ is equal to 200 , and $x>1$, then...

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Q JEE MAIN 2021

Let $L$ be a common tangent line to the curves $4 x^2+9 y^2=36$ and $(2 x)^2+(2 y)^2=31$. Then the square...

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Q JEE MAIN 2019

The tangent and the normal lines at the point $(\sqrt{3}, 1)$ to the circle $x^2+y^2=4$ and the $x$-axis form a...

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Q JEE MAIN 2021

Let $y=y(x)$ be the solution of the differential equation $x d y=\left(y+x^3 \cos x\right) d x$ with $y(\pi)=0$, then $y\left(\frac{\pi}{2}\right)$...

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Q JEE MAIN 2021

The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is

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Q JEE MAIN 2021

If ${ }^n P_r={ }^n P_{r+1}$ and ${ }^n C_r={ }^n C_{r+1}$, then the value of $r$ is equal to:

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let the equation of the pair of lines, $y=p x$ and $y=q x$, can be written as $(y-p x)(y-q x)=0$....

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Q JEE MAIN 2021

If the arithmetic mean and geometric mean of the $p^{\text {th }}$ and $q^{\text {th }}$ terms of the sequence...

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Q JEE MAIN 2019

A student scores the following marks in five tests : 45, 54, 41, 57, 43. His score is not known...

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Q JEE MAIN 2019

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis...

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Q JEE MAIN 2021

If $I_{m, n}=\int_0^1 x^{m-1}(1-x)^{n-1} d x$, for $m, n \geq 1$ and $...

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Q JEE MAIN 2019

The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0, 1, 2, 3,...

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Q JEE MAIN 2021

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha+\beta=1$ and $\alpha \beta=-1$. Let $p_n=(\alpha)^n+(\beta)^n, P_{n-1}=11$ and $p_{n+1}=29$ for...

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Q JEE MAIN 2019

Which one of the following statements is not a tautology?

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Q JEE MAIN 2021

Let the normals at all the points on a given curve pass through a fixed point $(a, b)$. If the...

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Q JEE MAIN 2021

Let zbe those complex numbers which satisfy $|z+5| \leq 4$ and $z(1+i)+\bar{z}(1-i) \geq-10, i-\sqrt{-1}$. If the maximum value of $|z+1|^2$...

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Q JEE MAIN 2019

The vector equation of plane through the line of intersection of the planes $x+y+z=1$ and $2 x+3 y+4 z=5$ which...

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Q JEE MAIN 2021

Let $f:[0,3] \rightarrow R$ be defined by $f(x)=\min \{x-[x], 1+[x]-x\}$ where $[x]$ is the greatest integer less than or equal...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If $y=y(x), y \in\left[0, \frac{\pi}{2}\right)$ is the solution of the dirrential equation $\sec y \frac{d y}{d x}-\sin x(x+y)-\sin (x-y)=0$, with...

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Q JEE MAIN 2021

A seven-digit number is formed using digits 3, 3, 4, 4, 4, 5, 5. The probability, that number so formed...

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Q JEE MAIN 2021

Let S={1,2,3,4,5,6,7}. Then the number of possible functions $f: S \rightarrow S$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for...

JEE Main Mathematics Hard

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Q JEE MAIN 2021

Let a plane $P$ pass through the point $(3,7,-7)$ and contain the line, $\frac{x-2}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$. If distance of the plane P...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let A_1 be the area of the region bounded by the curves $y=\sin x, y=\cos x$ and $y$-axis in the...

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Q JEE MAIN 2021

Let $F:[3,5] \rightarrow R$ be a twice differentiable function on $(3,5)$ such that $F(x)=e^{-x} \int_3^x\left(3 t^2+2 t+4 F^{\prime}(t)\right) d t$...

JEE Main Mathematics Hard

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Q JEE MAIN 2021

Let $f(x)=\int_0^x e^t f(t) d t+e^x$ be a differentiable function for all $x \in R$. Then $f(x)$ equals :

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $...

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Q JEE MAIN 2021

Let the domain of the function $f(x)=\log _4\left(\log _5\left(\log _3\left(18 x-x^2-77\right)\right)\right)$ be $(a, b)$. Then the value of the integral...

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Q JEE MAIN 2021

If $\log _3 2, \log _3\left(2^x-\frac{7}{2}\right)$ are in an arithmetic progression, then the value of $x$ is equal to $\_\_\_\_$...

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Q JEE MAIN 2021

If the mirror image of the point $(1,3,5)$ with respect to the plane $4 x-5 y+2 z=8$ is $...

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Q JEE MAIN 2021

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}$ and $\vec{c}=\hat{j}-\hat{k}$ be three vectors such that $\vec{a} \times \vec{b}=\vec{c}$ and $\vec{a} \cdot \vec{b}=1$. If the length...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

For real numbers $\alpha$ and $\beta$, consider the following system of linear equations: $x+y-z=2, x+2 y+\alpha z=1,2 x-y+z=\beta$. If the...

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Q JEE MAIN 2021

Let $...

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Q JEE MAIN 2021

Let $A(1,4)$ and $B(1,-5)$ be two points. Let $P$ be a point on the circle $(x-1)^2+(y-1)^2=1$ such that $...

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Q JEE MAIN 2021

Let $f(x)$ be a differentiable function at $x=a$ with $f^{\prime}(a)=2$ and $f(a)=4$. Then $\lim _{x \rightarrow 0} \frac{x f(a)-a f(x)}{x-a}$...

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Q JEE MAIN 2021

The probability that a randomly selected 2 -digit number belongs to the set $\left\{n \in N:\left(2^n-2\right)\right.$ is a multiple of...

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Q JEE MAIN 2021

The sum of the series $\sum_{n=1}^{\infty} \frac{n^2+6 n+10}{(2 n+1)!}$ is equal to:

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Q JEE MAIN 2021

Let $\alpha, \beta$ be two roots of the equation $x^2+(20)^{1 / 4} x+(5)^{\frac{1}{2}}=0$. Then $\alpha^8+\beta^8$ is equal to

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Q JEE MAIN 2021

Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes...

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Q JEE MAIN 2021

If $0

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Q JEE MAIN 2021

Let $f: R \rightarrow R$ be a function such that $f(2)=4$ and $f(2)=1$. Then, the value of $...

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Q JEE MAIN 2021

Two tangents are drawn from the point $P(-1,1)$ to the circle $x^2+y^2-2 x-6 y+6=0$. If these tangents touch the circle...

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Q JEE MAIN 2021

Consider the following system of equations : $$ \begin{aligned} & x+2 y-3 z=a \\ & 2 x+6 y-11 z=b \\...

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Q JEE MAIN 2021

Let the plane passing through the point (–1, 0, –2) and perpendicular to each of the planes 2x + y...

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Q JEE MAIN 2021

Let $y=y(x)$ be solution of the differential equation $\log _e\left(\frac{d y}{d x}\right)=3 x+4 y$, with $y(0)=0$. If $...

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Q JEE MAIN 2021

Let $f:\left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \rightarrow R$ be defined as $f(x)=\left\{\begin{array}{cc}(1+|\sin x|)^{\frac{3 a}{|\sin x|}}, & -\frac{\pi}{4}

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Q JEE MAIN 2021

If the locus of the mid-point of the line segment from the point $(3,2)$ to a point on the circle,...

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Q JEE MAIN 2021

Let $A=\left[\begin{array}{cc}1 & 2 \\ -1 & 4\end{array}\right]$. If $A^{-1}=\alpha I+\beta A, \alpha, \beta \in R, I$ is a $...

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Q JEE MAIN 2021

If $\sin \theta+\cos \theta=\frac{1}{2}$, then $16(\sin (2 \theta)+\cos (4 \theta)+\sin (6 \theta))$ is equal to :

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Q JEE MAIN 2021

Let slope of the tangent line to a curve at any point $P(x, y)$ be given by $\frac{x y^2+y}{x}$. If...

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Q JEE MAIN 2021

The compound statement $(P \vee Q) \wedge(\sim P) \Rightarrow Q$ is equivalent of:

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Q JEE MAIN 2021

If the coefficients of $x^7$ in $\left(x^2+\frac{1}{b x}\right)^{11}$ and $x^{-7}$ in $\left(x-\frac{1}{b x^2}\right)^{11}, b \neq 0$, are equal, then the...

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Q JEE MAIN 2021

Let $F_1(A, B, C)=(A \wedge \sim B) \vee[\sim C \wedge(A \vee B)] \vee \sim A$ and $...

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Q JEE MAIN 2021

A ray of light through $(2,1)$ is reflected at a point $P$ on the $y$-axis and then passes through the...

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Q JEE MAIN 2021

Let $L$ be a line obtained from the intersection of two planes $x+2 y+z=6$ and $y+2 z=4$. If point $...

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Q JEE MAIN_2021_

Let $E_1: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$. Let $E_2$ be ahother ellipse such that it touches the end points of major axis of...

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Q JEE MAIN 2021

If a tangent to the ellipse $x^2+4 y^2=4$ meets the tangents at the extremities of its major axis at $B$...

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Q JEE MAIN_2021_

Let a line $L$ : $2 x+y=k, k>0$ be a tangent to the hyperbola $x^2-y^2=3$. If $L$ is also a...

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Q JEE MAIN 2021

Let $x$ be a random variable such that the probability function of a distribution is given by $...

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Q JEE MAIN_2021_

Let $f: R \rightarrow R$ be defined as $...

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Q JEE MAIN 2021

If $P=\left[\begin{array}{cc}1 & 0 \\ 1 / 2 & 1\end{array}\right]$, then $P^{50}$ is :

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Q JEE MAIN_2021_

If the domain of the function $f(x)=\frac{\cos ^{-1} \sqrt{x^2-x+1}}{\sqrt{\sin ^{-1}\left(\frac{2 x-1}{2}\right)}}$ is the interval $(\alpha, \beta]$, then $\alpha+\beta$ is equal...

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Q JEE MAIN 2021

Consider function $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ and $\mathrm{g}: \mathrm{B} \rightarrow \mathrm{C}(\mathrm{A}, \mathrm{B}, \mathrm{C} \subseteq \mathrm{R})$ such that (gof) ${ }^{-1}$...

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Q JEE MAIN_2021

The number of solutions of $\sin ^7 x+\cos ^7 x=1, x \in[0,4 \pi]$ is equal to

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Q JEE MAIN 2021

The number of real solutions of the equation, $\mathrm{x}^2-|\mathrm{x}|-12=0$ is:

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Q JEE MAIN_2021_

Let $n$ denote the number of solutions of the equation $z^2+3 \bar{z}=0$, where $z$ is a complex number. Then the...

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Q JEE MAIN 2021

If $|\vec{a}|=2, \mid \vec{b}=5$ and $|\vec{a} \times \vec{b}|=8$, then $|\vec{a} \cdot \vec{b}|$ is equal to:

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Q JEE MAIN 2021

The number of distinct real roots of $...

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Q JEE MAIN_2021

Let the circle $\mathrm{S} ; 36 x^2+36 y^2-108 x+120 y+C=0$ be such that it neither intersects nor touches the coordinate...

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Q JEE MAIN 2021

If $[x]$ be the greatest integer less than or equal to $x$, then $\sum_{n=8}^{100}\left[\frac{(-1)^n n}{2}\right]$ is equal to:

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Q _JEE MAIN_2021_

Let $[x]$ denote the greatest integer less than or equal to $x$. Then, the values of $x \in R$ satisfying...

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Q JEE MAIN 2021

Let $a, b$ and $c$ be distinct positive numbers. If the vectors $a-a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}$ and $...

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Q JEE MAIN_22-

Let $A=\left[a_i\right]$ be a real matrix of order $3 \times 3$, such that $a_{i 1}+a_{i 2}+a_{i 3}=1$, for $i=1,2,3$. Then,...

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Q JEE MAIN _2021_

Which of the following Boolean expressions is not a tautology?

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Q JEE MAIN 2021

The value of the integral $\int_{-1}^1 \log \left(x+\sqrt{x^2+1}\right) d x$ is :

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Q JEE MAIN 2021

The lowest integer which is greater than $\left(1+\frac{1}{10^{100}}\right)^{10^{100}}$ is

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Q JEE MAIN_2021_

If the shortest distance between the straight lines $3(x-1)=6(y-2)=2(z-1)$ and $4(x-2)=2(y-\lambda)=(z-3)$, $\lambda \in \mathrm{R}$ is $\frac{1}{\sqrt{38}}$, then the integral value...

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Q JEE MAIN 2021

The value of $\cot \frac{\pi}{24}$ is :

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Q JEE MAIN_2021

The values of $\lambda$ and $\mu$ such that the system of equations $x+y+\mid z=6,3 x+5 y+5 z=26, x+2 y+\lambda z=\mu$...

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Q JEE MAIN 2021

Consider the statement "The match will be played only if the weather is good and ground is not wet". Select...

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Q JEE MAIN 2021

If the greatest value of the term independent of ' $x$ ' in the expansion of $...

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Q JEE MAIN 2021

If $f(x)=\left\{\begin{array}{cc}\int_0^x(5+|1-t|) d t, & x>2 \\ 5 x+1, & x \leq 2\end{array}\right.$, then

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Q JEE MAIN 2021

the first of the two samples in a group has 100 items with mean 15 and standard deviation 3 ....

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Q JEE MAIN 2021

The sum of all those terms which are rational numbers in the expansion of $\left(2^{1 / 3}+3^{1 / 4}\right)^{12}$ is:

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Q JEE MAIN 2021

If the value of $\lim _{x \rightarrow 0}(2-\cos x \sqrt{\cos 2 x})^{\left(\frac{x+2}{x^2}\right)}$ is equal to $e^a$, then a is equal...

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Q JEE MAIN 2021

Let $y=m x+c, m>0$ be the focal chord of $y^2=-64 x$, which is tangent to $(x+10)^2+y^2=4$. Then, the value of...

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Q JEE MAIN 2021

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are...

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Q JEE MAIN 2021

Let $a, b, c, d$ be in arithmetic progression with common difference $\lambda$. If $...

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Q JEE MAIN 2021

Let $T$ be the tangent to the ellipse $E: x^2+4 y^2=5$ at the point $P(1,1)$. If the area of the...

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Q JEE MAIN 2021

If the area of the bounded region $...

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Q JEE MAIN 2021

Let C be the set of all complex numbers. Let $...

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Q JEE MAIN 2021

The value of the definite integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{d x}{\left(1+e^{x \cos x}\right)\left(\sin ^4 x+\cos ^4 x\right)}$ is equal to :

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Q JEE MAIN 2021

Let $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+2 \hat{j}+3 \hat{k}$. Then the vector product $(\vec{a}+\vec{b}) \times((\vec{a} \times((\vec{a}-\vec{b}) \times \vec{b})) \times \vec{b})$ is equal...

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Q JEE MAIN 2021

If the shortest distance between the lines $\vec{r}_1=\alpha \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k}), \lambda \in R, \alpha>0$ and $...

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Q JEE MAIN 2021

The value of $\lim _{x \rightarrow \infty} \frac{1}{n} \sum_{j=1}^n \frac{(2 j-1)+8 n}{(2 j-1)+4 n}$ is equal to :

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Q JEE MAIN 2021

The number of rational terms in the binomial expansion of $\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)^{120}$

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Q JEE MAIN 2021

If the mean and variance of the following data : $6,10,7,13, a, 12, b, 12$ are 9 and $\frac{37}{4}$ respectively,...

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Q JEE MAIN 2021

Let $P$ be a plane passing through the points $(1,0,1),(1,-2,1)$ and $(0,1,-2)$. Let a vector $\overrightarrow{\mathbf{a}}=\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+\gamma \hat{\mathbf{k}}$ be...

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Q JEE MAIN 2021

let $A=\left(\begin{array}{ccc}1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1\end{array}\right)$ and $...

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Q JEE MAIN 2021

Let $\vec{a}, \vec{b}, \vec{c}$-be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$, with...

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Q JEE MAIN 2021

Let the tangent to the parabola $\mathrm{S}: \mathrm{y}^2=2 \mathrm{x}$ at the point $\mathrm{P}(2,2)$ meet the x -axis at Q and...

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Q JEE MAIN 2021

The probability of selecting integers $a \in[-5,30]$ such that $x^2+2(a+4) x-5 a+64>0$, for all $x \in R$, is :

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Q JEE MAIN 2021

Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that...

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Q JEE MAIN 2021

Let ' $a$ ' be a real number such that the function $f(x)=a x^2+6 x-15, x \in R$ is increasing...

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Q JEE MAIN 2021

Let $y=y(x)$ be the solution of the differential equation $e^x \sqrt{1-y^2} d x+\left(\frac{y}{x}\right) d y=0, y(1)=-1$. Then the value of...

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Q JEE MAIN 2021

The number of real roots of the equation $\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^2+x+1}=\frac{\pi}{4}$ is :

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Q JEE MAIN 2021

Let $\vec{a}-=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\vec{b}-=\hat{i}+\hat{j}$. If $\vec{c}-$ is a vector such that $\vec{a} \cdot \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2 \sqrt{2}$ and the angle...

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Q JEE-MAIN 2021

Let P be an arbitrary point having sum of the squares of the distance from the planes $\mathrm{x}+\mathrm{y}+\mathrm{z}=0, l \mathrm{x}-\mathrm{nz}=0$...

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Q JEE MAIN 2021

Let $A=\left[a_{i j}\right]$ be a $3 \times 3$ matrix, where $...

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Q JEE-MAIN 2021

Let $\vec{x}$ be a vector in the plane containing vectors $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-\hat{k}$. If the vector $\vec{x}$ is...

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Q JEE MAIN 2021

The coefficien of $x^{256}$ in the expansion of $(1-x)^{101}\left(x^2+x+1\right)^{100}$ is :

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Q JEE-MAIN 2021

Let $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ and $B=\left[\begin{array}{l}\alpha \\ \beta\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0\end{array}\right]$ such that $A B=B$ and...

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Q JEE-MAIN 2021

Let the coefficients of third, fourth and fifth terms in the expansion of $\left(x+\frac{a}{x^2}\right)^n, x \neq 0$ be in the...

JEE Main Mathematics Easy

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Q JEE-MAIN 2021

Consider a set of 3n numbers having variance 4. In this set, the mean of first 2n numbers is 6...

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Q JEE-MAIN 2021

Let $\tan \alpha, \tan \beta$ and $\tan \gamma, \alpha, \beta, \gamma \neq \frac{(2 \mathrm{n}-1) \pi}{2}, \mathrm{n} \in \mathrm{N}$ be the...

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Q JEE-MAIN 2021

Let $f:[-3,1] \rightarrow R$ given as $$ f(x)=\left\{\begin{array}{cc} \min \left\{(x+6), x^2\right\}, & -3 \leq x \leq 0 \\ \max \left\{\sqrt{x},...

JEE Main Mathematics Hard

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Q JEE MAIN 2021

Let $y=y(x)$ be the solution of the differential equation $...

JEE Main Mathematics Medium

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Q JEE-MAIN 2021

Let $f:[-1,1] \rightarrow R$ be defined as $f(x)=a x^2+b x+c$ for all $x \in[-1,1]$ where $a, b, c \in R$...

JEE Main Mathematics Hard

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Q JEE-MAIN 2021

If $1, \log _{10}\left(4^x-2\right)$ and $\log _{10}\left(4^x+\frac{18}{5}\right)$ are in arithmetic progression for a real number $x$, then the value of...

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Q JEE MAIN 2021

Let $[x]$ denote the greatest integer $\leq x$, where $x \in R$. If the domain of the real valued function...

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Q JEE MAIN 2021

The value of $\int_{-2}^2\left|3 x^2-3 x-6\right| d x$ is $\_\_\_\_$ .

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Q JEE MAIN 2021

If the curve, $y=y(x)$ represented by the solution of the differential equation $\left(2 x y^2-y\right) d x+x d y=0$, passes...

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Q JEE MAIN 2021

If $\lim _{x \rightarrow 0} \frac{a x-\left(e^{4 x}-1\right)}{a x\left(e^{4 x}-1\right)}$ exists and is equal to $b$, then the value of...

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Q JEE MAIN 2021

If in a triangle $A B C, A B=5$ units, $\angle B=\cos ^{-1}\left(\frac{3}{5}\right)$ and radius of circumcircle of $...

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Q JEE MAIN 2021

A line is a common tangent to the circle $(x-3)^2+ y^2=9$ and the parabola $y^2=4 x$. If the two points...

JEE Main Mathematics Hard

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Q JEE MAIN 2021

If $z$ and $\omega$ are two complex numbers such that $|z \omega|=1$ and arg $(z)-\arg (\omega) \frac{3 \pi}{2}$, then $...

JEE Main Mathematics Hard

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Q JEE MAIN 2021

If the remainder when x is divided by 4 is 3 , then the remainder when $(2020+\mathrm{x})^{2022}$ is divided by...

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Q JEE MAIN 2021

Let $A=\left[\begin{array}{ll}2 & 3 \\ a & 0\end{array}\right], a \in R$ be written as $P+Q$ where $P$ is a symmetric...

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Q JEE MAIN 2021

Let ${ }^{\mathrm{a}}=\hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}$. If the area of the parallelogram whose adjacent sides are represented...

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Q JEE MAIN 2021

What will be the projection of vector $\overrightarrow{\mathrm{A}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{k}$ on vector $\overrightarrow{\mathrm{B}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$ ?

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Q JEE MAIN 2021

A function $f$ is defined on $[-3,3]$ as $$ f(x)=\left\{\begin{array}{cc} \min \left\{|x|, 2-x^2\right\}, & -2 \leq x \leq 2 \\...

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Q JEE MAIN_2021_

Let three vectors $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be such that $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=\overline{\mathrm{c}}, \overline{\mathrm{b}} \times-\overline{\mathrm{c}}=\overline{\mathrm{a}}$ and $|\overline{\mathrm{a}}|=2$. Then which one...

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Q JEE MAIN 2021

The total number of two digit numbers ' n ', such that $3^n+7^n$ is a multiple of 10 , is...

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Q _JEE MAIN_2021_

If $\int_0^{100 \pi} \frac{\sin ^2 x}{e^{\left(\frac{x}{\pi}-\left[\frac{x}{\pi}\right]\right)}} d x=\frac{\alpha \pi^3}{1+4 \pi^2}, \alpha \in R$ where $[x]$ is the greatest integer less...

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Q JEE MAIN 2021

If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2+(3)^{1 / 4} x+3^{1 / 2}=0$, then the value...

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Q JEE MAIN 2021

The following system of linear equations 2x + 3y + 2z = 9 3x + 2y + 2z = 9...

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Q JEE MAIN_2021

Let a vector $\bar{a}$ be coplanar with vectors $\bar{b}=2 \hat{i}+\hat{j}+\hat{k}$ and $\bar{c}=\hat{i}-\hat{j}+\hat{k}$. If $\bar{a}$ is perpendicular to $...

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Q JEE MAIN 2021

The value of the integral $\int_{-1}^1 \log _e(\sqrt{1-x}+\sqrt{1+x}) d x$ is equal to :

JEE Main Mathematics Hard

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Q JEE MAIN_2021_

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 × 2 matrices. The...

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Q JEE MAIN 2021

Let x denote the total number of one-one functions from a set A with 3 elements to a set B...

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Q JEE MAIN_2021_

Let $y=y(x)$ be the solution of the differential equation $...

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Q _JEE MAIN_2021

Let $f: R \rightarrow R$ be defined as $...

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Q JEE MAIN 2021

The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are...

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Q JEE MAIN 2021

Let $a$ and $\beta$ be the roots of $x^2-6 x-2=0$. If $\alpha_n=\alpha^n -\beta^n$ for $n \geq 1$, then the value...

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Q JEE MAIN 2021

Let a be a positive real number such that $\int_0^a e^{x-[x]} d x=10 e-9$ where $[x]$ is the greatest integer...

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Q _JEE MAIN_2021

Let $S_n$ denote the sum of first $n$-terms of an arithmetic progression. If $S_{10}=530, S_5=140$, then $S_{20}-S_6$ is equal to

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Q JEE MAIN_2021

Let $L$ be the line of intersection of planes $\vec{r} \cdot(\hat{i}-\hat{j}+2 \hat{k})=2$ and $\vec{r} \cdot(2 \hat{i}+\hat{j}-\hat{k})=2$. If $P(\alpha, \beta, \gamma)$...

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Q JEE MAIN 2021

Let A be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that...

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Q JEE MAIN 2021

Let $y=y(x)$ be the solution of the differential equation $x d y-y d x=\sqrt{\left(x^2-y^2\right)} d x, x \geq 1$, with...

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Q JEE MAIN 2021

Let $P$ be $a$ plane containing the line $\frac{x-1}{3}=\frac{y+6}{4}=\frac{z+5}{2}$ and parallel to the line $\frac{x-3}{4}=\frac{y-2}{-3}=\frac{z+5}{7}$. If the point $(1,-1, \alpha)$...

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Q JEE MAIN 2021

The Boolean expression $(p \wedge \sim q) \Rightarrow(q \vee \sim p)$ is equivalent to :

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Q JEE MAIN 2021

$\lim _{n \rightarrow \infty}\left[\frac{1}{n}+\frac{n}{(n+1)^2}+\frac{n}{(n+2)^2}+\ldots . .+\frac{n}{(2 n-1)^2}\right]$ is equal to

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Q JEE MAIN_2021_

If $(2021)^{3762}$ is divided by 17 , then the remainder is $\_\_\_\_$ .

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Q JEE MAIN 2021

Let ${ }^n C_r$ denote the binomial coefficient of $\underline{X^r}$ in the expansion of $(1+\mathrm{x})^{\mathrm{n}}$. If $...

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Q JEE MAIN 2021

Let $f: R \rightarrow R$ satisfy the equation $f(x+y)=f(x)$. $f(y)$ for all $x, y \in R$ and $f(x) \neq 0$...

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Q JEE MAIN 2021

Let the mirror image of the point $(1,3, a)$ with respect to the plane $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})-\mathrm{b}=0$ be $(-3,5$, 2)....

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Q JEE MAIN 2021

Let $P(x)$ be real polynomial of degree 3 which vanishes at $x=-3$. Let $P(x)$ have local minima at $x=1$, local...

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Q JEE MAIN 2021

$$ \text { If } I_n=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot ^n x   d x, \text { then : } $$

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Q JEE-MAIN 2021

If the equation of plane passing through the mirror image of a point $(2,3,1)$ with respect to line $\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}$ and...

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Q JEE MAIN 2021

If $\lim _{x \rightarrow 0} \frac{\alpha x e^x-\beta \log _e(1+x)+\gamma x^2 e^{-x}}{x \sin ^2 x}=10, \alpha, \beta, \gamma \in R$,...

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Q JEE-MAIN 2021

If the Boolean expression $(p \wedge q)(p \otimes q)$ is a tautology, then and $\otimes$ are respectively given by

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Q JEE MAIN 2021

Let $\left\{a_n\right\}_{n=1}^{\infty}$ be a sequence such that $a_1=1, a_2=1$ and $a_{n+2}=2 a_{n+1}+a_n$ for all $n \geq 1$. Then the value...

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Q JEE MAIN 2021

The minimum value of $f(x)=a^{a x}+a^{1-a x}$, where $a, x \in R$ and $a>0$, is equal to :

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Q JEE MAIN 2021

The term independent of x in the expansion of $\left[\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right]^{10}, x \neq 1$ is...

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Q JEE MAIN 2021

For $k \in N$, let $\frac{1}{\alpha(\alpha+1)(\alpha+2) \ldots .(\alpha+20)}=\sum_{K=0}^{20} \frac{A_k}{\alpha+k}$, where $\alpha>0$. Then the value of $100\left(\frac{A_{14}+A_{15}}{A_{13}}\right)^2$ is equal to $\_\_\_\_$...

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Q JEE-MAIN 2021

Let the tangent to the circle $x^2+y^2=25$ at the point $R(3,4)$ meet $x$-axis and $y$-axis at point $P$ and $Q$,...

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Q JEE MAIN 2021

If $\sum_{r=1}^{10} r!\left(r^3+6 r^2+2 r+5\right)=\alpha(11!)$ then the value of $\alpha$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2021

Let $I$ be an identity matrix of order $2 \times 2$ and $P=\left[\begin{array}{ll}2 & -1 \\ 5 & -3\end{array}\right]$. Then...

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Q JEE MAIN 2021

Let a function $\mathrm{g}:[0,4] \rightarrow R$ be defined as $...

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Q JEE MAIN 2021

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P(x)=f\left(x^3\right)+x g\left(x^3\right)$ is divisible by $x^2+x$ +1 , then...

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Q JEE-MAIN 2021

The value of the limit $\lim _{\theta \rightarrow 0} \frac{\tan \left(\pi \cos ^2 \theta\right)}{\left(2 \pi \sin ^2 \theta\right)}$ equal to:

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Q JEE MAIN 2021

Let $f: R \rightarrow R$ be a function defined as $...

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Q JEE-MAIN 2021

Let $L$ be a tangent line to the parabola $y^2=4 x-20$ at $(6,20)$. If $L$ is also a tangent to...

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Q JEE MAIN 2021

If for the matrix, $A=\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right], A A^{\top}=I_2$, then the value of $\alpha^4+\beta^4$ is :

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Q JEE MAIN 2021

The area bounded by the curve $4 y^2=x^2(4-x)(x-2)$ is equal to:

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Q JEE-MAIN 2021

Consider the function $f: R \rightarrow R$ defined by $$ f(x)=\left\{\begin{array}{cl} \left(2 \sin \left(\frac{1}{x}\right)\right)|x|, & x \neq 0 \\ 0...

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Q JEE MAIN 2021

A function $f(x)$ is given by $f(x)=\frac{5^x}{5^x+5}$, then the sum of the series $$ f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\ldots \ldots+f\left(\frac{39}{20}\right) $$ is equal to:

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Q JEE MAIN 2021

If $15 \sin ^4 \alpha+10 \cos ^4 \alpha=6$, for some $\alpha \in \mathrm{R}$, then then value of $...

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Q JEE MAIN 2021

A pole stands vertically inside a triangular park $A B C$. Let the angle of elevation of the top of...

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Q JEE MAIN 2021

Define a relation $R$ over a class of $n \times n$ real matrices $A$ and $B$ as "ARB iff there...

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Q JEE-MAIN 2021

Let O be the origin. Let $\overline{\mathrm{OP}}=x \hat{\mathrm{i}}+y \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overline{\mathrm{PQ}}=\sqrt{20} \overline{\mathrm{OQ}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 x \hat{\mathrm{k}}, x, y \in \mathrm{R}, x>0$,...

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Q JEE MAIN 2021

Let a tangent be drawn to the ellipse $\frac{x^2}{27}+y^2=1$ at $(3 \sqrt{3} \cos \theta, \sin \theta)$ where $\theta \in\left(0, \frac{\pi}{2}\right)$....

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Q JEE-MAIN 2021

If $x, y, z$ are in arithmetic progression with common difference $d, x 3 d$, and the determinant of the...

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Q JEE MAIN 2021

Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and...

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If the sides AB, BC and CA of a triangle ABC have 3, 5 and 6 interior points respectively, then...

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Q JEE MAIN 2021

The contrapositive of the statement "If you will work, you will earn money" is :

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Let $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ be two non-zero vectors perpendicular to each other and $|\overline{\mathrm{a}}|=|\overline{\mathrm{b}}|$. If $|\overline{\mathrm{a}} \times \overline{\mathrm{b}}|=|\overline{\mathrm{a}}|$, then the...

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Q JEE-MAIN 2021

In the curve y = y (x) is the solution of the differential equation $...

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Let $S_1: x^2+y^2=9$ and $S_2:(x-2)^2+y^2=1$. Then the locus of center of a variable circle $S$ which touches $S_1$ internally and...

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Q JEE-MAIN 2021

Let $S_1, S_2$ and $S_3$ be three sets defined as $$ \begin{aligned} & S_1=\{z \in \mathbb{C}:|z-1| \leq \sqrt{2}\} \\ &...

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Q JEE MAIN 2021

Let in a series of $2 n$ observations, half of them are equal to a and remaining half are equal...

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If the curve $x^2+2 y^2=2$ intersects the line $x+y=1$ at two points $P$ and $Q$, then the angle subtended by...

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Q JEE MAIN 2021

Let a complex number be $\mathrm{w}=1-\sqrt{3} i$. Let another complex number $z$ be such that $|z w d|=1$ and $...

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Q JEE-MAIN 2021

The number of solutions of the equation $x+2 \tan x=\frac{\pi}{2}$ in the interval $[0,2 \pi]$ is :

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Q JEE MAIN 2021

Let $S_1$ be the sum of first $2 n$ terms of an arithmetic progression. Let $S_2$ be the sum of...

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Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability...

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Let $g(x)=\int_0^x f(t) d t$, where $f$ is continuous function in $[0,3]$ such that $\frac{1}{3} \leq f(t) \leq 1$ for...

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Q JEE MAIN 2021

$\operatorname{cosec}\left[2 \cot ^{-1}(5)+\cos ^{-1}\left(\frac{4}{5}\right)\right]$ is equal to :

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The number of solutions of the equation $$ \sin ^{-1}\left[x^2+\frac{1}{3}\right]+\cos ^{-1}\left[x^2-\frac{2}{3}\right]=x^2 $$ for $x \in[-1,1]$ and $[x]$ denotes the greatest...

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Q JEE-MAIN 2021

The value of $\lim _{n \rightarrow \infty} \frac{[r]+[2 r]+\ldots \ldots+[n r]}{n^2}$ where $r$ is non-zero real number and $[r]$ denotes...

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Q JEE MAIN 2021

In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140...

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Q JEE-MAIN 2021

The value of $\sum_{\mathrm{r}=0}^6\left({ }^6 \mathrm{C}_{\mathrm{r}} \cdot{ }^6 \mathrm{C}_{6-\mathrm{r}}\right)$ is equal to :

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Q JEE-MAIN 2021

Let y = y (x) be the solution of the differential equation

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Q JEE MAIN 2021

If P and Q are two statements, then which of the following compound statement is a tautology ?

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Consider a hyperbola $H_{\text {in }} x^2 2 y^2=4$. Let the tangent at a point $\mathrm{P}(4, \sqrt{6})$ meet the x...

JEE Main Mathematics Medium

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Q JEE-MAIN 2021

If the integral $\int_0^{10} \frac{[\sin 2 \pi x]}{e^{x-[x]}} d x=\alpha e^{-1}+\beta e^{-\frac{1}{2}}+\gamma$, where, $\alpha, \beta, \gamma$ are integers and $[x]$...

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Q JEE-MAIN 2021

Let $f: R \rightarrow R$ be defined as $f(x)=e^{-x} \sin x$. If $F:[0,1] \rightarrow R$ is a differentiable function such...

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Q JEE MAIN_2021_

If the equation of the plane passing through the line of intersection of the planes $...

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The minimum distance between any two points $P_1$ and $P_2$ while considering point $P_1$ on one circle and point $P_2$...

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Q JEE MAIN 2021

A plane passes through the points $\mathrm{A}(1,2,3), \mathrm{B}(2$, $3,1)$ and $\mathrm{C}(2,4,2)$. If O is the origin and P is (...

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Q JEE MAIN_2021_

If [ ] represents the greatest integer function, then the value of

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Q JEE MAIN_2021_1

IF A is

JEE Main Mathematics Medium

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Q JEE MAIN 2021

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral...

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If $\bar{a}=\alpha \hat{i}+b \hat{j}+3 \hat{k}, \bar{b}=\beta \hat{i}-\alpha \hat{j}-\hat{k}$ and $\bar{c}=\hat{i}-2 \hat{j}-\hat{k}$ such that $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=1$ and $\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}=-3$...

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Let there be three independent events $\mathrm{E}_1, \mathrm{E}_2$ and $E_3$. The probability that only $E_1$ occurs is $\alpha$, only $E_2$...

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Q JEE MAIN 2021

Let $f: R-\{3\} \rightarrow R-\{1\}$ be defined by $f(x)=\frac{x-2}{x-3}$. Let $g: R \rightarrow R$ be given as $g(x)=2 x$ -3...

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Q JEE MAIN_2021_

If $f(x)=\sin \left(\cos ^{-1}\left(\frac{1-2^{2 x}}{1+2^{2 x}}\right)\right)$ and its firs t derivative with respect to $x$ is $-\frac{b}{a} \log _e 2$...

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Q JEE MAIN 2021

Let the system of linear equations $...

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If the function $f(x)=\frac{\cos (\sin x)-\cos x}{x^4}$ is continuous at each point in its domain and $f(0)=\frac{1}{k}$, then $k$ is...

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Q JEE MAIN 2021

Consider a triangle having vertices $\mathrm{A}(-2,3), \mathrm{B}(1,9)$ and $\mathrm{C}(3,8)$. If a line L passing through the circum-centre of triangle $...

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Q JEE MAIN 2021

A hyperbola passes through the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and its transverse and conjugate axes coincide with major and...

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Q JEE MAIN_2021

The maximum value of z in the following equation $z=6 x y+y^2$, where $3 x+4 y \leq 100$ and $...

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Q JEE MAIN 2021

In a triangle ABC, if $|\overrightarrow{\mathrm{BC}}|=8,|\overrightarrow{\mathrm{CA}}|=7,|\overrightarrow{\mathrm{AB}}|=10$, then the projection of the vector $\overrightarrow{\mathrm{AB}}$ on $\overrightarrow{\mathrm{AC}}$ is equal to:

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For $\mathrm{p}>0$, a vector $\overrightarrow{\mathrm{v}}_2=2 \hat{\mathrm{i}}+(\mathrm{p}+1) \hat{\mathrm{j}}$ is obtained by rotating the vector $\overrightarrow{\mathrm{v}}_1=\sqrt{3} p \hat{\mathrm{i}}+\hat{\mathrm{j}}$ by an angle $\theta$...

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Q JEE MAIN 2021

Let $f: R \rightarrow R$ and $g \mid: R \rightarrow R$ be defined as $$ \begin{aligned} & f(x)=\left\{\begin{array}{ll} x+a, &...

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Q JEE MAIN 2021

Let $P=\left[\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{array}\right]$, where $...

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Q JEE MAIN_2021_

The value of

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Let a curve $y=y(x)$ be given by the solution of the differential equation $$ \cos \left(\frac{1}{2} \cos ^{-1}\left(e^{-x}\right)\right) d x=\sqrt{e^{2...

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Q JEE MAIN 2021

JEE Main Mathematics Medium

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Let $\mathrm{B}_i(\mathrm{i}=1,2,3)$ be three independent events in a sample space. The probability that only $\mathrm{B}_1$ occur is $\alpha$, only $\mathrm{B}_2$...

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Q JEE MAIN_2021_

Which of the following is true for y (x) that satisfies the differential equation

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If $a, \beta \in R$ are such that $1-2 i$ (here $i^2=-1$ ) is a root of $z^2+\alpha z+\beta=0$, then...

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Q JEE MAIN 2021

The number of solutions of the equation $\log _{(x+1)}\left(2 x^2+7 x+5\right)+\log _{(2 x+5)}(x+1)^2-4=0, x>0$, is

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Q JEE MAIN 2021

Let $A=\left[\begin{array}{l}a_1 \\ a_2\end{array}\right]$ and $B=\left[\begin{array}{l}b_1 \\ b_2\end{array}\right]$ be two $2 \times 1$ matrices with real entries such that $\mathrm{A}=\mathrm{XB}$,...

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Q JEE MAIN_2021_

Which of the following statements is incorrect for the function $\mathrm{g}(\mathrm{a})$ for $\alpha \in \mathrm{R}$ such that

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Q JEE MAIN 2021

Let three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and, $\vec{b}, \vec{a} \cdot \vec{c}=7$...

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Q JEE MAIN 2021

Let $\mathrm{A}=\left\{\mathrm{a}_{\mathrm{ij}}\right\}$ be a $3 \times 3$ matrix, where $\mathrm{a}_{\mathrm{ij}}=\left\{\begin{array}{cll}(-1)^{\mathrm{j}-\mathrm{i}} & \text { if } & \mathrm{i}\mathrm{j},\end{array}\right.$, hen $...

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Q JEE MAIN 2021

$\lim _{n \rightarrow \infty} \tan \left\{\sum_{r=1}^n \tan ^{-1}\left(\frac{1}{1+r+r^2}\right)\right\}$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2021

JEE Main Mathematics Medium

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Choose the incorrect statement about the two circles whose equations are given below. $$ \begin{aligned} & x^2+y^2-10 x-10 y+41=0 \text...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

The minimum value of a for which the equation $\frac{4}{\sin x}+\frac{1}{1-\sin x}=\alpha$ has at least one solution in $\left(0, \frac{\pi}{2}\right)$...

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Q JEE MAIN 2021

In a triangle ABC , if $|\overrightarrow{\mathrm{BC}}|=3,|\overrightarrow{\mathrm{CA}}|=5$ and $|\overrightarrow{\mathrm{BA}}|=7$, then the projection of the vector $\overrightarrow{\mathrm{BA}}$ on $\overrightarrow{\mathrm{BC}}$ is equal...

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Q JEE MAIN_2021_

VALUE OF

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If the distance of the point $(1,-2,3)$ from the plane $x+2 y-3 z+10=0$ measured parallel to the line, $\frac{x-1}{3}=\frac{2-y}{m}=\frac{z+3}{1}$ is...

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Q JEE MAIN 2021

If sum of the first 21 terms of the series $...

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Q JEE MAIN 2021

If one of the diameters of the circle $x^2+y^2-2 x-6 y+6=0$ is a chord of another circle ' $C$ ',...

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Q JEE MAIN_2021_

Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a...

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Q JEE MAIN 2021

The value of $k \in R$, for which the following system of linear equations $$ \begin{aligned} & 3 x-y+4 z=3...

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Q JEE MAIN 2021

Let $\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\vec{b}=2 \hat{i}-3 \hat{j}+5 \hat{k}$ If $\vec{r} \times \vec{a}=\vec{b} \times \vec{r} \cdot(a \hat{i}+2 \hat{j}+\hat{k})=3$ and $...

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Q JEE MAIN 2021

Let M be any 3 × 3 matrix with entries from the set {0, 1, 2}. The maximum number of...

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Q JEE MAIN 2021

Let $P$ be a variable point on the parabola $y=4 x^2+1$. Then, the locus of the mid-point of the point...

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Q JEE MAIN_2021_

The line 2x – y + 1 = 0 is a tangent to the circle at the point (2, 5)...

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Q JEE MAIN 2021

The integral $...

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Let C 1 be the curve obtained by the solution of differential equation $2 x y \frac{d y}{d x}=y^2-x^2, x>0$....

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Q JEE MAIN_2021

The area of the triangle with vertices A (z), B (iz) and C (z + iz) is :

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Let $g(t)=\int_{-\pi / 2}^{\pi / 2}\left(\cos \frac{\pi}{4} t+f(x)\right) d x$, where $f(x)=\log _e\left(x+\sqrt{x^2+1}\right), x \in R$. Then which one of...

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Q JEE MAIN 2021

If the foot of the perpendicular from point $(4,3,8)$ on the line $\mathrm{L}_1: \frac{\mathrm{x}-\mathrm{a}}{\ell}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{z}-\mathrm{b}}{4}, \ell \neq 0$ is $(3,5$, 7),...

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Q JEE MAIN_2021_

The sum of possible values of $x$ for $\tan ^{-1}(x+1)+\cot ^{-1}\left(\frac{1}{x-1}\right)=\tan ^{-1}\left(\frac{8}{31}\right)$ is :

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Q JEE MAIN 2021

Let $A$ be a $3 \times 3$ matrix with $\operatorname{det}(A)=4$. Let $R_i$ denote the ith row of $A$. If a...

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Q JEE MAIN 2021

-If the mean and variance of six observations $7,10,11,15, a, b$ are 10 and $\frac{20}{3}$, respectively, then the value of...

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Q JEE MAIN 2021

Consider the line $L$ given by the equation $\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1}$. Let $Q$ be the mirror image of the point $(2,3$, -1)...

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Q JEE MAIN_2021_

In a school, there are three types of games to be played. Some of the students play two types of...

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Let $\mathrm{P}(\mathrm{x})=\mathrm{x}^2+\mathrm{bx}+\mathrm{c}$ be a quadratic polynomial with real coefficients such that $=\int_0^1 P(x) d x=1$ and $\mathrm{P}(\mathrm{x})$ leaves remainder 5...

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Q JEE MAIN 2021

Let $y=y(x)$ satisfies the equation $\frac{d y}{d x}-|A|=0$, for all $x>0$ where $...

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Let $f: S \rightarrow$ Swhere $S=(0, \infty)$ be a twice differentiable function such that $\mathrm{f}(\mathrm{x}+1)= \mathrm{xf}(\mathrm{x})$. If $...

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Q JEE MAIN 2021

Let in a right angled triangle, the smallest angle be $\theta$. If a triangle formed by taking the reciprocal of...

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Q JEE MAIN 2021

Let $f$ be a real valued funditon, defined on $R-\{-1$, $1\}$ and given by $$ f(x)=3 \log _e\left|\frac{x-1}{x+1}\right|-\frac{2}{x-1} $$ Then...

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Q JEE MAIN 2021

The sum of all the local minimum values of the twice differentiable function $f: R \rightarrow R$ defined by $...

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Q JEE MAIN 2021

Let $A, B$ and $C$ be three events such that the probability that exactly one of $A$ and $B$ occurs...

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Let $A(-1,1), B(3,4)$ and $C(2,0)$ be given three points. A line $y=m x, m>0$, intersects lines $A C$ and BC...

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Q JEE MAIN 2021

If $f: R \rightarrow R$ is given by $f(x)=x+1$, then the value of $\lim _{n \rightarrow \infty}\left[f(x)+f\left(\frac{5}{n}\right)+f\left(\frac{10}{n}\right)+\ldots+f\left(\frac{5(n-1)}{n}\right)\right]$, is :

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Q JEE MAIN 2021

Lef $f: R-\left\{\frac{\alpha}{6}\right\} \rightarrow R$ be defined by $f(x)=\frac{5 x+3}{6 x-\alpha}$. Then the value of $\alpha$ for which (fof) $(x)=x$,...

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Q JEE MAIN 2021

If the real part of the complex number $(1-\cos \theta+2 i \sin \theta)^{-1}$ is $\frac{1}{5}$ for $\theta \in(0, \pi)$, then...

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Q JEE MAIN 2021

If $\int_{-a}^a(|x|+|x-2|) d x=22(a>2)$ and $[x]$ denotes the greatest integer $\leq x$, then $\int_a^{-a}(x+[x]) d x$ is equal to $\_\_\_\_$...

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Q JEE MAIN 2021

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of $x$ which satisfy...

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Q JEE MAIN 2021

If the least and the largest real values of $a$, for which the equation $z+\alpha|z-1|+2 i=0(z \in C$ and $i=\sqrt{-1})$...

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Q JEE MAIN 2021

If the point of intersections of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the circle $x^2+y^2=4 b, b>4$ lie on the curve $...

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Q JEE MAIN 2021

The locus of the mid-point of the line segment joining the focus of the parabola $y^2=4 a x$ to a...

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Q JEE MAIN 2021

Consider a rectangle ABCD having $5,7,6,9$ points in the interior of the line segments $...

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If $e^{\left(\cos ^2 x+\cos ^4 x+\cos ^6 x+-\infty\right) \log _4 2}$ satisfies the equation $t^2-9 t+8=0$, then the value of...

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Q JEE MAIN 2021

The least value of $|z|$ where $z$ is complex number which satisfies the inequality $$ \exp \left(\frac{(|z|+3)(|z|-1)}{||z|+1|} \log _e 2\right)...

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Q JEE MAIN 2021

Let the lengths of intercepts on $x$-axis and $y$-axis made by the circle $x^2+y^2+a x+2 a y+c=0,(a< 0)$ be $...

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Q JEE MAIN 2021

$\lim _{x \rightarrow 0} \frac{\int_0^{x^2}(\sin \sqrt{t}) d t}{x^3}$ is equal to :

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Q JEE MAIN 2021

Two vertical poles are 150 m apart and the height of one is three times that of the other. If...

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Let $A \mid=\{2,3,4,5, \ldots ., 30\}$ and ${ }^{\prime} \simeq$ ' be an equivalence relation on $\mathrm{A} \times \mathrm{A}$, defined...

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Q JEE MAIN 2021

Let $p$ and $q$ be two positive numbers such that $p+q=2$ and $p^4+q^4=272$. Then $p$ and $q$ are roots of...

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Q JEE MAIN 2021

If $y=y(x)$ is the solution of the differential equation $\frac{d y}{d x}+(\tan x) y=\sin x, 0 \leq x \leq \frac{\pi}{3}$,...

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Q JEE MAIN 2021

Let C be the locus of the mirror image of a point on the parabola $\mathrm{y}^2=4 \mathrm{x}$ with respect to...

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Q JEE MAIN 2021

Consider the integral $$ \int_0^{10} \frac{[x] e^{[x]}}{e^{x-1}} d x $$ where $[x]$ denotes the greatest integer less than or equal...

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Q JEE MAIN 2021

The area (in sq. units) of the part of the circle $x^2+y^2=36$, which is outside the parabola $y^2=9 x$, is...

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Q JEE MAIN 2021

If $(x, y, z)$ be an arbitrary point lying on a plane $P$ which passes through the point $(42,0,0),(0,42,0)$ and...

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Q JEE MAIN 2021

A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and...

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Q JEE MAIN 2021

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2...

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Let $f ; \mathrm{R} \rightarrow \mathrm{R}$ be defined as $f(\mathrm{x})=2 \mathrm{x}-1$ and $\mathrm{g}: \mathrm{R}-\{1\} \rightarrow \mathrm{R}$ be defined as $\mathrm{g}(\mathrm{x})=\frac{\mathrm{x}-\frac{1}{2}}{\mathrm{x}-1}$...

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Q JEE MAIN 2021

Let $\alpha \in \mathrm{R}$ be such that the function $$ f(x)= \begin{cases}\frac{\cos ^{-1}\left(1-\{x\}^2\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^3}, & x \neq 0 \\...

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Q JEE MAIN 2021

Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions,...

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Q JEE MAIN 2021

The function $f(x)=\frac{4 x^3-3 x^2}{6}-2 \sin x+(2 x-1) \cos x:$

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Q JEE MAIN 2021

The maximum value of $$ f(x)=\left|\begin{array}{ccc} \sin ^2 & 1+\cos ^2 x & \cos 2 x \\ 1+\sin ^2 x...

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Q JEE MAIN 2021

If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $...

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Q JEE MAIN_2021_

If the fourth term in the expansion of $\left(x+x^{\log _2 x}\right)^7$ is 4480 , then the value of $x$ where...

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Q _JEE MAIN_2021_

Two dices are rolled. If both dices have six faces numbered 1,2,3,5,7 and 11, then the probability that the sum...

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If the Boolean expression $(p=q) \Leftrightarrow\left(q^*(\sim p)\right)$ is a tautology, then the Boolean expression $\mathrm{p}^*(\sim \mathrm{q})$ is equivalent toni

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Q JEE MAIN 2021

The lines $x=a y-1=z-2$ and $x=3 y-2=b z-2,(a b \neq 0)$ are coplanar, if .

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Q JEE MAIN_2021_

If $A=\left(\begin{array}{cc}0 & \sin \alpha \\ \sin \alpha & 0\end{array}\right)$ and $\operatorname{det}=\left(A^2-\frac{1}{2} I\right)=0$ then a possible value of $\alpha$ is

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Q JEE MAIN 2021

Consider the following three statements : (A) If 3 + 3 = 7 then 4 + 3 = 8. (B)...

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Q JEE MAIN_2021

The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is :

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Q JEE MAIN 2021

Let $r_1$ and $r_2$ be the radii of the largest and smallest circles, respectively, which pass through the point $(-4,1)$...

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Q _JEE MAIN_2021_

If $\cot ^{-1}(\alpha)=\cot ^{-1} 2+\cot ^{-1} 8+\cot ^{-1} 18+\cot ^{-1} 32+$ sum Upto 100 terms, then $\alpha$ is is

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Q JEE MAIN 2021

The value of $\left(2 \tan ^{-1}\left(\frac{3}{5}\right)+\sin ^{-1}\left(\frac{5}{13}\right)\right)$ is equal to:

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The system of equations $\mathrm{kx}+\mathrm{y}+\mathrm{z}=1, \mathrm{x}+\mathrm{ky}+\mathrm{z} =k$ and $x+y+z k=k^2$ has no solution if $k$ is equal to i

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Q JEE MAIN 2021

For the natural numbers $m, n$, if $(1-y)^m(1+y)^n=1+a_1 y+a_2 y^2+\ldots . .+a_{m+n} y^{m+n}$ and $a_1=a_2=10$, then the value of $(m+n)$...

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Q JEE MAIN_2021_

In a triangle PQR , the co-ordinates of the points P and $Q$ are ( $-2,4$ ) and ( $4,-2$...

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Q JEE MAIN 2021

The number of the real roots of the equation $(x+1)^2+|x-5|=\frac{27}{4}$ is $\_\_\_\_$ .

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Q _JEE MAIN_2021

Let $\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}$ and $\vec{b}=7 \hat{i}+\hat{j}-6 \hat{k}$ If $\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})=-3$ then $...

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Q JEE MAIN 2021

Let $\mathrm{i}=\sqrt{-1}$. If $\frac{(-1+\mathrm{i} \sqrt{3})^{21}}{(1-\mathrm{i})^{24}}+\frac{(1+\mathrm{i} \sqrt{3})^{21}}{(1+\mathrm{i})^{24}}=\mathrm{k}$, and $\mathrm{n}=$ [| $\mathrm{k} \mid]$ be the greatest integral part of $|\mathrm{k}|$. Then $\sum_{j=0}^{n+5}(j+5)^2-\sum_{j=0}^{n+5}(j+5)$...

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Q JEE MAIN_2021

The inverse of $y=5^{\log x}$ is :

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Q JEE MAIN 2021

The students $S_1, S_{2, \ldots \ldots}, S_{10}$ are to be divided into 3 groups $\mathrm{A}, \mathrm{B}$ and C such that...

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Q JEE MAIN 2021

The sum of first four terms of a geometric progression (G.P.) is $\frac{65}{12}$ and the sum of their respective reciprocals...

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Q JEE MAIN 2021

If the variance of 10 natural numbers 1, 1, 1,...., 1, k is less than 10, then the maximum possible...

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Q JEE Main 2019

Consider the set of all lines $p x+q y+r=0$ such that $3 p+2 q+4 r=0$. Which one of the following...

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Q JEE MAIN 2021

If the area of the triangle formed by the positive xaxis, the normal and the tangent to the circle (...

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Q JEE MAIN 2021

Let a point P be such that its distance from the point $(5,0)$ is thrice the distance of P from...

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Q JEE MAIN 2021

If $a+\alpha=1, b+\beta=2$ and $a f(x)+\alpha f\left(\frac{1}{x}\right)=b x+\frac{\beta}{x}, x \neq 0$, then the value of expression $\frac{f(x)+f\left(\frac{1}{x}\right)}{x+\frac{1}{x}}$ is $\_\_\_\_$ .

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Q JEE MAIN 2021

Let $\lambda$ be an interger. If the shortest distance between the lines $x-\lambda=2 y-1=-2 z$ and $x=y+ 2 \lambda=z-\lambda$ is...

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Q JEE MAIN 2021

For integers $n$ and $r$, let $...

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Q JEE MAIN 2021

The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in...

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Q JEE MAIN 2021

For the system of linear equations : $$ x-2 y=I, x-y+k z=-2, k y+4 z=6, k \in R, $$ consider...

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Q JEE MAIN 2021

Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be in arithmetic progression. Let the centroid of the triangle with vertices $(\mathrm{a}, \mathrm{c}),(2, \mathrm{~b})$ and...

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Q JEE MAIN 2021

Let $A$ and $B$ be $3 \times 3$ real matrices such that $A$ is symmetric matrix and $B$ is skew-symmetric...

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Q JEE MAIN 2021

Let $f(x)$ be a differentiable function defined on $[0,2]$ such that $f^{\prime}(x)=f^{\prime}(2-x)$ for all $x \in(0,2), f(0)=1$ and $f(2)=e^2$. Then...

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Q JEE MAIN 2021

If a curve $y=f(x)$ passes through the point $(1,2)$ and satisfies $x \frac{d y}{d x}+y=b x^4$, then for what value...

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Q JEE MAIN 2021

The area of the region : $$ R=\left\{(x, y): 5 \times 2 \leq y \leq 2 x^2+9\right\} \text { is...

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Q JEE MAIN 2021

If the curve $y=a x^2+b x+c, x \in R$, passes through the point $(1,2)$ and the tangent line to this...

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Q JEE MAIN 2021

The negative of the statement $\sim p \wedge(p \vee q)$ is

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Q JEE MAIN 2021

A possible value of $\tan \left(\frac{1}{4} \sin ^{-1} \frac{\sqrt{63}}{8}\right)$ is :

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Q JEE MAIN 2021

The value of the integral, $\int_1^3\left[x^2-2 x-2\right] d x$, where $[x]$ denotes the greatest integer less than or equal to...

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Q JEE MAIN 2021

For which of the following curves, the line $x+\sqrt{3} y=2 \sqrt{3}$ is the tangent at the point $...

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Q JEE MAIN 2021

Let $f$ be a twice differentiable function defined on $R$ such that $f(0)=1, f^{\prime}(0)=2$ and $f^{\prime}(x) \neq 0$ for all...

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Q JEE MAIN 2021

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as, $$ f(x)= \begin{cases}-55 x, & \text { if } x4\end{cases} $$ Let...

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Q JEE MAIN 2021

If $n \geq 2$ is a positive integer, then the sum of the series $...

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Q JEE MAIN 2021

The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight...

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Q JEE MAIN 2021

If $P$ is a point on the parabola $y=x^2+4$ which is closest to the straight line $y=4 x-1$, then the...

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Q JEE MAIN 2021

The vector equation of the plane passing through the intersection of the planes $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=1$ and $\vec{r} \cdot(\hat{i}-2 \hat{j})=-2$, and...

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Q JEE MAIN 2021

Let $\mathrm{a}, \mathrm{b} \in \mathrm{R}$. If the mirror image of the point $\mathrm{P}(\mathrm{a}, 6$, 9) with respect to the line...

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Q JEE MAIN 2021

For the statements p and q, consider the following compound statements :\ Then which of the following statements is correct?

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Q JEE MAIN 2021

The number of solutions of the equation $|\cot x|=\cot x+\frac{1}{\sin x}$ in the interval $[0,2 \pi]$ is

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Q JEE MAIN 2021

A square $A B C D$ has all its vertices on the curve $x^2 y^2=1$. The midpoints of its sides...

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Q JEE MAIN 2021

If $f(x)=\int \frac{5 x^8+7 x^6}{\left(x^2+1+2 x^7\right)^2} d x,(x \geq 0), f(0)=0$ and $f(1) =\frac{1}{\mathrm{~K}}$, then the value of K is

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Q JEE MAIN 2021

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60...

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Q JEE MAIN 2021

The equation of the planes parallel to the plane x – 2y + 2z – 3 = 0 which are...

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Q JEE MAIN 2021

Let $z_1, z_2$ be the roots of the equation $z^2+a z+12 =0$ and $\mathrm{Z}_1, \mathrm{Z}_2$ form an equilateral triangle with...

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Q JEE MAIN 2021

The missing value in the following figure is

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Q JEE MAIN 2021

Let $f(x)$ and $g(x)$ be two functions satisfying $f\left(x^2\right)+ g(4-x)=4 x^3$ and $g(4-x)+g(x)=0$, then the value of $...

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Q JEE MAIN 2021

Let the plane $a x+b y+c z+d=0$ bisect the line joining the points $(4,-3,1)$ and $(2,3,-5)$ at the right angles....

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Q JEE MAIN 2021

The number of times the digit 2 will be written when listing the integers from 1 to 1000 is

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Q JEE MAIN 2021

The value of $3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\infty}}}}$ is equal to

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Q JEE MAIN 2021

The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3...

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Q JEE MAIN 2021

Let $A+2 B=\left[\begin{array}{ccc}1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1\end{array}\right]$ and...

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Q JEE MAIN 2021

If $\quad f(x)=\left\{\begin{array}{cl}\frac{1}{|x|} ; & |x| \geq 1 \\ a x^2+b ; & |x|

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Q JEE MAIN 2021

$\frac{1}{3^2-1}+\frac{1}{5^2-1}+\frac{1}{7^2-1}+\ldots \ldots+\frac{1}{(201)^2-1}$ is equal to

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Q JEE MAIN 2021

The real valued function $f(x)=\frac{\operatorname{cosec}^{-1} x}{\sqrt{x-[x]}}$, where $[\mathrm{x}]$ denotes the greatest integer less than or equal to x , is...

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Q JEE MAIN 2021

If $\alpha, \beta$ are natural numbers such that $100^\alpha-199 \beta=$ (100) $(100)+(99)(101)+(98)(102)+\ldots . .+(1)$ (199), then the slope of the...

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Q JEE MAIN 2021

For the four circles $\mathrm{M}, \mathrm{N}, \mathrm{O}$ and P , following four equations are given : Circle $\mathrm{M}: \mathrm{x}^2+\mathrm{y}^2=1$ Circle...

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Q JEE MAIN 2021

If the equation $\mathrm{a}|\mathrm{z}|^2+\overline{\alpha \overline{\mathrm{z}}+\alpha \overline{\mathrm{z}}}+\mathrm{d}=0$ represents a circle where $a, d$ are real constant when which of the following...

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Q JEE MAIN 2021

A vector $\vec{a}$ has components $3 p$ and 1 with respect to a rectangular cartesian system. This system is rotated...

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Q JEE MAIN 2021

If $\lim _{x \rightarrow 0} \frac{\sin ^{-1} x-\tan ^{-1} x}{3 x^2}$ is equal to $L$, then the value of $...

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Q JEE MAIN 2021

The equation of one of the straight lines which passes through the point $(1,3)$ and makes an angles $...

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Q JEE MAIN 2021

The integral $\int \frac{(2 x-1) \cos \sqrt{(2 x-1)^2+5}}{\sqrt{4 x^2-4 x+6}} d x$ is equal to (where c is a constant...

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Q JEE MAIN 2021

Let $\alpha, \beta, \gamma$ be the real roots of the equation, $\mathrm{x}^2+\mathrm{ax}^2 +b x+c=0,(a, b, c \in R$ and $...

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Q JEE MAIN 2021

Choose the correct statement about two circles whose equation are given below : $...

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Q JEE MAIN 2021

The solutions of the equation

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Q JEE MAIN 2021

Let $\left(1+x+2 x^2\right) 20=a_0+a_1 x+a_2 x_2+\ldots+a_{40} x^{40}$. then $\mathbf{a}_1+\mathbf{a}_3+\mathbf{a}_5+\ldots+\mathbf{a}_{37}$ is equal to

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Q JEE MAIN 2021

The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y...

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Q JEE MAIN 2021

The differential equation satisfied by the system of parabolas $y^2=4 a(x+a)$ is :

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Q JEE-MAIN 2021

Let $z$ and $\omega$ be two complex numbers such that $\omega=z \bar{z}-2 z+2,\left|\frac{z+i}{z-3 i}\right|=1$ and $\operatorname{Re}(\omega)$ has minimum value. Then,...

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Q JEE-MAIN 2021

Let the curve $y=y(x)$ be the solution of the differential equation, $\frac{d y}{d x}=2(x+1)$. If the numerical value of area...

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Q JEE-MAIN 2021

If the normal to the curve $y(x)=\int_0^x\left(2 t^2-15 t+10\right) d t$ at a point $(a, b)$ is parallel to the...

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Q JEE-MAIN 2021

Let

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Q JEE-MAIN 2021

The total number of 3 × 3 matrices A having enteries from the set (0, 1, 2, 3) such that...

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Q JEE-MAIN 2021

If $\lim _{x \rightarrow 0} \frac{a e^x-b \cos x+c e^{-x}}{x \sin x}=2$, then $a+b+c$ is equal to $\_\_\_\_$ .

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Q JEE-MAIN 2021

Let A B C D be a square of side of unit length. Let a circle $C_1$ centered at $A$...

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Q JEE-MAIN 2021

Let $f:(0,2) \rightarrow \mathbb{R}$ be defined as

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Q JEE-MAIN 2021

Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26,...

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Q JEE MAIN 2021

Let $S_k=\sum_{r=1}^k \tan ^{-1}\left(\frac{6^r}{2^{2 r+1}+3^{2 r+1}}\right)$. Then $\lim _{k \rightarrow \infty} S_k$ is equal to :

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Q JEE MAIN 2021

The number of roots of the equation, $$ (81)^{\sin ^2 x}+(81)^{\cos ^2 x}=30 $$ in the interval $[0, \pi]$ is...

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Q JEE MAIN 2021

The locus of the midpoints of the chord of the circle, $x^2+y^2=25$ which is tangent to the hyperbola, $\frac{x^2}{9}-\frac{y^2}{16}=1$ is...

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Q JEE MAIN 2021

If $y=y(x)$ is the solution of the differential equation, $\frac{d y}{d x}+2 y \tan x=\sin x, y\left(\frac{\pi}{3}\right)=0$, then the maximum...

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Q JEE MAIN 2021

Let $[x]$ denote greatest integer less than or equal to $x$. If for $n \in \mathbb{N},\left(1-x+x^3\right)^n=\sum_{j=0}^{3 n} a_j x^j m$...

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Q JEE MAIN 2021

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The...

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Q JEE MAIN 2021

The range of $a \in \mathbb{R}$ for which the function $$ f(x)=(4 a-3)\left(x+\log _e 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^2\left(\frac{x}{2}\right) $$...

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Q JEE MAIN 2021

Let $P$ be a plane $l x+m y+n z=0$ containing the line, $\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}$. IF plane $P$ divides the line segment...

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Q JEE MAIN 2021

If $n$ is the number of irrational terms in the expansion of $\left(3^{1 / 4}+5^{1 / 8}\right)^{60}$, then $(n-1)$ is...

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Q JEE MAIN 2021

Let a complex number $\mathrm{z},|\mathrm{z}| \neq 1$, satisfy $\log _{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1)^2}\right) \leq 2$. Then, the largest value of $|z|$ is equal...

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Q JEE MAIN 2021

Which of the following Boolean expression is a tautology ?

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Q JEE MAIN 2021

Let the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be defined as

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Q JEE MAIN 2021

If $\sqrt{3}\left(\cos ^2 x\right)=(\sqrt{3}-1) \cos x+1$, the number of solutions of the given equation when $\mathrm{x} \in\left[0, \frac{\pi}{2}\right]$ is

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Q JEE MAIN 2021

Let the position vectors of two points $P$ and $Q$ be $3 \hat{i}+\hat{j}+2 \hat{k}$ and $\hat{i}+2 \hat{j}-4 \hat{k}$, respectively. Let...

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Q JEE MAIN 2021

If the three normasl drawn to the prabola, $y^2=2 x$ pass through the point $(a, 0) a \neq 0$, then...

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Q JEE MAIN 2021

The value of the integral $\int_0^\pi|\sin 2 x| d x$ is

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Q JEE MAIN 2021

Let $A=\left[\begin{array}{cc}i & -i \\ -i & i\end{array}\right], i=\sqrt{-1}$ Then, the system of linear equation $A^8=\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}8 \\ 64\end{array}\right]$...

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Q JEE MAIN 2021

The area bounded by the lines y = ||x – 1| – 2| is

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Q JEE MAIN 2021

If for $x \in\left(0, \frac{\pi}{2}\right), \log _{10} \sin x+\log _{10} \cos x=-1$ and $\log _{10}(\sin x+\cos x)=\frac{1}{2}\left(\log _{10} n-1\right), n>0$,...

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Q JEE MAIN 2021

Let $(\lambda, 2,1)$ be a point on the plane which passes through the point $(4,-2,2)$. If the plane is perpendicular...

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Q JEE MAIN 2021

Consider three observations a, b and c such that b = a + c. If the standard deviation of a...

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Q JEE MAIN 2021

If $y=y(x)$ is the solution of the equation $...

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Q JEE MAIN 2021

Let $m, n \in N$ and $\operatorname{gcd}(2, n)=1$, If $$ 30\binom{30}{0}+29\binom{30}{1}+\ldots+2\binom{30}{28}+1\binom{30}{29}=n \cdot 2^m $$ then $n+m$ is equal to (Here...

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Q JEE MAIN 2021

The sum of 162th power of the roots of equation $x^3-2 x^2+2 x-1=0$ is

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Q JEE MAIN 2021

The number of solutions of the equation $\log _4(x-1)=\log _2(x-3)$ is

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Q JEE MAIN 2021

If for $\mathrm{a}>0$, the feet of perpendiculars from the points $\mathrm{A}(\mathrm{a},-2 \mathrm{a}, 3)$ and $\mathrm{B}(0,4,5)$ on the plane $l \mathrm{x}+\mathrm{my}+\mathrm{nz}=0$...

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Q JEE MAIN 2021

The number of integral values of ' k ' for which the equation $3 \sin x+4 \cos x=k+1$ has a...

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Q JEE MAIN 2021

The difference between degree and order of a differential equation that represents the family of curves given by $y^2=a\left(x+\frac{\sqrt{a}}{2}\right), a>0$...

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Q JEE MAIN 2021

Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3} \hat{i}+\hat{j}$ by an angle $45^{\circ}$ about the...

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Q JEE MAIN 2021

Let R = {(P,Q) | P and Q are at the same distance from the origin} be a relation, then...

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Q JEE MAIN 2021

The value of $\int_{-\pi / 2}^{\pi / 2} \frac{\cos ^2 x}{1+3 x} d x$ is

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Q JEE MAIN 2021

The value of $$ \left|\begin{array}{lll} (a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4...

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Q JEE MAIN 2021

Consider the three planes $$ \begin{aligned} & P_1: 3 x+15 y+21 z=9 \\ & P_2: x \cdot 3 y \cdot...

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Q JEE MAIN 2021

The triangle of maximum area that can be inscribed in a given circle of radius ' r ' is :

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Q JEE MAIN 2021

The intersection of three lines x – y = 0, x + 2y = 3 and 2x + y =...

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Q JEE MAIN 2021

Let $f(x)=\sin ^{-1} x$ and $g(x)=\frac{x^2-x-2}{2 x^2-x-6}$. If $g(2)=\lim _{x \rightarrow 2} g(x)$, then the domain of the function fog...

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Q JEE MAIN 2021

The value of ${ }^{-15} C_1+2 .{ }^{15} C_2-3 .{ }^{15} C_3+\ldots \ldots$. $...

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Q JEE MAIN 2021

The maximum slope of the curve $y=\frac{1}{2} x^4-5 x^3+ 18 x^2-19 x$ occurs at the point

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A natural number has prime factorization given by $\mathrm{n}=2^{\mathrm{x}} 3^{\mathrm{y}} 5^{\mathrm{z}}$, where y and z are such that $\mathrm{y}+\mathrm{z}=5$ and...

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Q JEE MAIN 2021

Let f be any function defined on R and let it satisfy the condition : $$ |f(x)-f(y)| \leq\left|(x-y)^2\right|, \forall(x, y)...

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Q JEE MAIN 2021

For $x>0$, if $f(x)=\int_1^x \frac{\log _e t}{(1+t)} d t$ then $f(e)+f\left(\frac{1}{e}\right)$ is equal to

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Q JEE MAIN 2021

The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits...

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Q JEE MAIN 2021

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Q JEE MAIN 2021

Let $f: R \rightarrow R$ be defined as $$ f(x)=\left\{\begin{array}{cc} 2 \sin \left(-\frac{\pi x}{2}\right), & \text { if } x1...

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Q JEE MAIN 2021

If $\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2 x}} d x=a \sin ^{-1}\left(\frac{\sin x+\cos x}{b}\right)+c$, where $c$ is a constant of integration,...

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Q JEE MAIN 2021

If the tangent to the curve $y=x^3$ at the point $P\left(t, t^3\right)$ meets the curve again at $Q$, then the...

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Q JEE MAIN 2021

Let $A=\{1,2,3, \ldots, 10\}$ and $f: A \rightarrow A$ be defined as $...

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Q JEE MAIN 2021

If vectors $\vec{a}_1=x \hat{i}-\hat{j}+\hat{k}$ and $\vec{a}_2=\hat{i}+y \hat{j}+z \hat{k}$ are collinear, then a possible unit vector parallel to the vector $...

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Q JEE MAIN 2021

If $(1,5,35),(7,5,5),(1, \lambda, 7)$ and $(2 \lambda, 1,2)$ are coplanar, then the sum of all possible values of $\lambda$ is

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Q JEE MAIN 2021

The rate of growth of bacteria in a culture is proportional to the number of bacteris present and the bacteria...

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The maximum value of the term independent of ' t ' in the expansion of $\left(t^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{t}\right)^{10}$ where $x \in (0,1)$...

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Q JEE MAIN 2021

The value of $\lim _{h \rightarrow 0} 2\left\{\frac{\sqrt{3} \sin \left(\frac{\pi}{6}+h\right)+\cos \left(\frac{\pi}{6}+h\right)}{\sqrt{3} h(\sqrt{3} \cos h-\sin h)}\right\}$ is

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Q JEE MAIN 2021

The distance of the point $(1,1,9)$ from the point of intersection of the line $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}$ and the plane $x+y+z=17$ is

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Q JEE MAIN 2021

The sum of the infinite series $1+\frac{2}{3}+\frac{7}{3^2}+\frac{12}{3^3}+\frac{17}{3^4}+\frac{22}{3^5}+\ldots$ is equal to

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Q JEE MAIN 2021

In the circle given below, let $\mathrm{OA}=1$ unit, $\mathrm{OB}=13$ unit and $\mathrm{PQ} \perp \mathrm{OB}$. Then, the area of the triangle...

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Q JEE MAIN 2021

If $f: \mathrm{R} \rightarrow \mathrm{R}$ is a function defined by $f(\mathrm{x})=[\mathrm{x}-1] \cos \left(\frac{2 \mathrm{x}-1}{2}\right) \pi$, where $[$.$...

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Q JEE MAIN 2021

In a increasing geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of...

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Q JEE MAIN 2021

The system of linear equations 3x – 2y – kz = 10 2x – 4y – 2z = 6 x...

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Q JEE MAIN 2021

Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of...

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Q JEE MAIN 2021

The locus of the point of intersection of the lines $(\sqrt{3}) k x+k y-4 \sqrt{3}=0$ and $\sqrt{3 x}-y-4(\sqrt{3}) k=0$ is...

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Q JEE MAIN 2021

If the system of equations kx + y + 2z = 1 3x – y – 2z = 2 –2x...

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Q JEE MAIN 2021

The population $P=P(t)$ at time ' $t$ ' of a certain species follows the differential equation $\frac{d P}{d t}=0.5 P-450$....

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Q JEE MAIN 2021

The number of elements in the set $\{x \in \mathbb{R}:(|x|-3)|x+4|=6\}$ is equal to

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Q JEE MAIN 2021

A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to...

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Q JEE MAIN 2021

Let $\vec{a}=\hat{i}+2 \hat{j}-\hat{k}, \vec{b}=\hat{i}-\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $...

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Q JEE MAIN 2021

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3,...

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Q JEE MAIN 2021

The equation of the plane passing through the point (1, 2, –3) and perpendicular to the planes 3x + y...

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Q JEE MAIN 2021

If $A=\left[\begin{array}{cc}0 & -\tan \left(\frac{\theta}{2}\right) \\ \tan \left(\frac{\theta}{2}\right) & 0\end{array}\right]$ and $\left(I_2+A\right)\left(I_2-A\right)^{-1}= \left[\begin{array}{cc}a & -b \\ b & a\end{array}\right]$, then...

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Q JEE MAIN 2021

If $\vec{a} \quad$ and $\vec{b}$ are perpendicular, then $\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})))$ is equal to.

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Q JEE MAIN 2021

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line...

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Q JEE MAIN 2021

Let $A=\left[\begin{array}{lll}x & y & z \\ y & z & x \\ z & x & y\end{array}\right]$, where $...

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Q JEE MAIN 2021

Let $\mathrm{A}_1, \mathrm{~A}_2, \mathrm{~A}_3$, $\_\_\_\_$ be squares such that for each $n \geq 1$, the length of the side of...

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Q JEE MAIN 2021

The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points...

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Q JEE MAIN 2021

The statement among the following that is a tautology is :

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Q JEE MAIN 2021

The number of points, at which of function $f(x)=|2 x+1|-3|x+2|+\left|x^2+x-2\right|, x \in R$ is not differentiable, is $\_\_\_\_$

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Q JEE MAIN 2021

let $f(x)$ be polynomial of degree 6 in $x$, in which the coefficient of $x^6$ is unity and it has...

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Q JEE MAIN 2021

If Rolle's theorem holds for the function $f(x)=x^3-a x^2+b x-4, x \in[1,2]$ with $f^{\prime}\left(\frac{4}{3}\right)=0$, then ordered pair $(a, b)$ is...

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Q JEE MAIN 2021

The statement $A \rightarrow(B \rightarrow A)$ is equivalent to :

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Q JEE MAIN 2021

If a curve passes through the origin and the slope of the tangent to it at any point $(x, y)$...

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Q JEE MAIN 2021

The integer ' $k$ ' for which the inequality $x^2-2(3 k-1) x+8 k^2-7>0$ is valid for every $x$ in $R$,...

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Q JEE MAIN 2021

The total number of positive integral solutions (x, y, z) such that xyz = 24 is :

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Q JEE MAIN 2021

The coefficients $a, b$ and $c$ of the quadratic equation, $a x^2+b x+c=0$ are obtained by throwing a dice three...

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Q JEE MAIN 2021

$\lim _{n \rightarrow \infty}\left(1+\frac{1+\frac{1}{2}+\ldots \ldots+\frac{1}{n}}{n^2}\right)^n$ is equal to :

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Q JEE MAIN 2021

If the curves, $\frac{x^2}{a}+\frac{y^2}{b}=1$ and $\frac{x^2}{c}+\frac{y^2}{d}=1$ intersect each other at an angle of $90^{\circ}$, then which of the following relations...

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Q JEE MAIN 2021

The image of the point (3, 5) in the line x – y + 1 = 0, lies on :

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Q JEE MAIN 2021

Let the lines $(2-i) z=(2+i) \bar{z}$ and $(2+i) z+(i-2) \bar{z}-4 i=0$, (here $i^2=-1$ ) be normal to a circle $C$....

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Q JEE MAIN 2021

All possible values of $\theta \in[0,2 \pi]$ for which $\sin 2 \theta+\tan 2 \theta>0$ lie in :

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Q JEE MAIN 2021

A tangent is drawn to the parabola $y^2=6 x$ which is perpendicular to the line $2 x+y=1$. Which of the...

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Q JEE MAIN 2021

The value of $\int_{-1}^1 x^2 e^{\left[x^3\right]} d x$ where $[t]$ denotes the greatest integer $\leq t$, is:

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Q JEE MAIN 2021

The value of the integral $...

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Q JEE MAIN 2021

Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l+\mathrm{m}-\mathrm{n}=0$ and $l^2+\mathrm{m}^2-\mathrm{n}^2 =0$. Then then...

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Q JEE MAIN 2021

The equation of the line through the point $(0,1,2)$ and perpendicular to the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is:

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Q JEE MAIN 2021

Let $f, g: N \rightarrow N$ such that $f(n+1)=f(n)+f(1) \forall n \in N$ and $g$ be any arbitrary function. Which...

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Q JEE MAIN 2021

If $0

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Q JEE MAIN 2021

When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}$ and the probability that...

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Q JEE MAIN_2022

Let Q and R be two points on the line $\frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2}$ at a distance $\sqrt{26}$ from the point $P(4,2,7)$. Then...

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Q JEE MAIN_2022

Let the function $f(x)=2 x^2-$ logex, $x>0$, be decreasing in (0, a) and increasing in (a, 4). A tangent to...

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Q JEE MAIN_2022

The equations of the sides $A B, B C$ and $C A$ of a triangle $A B C$ are $...

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Q JEE MAIN_2022

Let a curve $y=y(x)$ pass through the point $(3,3)$ and the area of the region under this curve, above the...

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Q JEE MAIN_2022

1`$x$ and $x^2$ in the expansion of $(1+x)^p(1-x)^q, p, q \leq 15$, are -3 and -5 respectively, then the coefficient...

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Q JEE MAIN_2022

The number of distinct real roots of the equation $x^5\left(x^3-x^2-x+1\right)+x\left(3 x^3-4 x^2-2 x+4\right)-1=0$ is

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Q JEE MAIN_2022

The series of positive multiples of 3 is divided into sets : {3}, {6, 9,12}, {15, 18, 21, 24, 27},......

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Q JEE MAIN_2022

The number of 5-digit natural numbers, such that the product of their digits is 36, is

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Q JEE MAIN_2022

If for some p, q, r in R, not all have same sign, one of the roots of the equation...

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Q JEE MAIN_2022

The statement $(\sim(p \Leftrightarrow \sim q)) \wedge q$ is:

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Q JEE MAIN_2022

$\tan \left(2 \tan ^{-1} \frac{1}{5}+\sec ^{-1} \frac{\sqrt{5}}{2}+2 \tan ^{-1} \frac{1}{8}\right)$ is equal to:

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Q JEE MAIN_2022

Let $S=\left\{\theta \in[0,2 \pi]: 8^{2 \sin ^2 \theta}+8^{20 \cos ^2 \theta}=16\right\}$. Then $n(S)+\sum_{\theta \in S}\left(\sec \left(\frac{\pi}{4}+2 \theta\right) \operatorname{cosec}\left(\frac{\pi}{4}+2 \theta\right)\right)$ is...

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Q JEE MAIN_2022

Let $E_1, E_2, E_3$ be three mutually exclusive events such that $P\left(E_1\right)=\frac{2+3 p}{6}, P\left(E_2\right)=\frac{2-p}{8}$ and $P\left(E_3\right)=\frac{1-p}{2}$. If the maximum and...

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Q JEE MAIN_2022

The mean and variance of a binomial distribution are $\alpha$ and $\frac{\alpha}{3}$ respectively. If $P(X=1)=\frac{4}{243}$, then $P(X=4$ or 5) is...

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Q JEE MAIN_2022

Let $\vec{a}=\alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0$. If the projection of $\vec{a} \times \vec{b}$ on the vector $...

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Q JEE MAIN_2022

The length of the perpendicular from the point (1, –2, 5) on the line passing through (1, 2, 4) and...

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Q JEE MAIN_2022

Let the tangent drawn to the parabola $\mathrm{y}^2=24 \mathrm{x}$ at the point $(\alpha, \beta)$ is perpendicular to the line $...

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Q _JEE MAIN_2022

A point P moves so that the sum of squares of its distances from the points (1, 2) and (–2,...

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Q JEE MAIN_2022

If $\frac{d y}{d x}+2 y \tan x=\sin x, 0

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Q JEE MAIN_2022

If $a=\lim _{n \rightarrow \infty} \sum_{k=1}^n \frac{2 n}{n^2+k^2}$ and $f(x)=\sqrt{\frac{1-\cos x}{1+\cos x}}, x \in(0,1)$, then :

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Q JEE MAIN 2022

The line of shortest distance between the lines $\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}$ and $\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}$ makes an angle of $\cos ^{-1}\left(\sqrt{\frac{2}{27}}\right)$ with the plane...

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Q JEE MAIN_2022

Let $f(x)=\left\{\begin{array}{ll}x^3-x^2+10 x-7 & , x \leq 1 \\ -2 x+\log _2\left(b^2-4\right) & , x>1\end{array}\right.$. Then the set of all...

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Q JEE MAIN_2022

If $f(x)=\left\{\begin{array}{cc}x+a, & x \leq 0 \\ |x-4|, & x>0\end{array}\right.$ and $g(x)=\left\{\begin{array}{cc}x+1, & x

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Q JEE MAIN 2022

Let the equation of two diameters of a circle $x^2+y^2-2 x+2 f y+1=0$ be $2 p x-y=1$ and $...

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Q JEE MAIN_2022

If the function $f(x)=\left\{\frac{\log e\left(1-x+x^2\right)+\log e\left(1+x+x^2\right)}{\sec x-\cos x}, x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)-\{0\}\right.$ is continuous at $x=0$, then $k$ is equal to:...

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Q JEE MAIN 2022

If $\lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)]=33 . \lim _{n \rightarrow \infty} \frac{1}{n^{k+1}}\left[1^k+2^k+3^k+\ldots . .+n^k\right]$, then the integral...

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Q JEE MAIN_2022

Consider two G.Ps. $2,2^2, 2^3$, ann and $4,4^2, 4^3$ of 60 and $n$ terms respectively. If the geometric mean of...

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Q JEE MAIN 2022

Let $...

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Q JEE MAIN 2022

Let $a_1=b_1=1, a_n=a_{n-1}+2$ and $b_n=a_n+b_{n-1}$ for every natural number $n \geq 2$. Then $\sum_{n=1}^{15} a_n \cdot b_n$ is equal to$\_\_\_\_$...

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Q JEE MAIN_2022

The odd natural number $a$, such that the area of the region bounded by $y=1, y=3, x \mid=0, x=y^a$ is...

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Q JEE MAIN 2022

Let $a, b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $...

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Q JEE MAIN 2022

If the maximum value of the term independent of $t$ in the expansion of $\left(t^2 x^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{t}\right)^{15}, x \geq 0$ is...

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Q JEE MAIN_2022

Let A be a 2 × 2 matrix with det (A) = –1 and det ((A + I) (Adj (A)...

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Q JEE MAIN 2022

The letters of the word ‘MANKIND’ are written in all possible orders and arranged in serial order as in an...

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Q JEE MAIN_2022

If the system of linear equations. $$ \begin{aligned} & 8 x+y+4 z=-2 \\ & x+y+z=0 \\ & \lambda x-3 y=\mu...

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Q JEE MAIN 2022

Let $A=\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right]$ and $B=A-I$....

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Q JEE MAIN 2022

Which of the following statements is a tautology?

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Q JEE MAIN_2022

Let $O$ be the origin and $A$ be the point $z 1=1+2 i$. If $B$ is the point $...

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Q JEE MAIN 2022

A tower $P Q$ stands on a horizontal ground with base $Q$ on the ground. The point $R$ divides the...

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Q JEE MAIN 2022

The number of solutions of $|\cos x|=\sin x$, such that $-4 \pi \leq x \leq 4 \pi$ is in

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Q JEE MAIN 2022

Let $f: R \rightarrow R$ be a continuous function such that $f(3 x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to:

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Q JEE MAIN 2022

If the numbers appeared on the two throws of a fair six faced die are $a$ and $b$, then the...

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Q JEE MAIN 2022

If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then...

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Q JEE MAIN 2022

Let ABC be a triangle such that $\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{CA}}=\overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{c}},|\overrightarrow{\mathrm{a}}|=6 \sqrt{2},|\overrightarrow{\mathrm{~b}}|=2 \sqrt{3}$ and $\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=12$ Consider the statements: $...

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Q JEE MAIN 2022

Let P be the plane containing the straight line $\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}$ and perpendicular to the plane containing the straight lines $\frac{x}{2}=\frac{y}{3}=\frac{z}{5}$...

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Q JEE MAIN 2022

Let the locus of the centre $(\alpha, \beta), \beta>0$, of the circle which touches the circle $x^2+(y-1)^2=1$ externally and also...

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Q JEE MAIN 2022

A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x-y-2=0$ at the...

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Q JEE MAIN 2022

The general solution of the differential equation $\left(x-y^2\right) d x+y\left(5 x+y^2\right) d y=0$ is :

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Q JEE MAIN 2022

The slope of the tangent to a curve $C: y=y(x)$ at any point $[x, y)$ on it is $...

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Q JEE MAIN 2022

For any real number $x$, let $[x]$ denote the largest integer less than equal to $x$. Let $f$ be a...

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Q JEE MAIN 2022

The area of the region given by $A=\left\{(x, y): x^2 \leq y \leq \min \{x+2,4-3 x\}\right\}$ is i.

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Q JEE MAIN 2022

The curve $y(x)=a x^3+b x^2+c x+5$ touches the $x$-axis at the point $P(-2,0)$ and cuts the $y$-axis at the point...

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Q JEE MAIN 2022

If the absolute maximum value of the function $f(x)=\left(x^2-2 x+7\right) e^{\left(4 x^3-12 x^2-180 x+31\right)}$ in the interval $[-3,0]$ is $f(\alpha)$,...

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Q JEE MAIN 2022

If $\lim _{n \rightarrow \infty}\left(\sqrt{n^2-n-1}+n \alpha+\beta\right)=0$ then $8(\alpha+\beta)$ is equal to :

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Q JEE MAIN 2022

The number of $\theta \in(0,4 \pi)$ for which the system of linear equations $...

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Q JEE MAIN 2022

For $\mathrm{n} \in \mathrm{N}$, let $\mathrm{S}_{\mathrm{n}}=\left\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}-3+2 \mathrm{i}|=\frac{\mathrm{n}}{4}\right\}$ and $\mathrm{T}_{\mathrm{n}}=\left\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}-2+3 \mathrm{i}|=\frac{1}{4}\right\}$. Then the number of elements in...

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Q JEE MAIN 2022

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^4+x^3+x^2+x+1=0$, then $\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}$ is equal to

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Q JEE MAIN 2022

The total number of functions, $f:\{1,2,3,4\} \cdot\{1,2,3,4,5,6\}$ such that $f(1)+f(2)=f(3)$, is equal to :

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Q MEMORY BASED QUESTION 2026

If $A=\left[\begin{array}{ll}\alpha & 2 \\ 1 & 2\end{array}\right], B=\left[\begin{array}{ll}1 & 1 \\ \beta & 1\end{array}\right]$ and $...

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Q Memory Based Question 2026

The value of $\operatorname{cosec} 10^{\circ}-\sqrt{3} \sec 10^{\circ}$

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Q JEE MAIN 2022

A transformer operating at primary voltage 8 kV and secondary voltage 160 V serves a load of 80 kW. Assuming...

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Q JEE MAIN 2022

Let $S=\left\{E, E_2 \ldots E_8\right\}$ be a sample space of random experiment such that $P\left(E_n\right)=\frac{n}{36}$ for every $...

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Q JEE MAIN 2022

Let a circle $C$ of radius 5 lie below the $x$-axis. The line $L_1=4 x+3 y-2$ passes through the centre...

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Q JEE MAIN 2022

If the area of the region $\left\{(x, y): x^{\frac{2}{3}}+y^{\frac{2}{3}} \leq 1, x+y \leq 0, y \geq 0\right\}$ is $A$, then...

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Q JEE MAIN 2022

If $y(x)=\left(x^{x^x}\right), x>0$ then $\frac{d^2 x}{d y^2}+20$ at $x=1$ is equal to :

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Q JEE MAIN 2022

Let $[t]$ denote the greatest integer $\leq t$ and $\{t\}$ denote the fractional part of $t$. Then integral value of...

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Q JEE MAIN 2022

If the sum of the coefficients of all the positive powers of $x$, in the binomial expansion of $\left(x^n+\frac{2}{x^5}\right)^7$ is...

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Q JEE MAIN 2022

Let A be a matrix of order 2 × 2, whose entries are from the set {0, 1, 2, 3,...

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Q JEE MAIN 2022

Let $\alpha, \beta$ be the roots of the equation $x^2-4 \lambda x+5=0$ and $\alpha, \gamma$ be the roots of the...

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Q JEE MAIN 2022

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$. Define $$ f: S \rightarrow S \text { as } f(n)=\left\{\begin{array}{cc} 2 n, & \text { if }...

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Q JEE MAIN 2022

Which of the following statement is a tautology?

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Q JEE MAIN 2022

$\alpha=\sin 36^{\circ}$ is a root of which of the following equation

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Q JEE MAIN 2022

The value of $\cot \left(\sum_{n=1}^{50} \tan ^{-1}\left(\frac{1}{1+n+n^2}\right)\right)$ is :

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Q JEE MAIN 2022

Let $f(x)$ and $g(x)$ be two real polynomials of degree 2 and 1 respectively. If $f(g(x))=8 x^2-2 x$, and $...

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Q JEE MAIN 2022

If a point $A(x, y)$ lies in the region bounded by the $y$-axis, straight lines $2 y+x=6$ and $...

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Q JEE MAIN 2022

Let $M=\left[\begin{array}{cc}0 & -\alpha \\ \alpha & 0\end{array}\right]$, where $\alpha$ is a non-zero real number an $N=\sum_{k=1}^{49} M^{2 k}$. If...

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Q JEE MAIN 2022

The mean and variance of the data $4,5,6,6,7,8, x, y$ where $x

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Q JEE MAIN 2022

Let $\vec{a}$ and $\vec{b}$ be the vectors along the diagonal of a parallelogram having area $2 \sqrt{2}$. Let the angle...

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Q JEE MAIN 2022

The maximum number of compound propositions, out of $...

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Q JEE MAIN 2022

The shortest distance between the lines $\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1}$ and $\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3}$ is :

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Q JEE MAIN 2022

Let $S=\{1,2,3,4\}$. Then the number of elements in the set $\{f: S \times S \rightarrow S: f$ is onto and...

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Q JEE MAIN 2022

The total number of four digit numbers such that each of the first three digits is divisible by the last...

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Q JEE MAIN 2022

Sum of squares of modulus of all the complex numbers $z$ satisfying $\bar{z}=i z^2+z^2-z$ is equal to

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Q JEE MAIN 2022

Given $\pi a^2-\pi a b=30 \pi$ and $\pi a b-\pi b^2=18 \pi$ on subtracting, we get $(a-b)^2=a^2-2 a b+b^2=12$ Let...

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Q JEE MAIN 2022

Let $A=\left(\begin{array}{cc}1+i & 1 \\ -i & 0\end{array}\right)$ where $i=\sqrt{-1}$. Then, the number of elements in the set $...

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Q JEE MAIN 2022

If the system of linear equations $2 x-3 y=\gamma+5, \alpha x+5 y=\beta+1$ where $\alpha, \beta, \gamma \in \mathbf{R}$ has infinitely...

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For real numbers a, b (a > b > 0), let

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If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple...

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The number of solutions of the equation $\sin x=\cos ^2 x$ in the interval $(0,10)$ is $\_\_\_\_$ .

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Let for $\mathrm{n}=1,2, \ldots, 50, \mathrm{~S}_{\mathrm{n}}$ be the sum of the infinite geometric progression whose first term is $\mathrm{n}^2$ and...

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Let a line $L_1$ be tangent to the hyperbola $\frac{x^2}{16}-\frac{y^2}{4}=1$ and let $L_2$ be the line passing through the origin...

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Let 3, 6, 9, 12, ... upto 78 terms and 5, 9, 13, 17, ... upto 59 terms be two...

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If $\lim _{x \rightarrow 1} \frac{\sin \left(3 x^2-4 x+1\right)-x^2+1}{2 x^3-7 x^2+a x+b}=-2$, then the value of $(a-b)$ is equal to

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Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with...

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Let $y=y(x), x>1$, be the solution of the differential equation $(x-1) \frac{d y}{d x}+2 x y=\frac{1}{x-1}$, with $y(2)=\frac{1+e}{2 e^4}$. If...

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The integral $\frac{24}{\pi} \int_0^{\sqrt{2}} \frac{\left(2-x^2\right) d x}{\left(2+x^2\right) \sqrt{4+x^4}}$ is equal to

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If $a_1(>0), a_2, a_3, a_4, a_5$ are in a G.P., $a_2+a_4=2 a_3+1$ and $3 a_2+a_3=2 a_4$, then $a_2+a_4+2 a_6$ is...

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If one of the diameters of the circle $x^2+y^2-2 \sqrt{2} x-6 \sqrt{2} y+14=0$ is a chord of the circle $...

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Let $\vec{a}=\hat{i}-2 \hat{j}+3 \hat{k}, \vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}+(\vec{b} \times \vec{c})=\overrightarrow{0}$ and $\vec{b} \cdot \vec{c}=5$. Then,...

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Let the hyperbola H : $\frac{x^2}{a^2}-y^2=1$ and the ellipse E : $3 x^2+4 y^2=12$ be such that the length of...

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If $\left({ }^{40} \mathrm{C}_0\right)+\left({ }^{41} \mathrm{C}_1\right)+\left({ }^{42} \mathrm{C}_2\right)+\ldots .+\left({ }^{60} \mathrm{C}_{20}\right)=\frac{m}{n}{ }^{60} \mathrm{C}_{20}, m$ and $n$ are coprime, then $m+n$...

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Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62, and their...

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The total number of 3–digit numbers, whose greatest common divisor with 36 is 2, is ______ .

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Let $n \geq 5$ be an integer. If $9^n-8 n-1=64 \alpha$ and $6^n-5 n-1=25 \beta$, then $\alpha-\beta$ is equal to...

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In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4...

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Let the image of the point $P(1,2,3)$ in the line $L: \frac{x-6}{3}=\frac{y-1}{2}=\frac{z-2}{3}$ be $Q$. let $R(\alpha, \beta, \gamma)$ be a...

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Let $...

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If $\cot \alpha=1$ and $\sec \beta=-\frac{5}{3}$, where $\pi

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Let a circle C : $(x-h)^2+(y-k)^2=r^2, k>0$,, touch the x-axis at (1, 0). If the line x + y =...

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If $z^2+z+1=0, z \in \mathbb{C}$, then $\left|\sum_{n=1}^{15}\left(z^n+(-1)^n \frac{1}{z^n}\right)^2\right|$ is equal to

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Negation of the Boolean statement $(p \vee q) \Rightarrow((\sim r) \vee p)$ is equivalent to :

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Let $p$ and $q$ be two real numbers such that $p+q=3$ and $p^4+q^4=369$. Then $\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}$ is equal to . $\_\_\_\_$

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The area (in sq. units) of the region enclosed between the parabola $y^2$ = 2x and the line x +...

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Let the foot of the perpendicular from the point $(1,2,4)$ on the line $\frac{x+2}{4}=\frac{y-1}{2}=\frac{z+1}{3}$ be $P$. Then the distance of...

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Let $\vec{a}$ be a vector which is perpendicular to the vector $3 \hat{i}+\frac{1}{2} \hat{j}+2 \hat{k}$. If $...

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From the base of a pole of height 20 meter, the angle of elevation of the top of a tower...

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The set of values of $k$ for which the circle $C: 4 x^2+4 y^2-12 x+8 y+k=0$ lies inside the fourth...

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The remainder on dividing $1+3+3^2+3^3+\ldots+3^{2021}$ by 50 is $\_\_\_\_$

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The value of $\lim _{n \rightarrow \infty} 6 \tan \left\{\sum_{r=1}^n \tan ^{-1}\left(\frac{1}{r^2+3 r+3}\right)\right\}$

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The number of values of $a \in \mathbb{N}$ such that the variance of $3,7,12 a, 43-a$ is a natural number...

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The sum of all the elements of the set $\{\alpha \in\{1,2, \ldots, 100\}: \operatorname{HCF}(\alpha, 24)=1\}$ is $\_\_\_\_$

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The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to:

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The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2,...

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The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3,...

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Let the plane ax + by + cz = d pass through (2, 3, –5) and is perpendicular to the...

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Let $Q$ be the mirror image of the point $P(1,2,1)$ with respect to the plane $x+2 y+2 z=16$. Let $T$...

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If vertex of a parabola is (2, –1) and the equation of its directrix is 4x – 3y = 21,...

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The distance of the origin from the centroid of the triangle whose two sides have the equations $x-2 y+1=0$ and...

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Let $\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z+3}{-1}$ lie on the plane $p x-q y+z=5$, for some $p, q \in \mathbb{R}$. The shortest distance of the...

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Let a triangle $A B C$ be inscribed in the circle $x^2-\sqrt{2}(x+y)+y^2=0$ such that $\angle B A C=\frac{\pi}{2}$. If the...

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Let $P: y^2=4 a x, a>0$ be a parabola with focus $S$. Let the tangents to the parabola $P$ make...

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If $y=y(x)$ is the solution of the differential equation $\left(1+e^{2 x}\right) \frac{d y}{d x}+2\left(1+y^2\right) e^x=0$ and $y(0)=0$, then $...

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If $\int_0^2\left(\sqrt{2 x}-\sqrt{2 x-x^2}\right) d x=\int_0^1\left(1-\sqrt{1-y^2}-\frac{y^2}{2}\right) d y+\int_1^2\left(2-\frac{y^2}{2}\right) d y+1$

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Let $f$ be a real valued continuous function on $[0,1]$ and $f(x)=x+\int_0^1(x-t) f(t) d t$. Then which of the following...

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Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=(x-3)^{n_1}(x-5)^{n_2}, n_1, n_2 \in N$. The, which of the following...

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The value of $\lim _{x \rightarrow 1} \frac{\left(x^2-1\right) \sin ^2(\pi x)}{x^4-2 x^3+2 x-1}$ is equal to:

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Let $\mathrm{S}=\left\{\left(\begin{array}{cc}-1 & \mathrm{a} \\ 0 & \mathrm{~b}\end{array}\right) ; \mathrm{a}, \mathrm{b} \in\{1,2,3, \ldots 100\}\right\}$ and let $\mathrm{T}_{\mathrm{n}}=\left\{\mathrm{A} \in \mathrm{S}: \mathrm{A}^{\mathrm{n}(\mathrm{n}+1)}=\mathrm{I}\right\}$....

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The sum of the infinite series $1+\frac{5}{6}+\frac{12}{6^2}+\frac{22}{6^3}+\frac{35}{6^4}+\frac{51}{6^5}+\frac{70}{6^6}+\ldots .$. is equal to :

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Let $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfy $f(x+y)=2 x f(y)+4 y f(x), \forall x, y \in \mathbb{R}$. If $f(2)=3$, then 14...

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Let $\vec{a}=\alpha \hat{i}+2 \hat{j}-\hat{k}$ and $\vec{b}=-2 \hat{i}+\alpha \hat{j}+\hat{k}$, where $a \in \mathbf{R}$. If the area of the parallelogram whose adjacent...

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Let $S=\{z \in \mathbb{C}:|z-3| \leq$ and $z(4+3 i)+\bar{z}(4-3 i) \leq 24\}$. if $\alpha+\mathrm{i} \beta$ is the point in S which...

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Let $\mathrm{A}=\left(\begin{array}{cc}2 & -1 \\ 0 & 2\end{array}\right)$. If $\mathrm{B}=1-{ }^5 \mathrm{C}_1(\operatorname{adj} \mathrm{~A})+{ }^5 \mathrm{C}_2(\operatorname{adj} \mathrm{~A})^2-\ldots \ldots-{ }^5 \mathrm{C}_5(\operatorname{adj} \mathrm{~A})^5$,...

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Let $r \in\{p, q, \sim p, \sim q\}$ be such that the logical statement $...

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Let $\mathrm{a}>0, \mathrm{~b}>0$. Let e and $\ell$ respectively be the eccentricity and length of the latus rectum of the hyperbola...

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Let $\arg (z)$ represent the principal argument of the complex number $z$. The, $|z|=3$ and $\arg (z-1)-\arg (z+1) =\frac{\pi}{4}$ intersect:

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Let a triangle be bounded by the lines $L_{1 i} 2 x+5 y=10 ; L_{2 i}-4 x+3 y=12$ and the...

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Let $\alpha$ be a root of the equation $1+x^2+x^4=0$. Then the value of $\alpha^{1011}+\alpha^{2022}-\alpha^{3033}$ is equal to:

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Let the slope of the tangent to a curve $y=f(x)$ at $(x, y)$ be given by $2 \tan (\cos x-y)$....

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Let $l_1$ be the line in xy - plane with x and y intercepts $\frac{1}{8}$ and $\frac{1}{4 \sqrt{2}}$ respectively, and...

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If the inverse trigonometric functions take principal values, then $\cos ^{-1}\left(\frac{3}{10} \cos \left(\tan ^{-1}\left(\frac{4}{3}\right)\right)+\frac{2}{5} \sin \left(\tan ^{-1}\left(\frac{4}{3}\right)\right)\right)$ is equal to...

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Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be $\frac{5}{4}$. If the equation of the normal at the point $\left(\frac{8}{\sqrt{5}}, \frac{12}{5}\right)$...

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Let $\mathrm{x}=\mathrm{x}(\mathrm{y})$ be the solution of the differential equation $2 \mathrm{ye}^{\mathrm{x} / \mathrm{y}^2} \mathrm{dx}+\left(\mathrm{y}^2-4 \mathrm{xe}^{\mathrm{x} / \mathrm{y}^2}\right) \mathrm{dy}=0$ such that...

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Let $f(x)=\left\lfloor(x-1)\left(x^2-2 x-3\right)\right\rfloor+x-3, x \in R$. If $m$ and $M$ are respectively the number of points of local minimum and...

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The slope of normal at any point (x, y), x > 0, y > 0 on the curve y =...

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$16 \sin \left(20^{\circ}\right) \sin \left(40^{\circ}\right) \sin \left(80^{\circ}\right)$ is equal to :

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The total number of three-digit numbers, with one digit repeated exactly two times, is

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The area of the bounded region enclosed by the curve $y=3-\left|x-\frac{1}{2}\right|-|x+1|$ and the $x$-axis is

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The mean and standard deviation of 50 observations are 15 and 2 respectively. It was found that one incorrect observation...

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Consider the following statements : A : Rishi is a judge. B : Rishi is honest. C : Rishi is...

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Let $f: R \rightarrow R$ be continuous function satisfying $f(x)+f(x+k)=n$, for all $x \in R$ where $k>0$ and $n$ is...

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Let $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}, \vec{b}=2 \hat{i}-3 \hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$ be three given vectors. Let $\vec{v}$ be a vector in the plane...

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Let $\vec{b}=\hat{i}+\hat{j}+\lambda \hat{k}, \lambda \in R$. If $\vec{a}$ is a vector such that $\vec{a} \times \vec{b}=13 \hat{i}-\hat{j}-4 \hat{k}$ and $...

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If $y=\tan ^{-1}\left(\sec x^3-\tan x^3\right)$. $\frac{\pi}{2}$< $x^3$< $\frac{3 \pi}{2}$, then

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If the mean deviation about the mean of the numbers $1,2,3$, $\_\_\_\_$ $n$, where $n$ is odd, is $\frac{5(n+1)}{n}$, then...

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If the lines and $\vec{r}=(\hat{i}-\hat{j}+\hat{k})+\lambda(3 \hat{j}-\hat{k})$ and $\vec{r}=(\alpha \hat{i}-\hat{j})+\mu(2 \hat{i}-3 \hat{k})$ are co-planar, then distance of the plane containing these...

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Let $f: R \rightarrow R$ be a differentiable function such that $f\left(\frac{\pi}{4}\right)=\sqrt{2}, f\left(\frac{\pi}{2}\right)=0$ and $f\left(\frac{\pi}{2}\right)=1$ and let $...

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Let $f, g: R \rightarrow R$ be functions defined by $f(x)=\left\{\begin{array}{cc}{[x],} & x

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If the plane 2x + y – 5z = 0 is rotated about its line of intersection with the plane...

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If the sum of the coefficients of all the positive even powers of $x$ in the binomial expansion of $...

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If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the...

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The value of $b>3$ for which $12 \int_3^b \frac{1}{\left(x^2-1\right)\left(x^2-4\right)} d x=\log _e\left(\frac{49}{40}\right)$, is equal to

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The normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{9}=1$ at the point $(8,3 \sqrt{3})$ on it passes through the point :

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Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that $|(\hat{a}+\hat{b})+2(\hat{a} \times \hat{b})|=2$. If $\theta \in(0, \pi)$ is the angle...

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Let $f(x)=\left[2 x^2+1\right]$ and $g(x)=\left\{\begin{array}{ll}2 x-3, & x

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The locus of the mid point of the line segment joining the point (4, 3) and the points on the...

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The term independent of $x$ in the expression of $\left(1-x^2+3 x^3\right)\left(\frac{5}{2} x^3-\frac{1}{5 x^2}\right)^{11}, x \neq 0$ is

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Let $A=\left(\begin{array}{ll}2 & -2 \\ 1 & -1\end{array}\right)$ and $B=\left(\begin{array}{ll}-1 & 2 \\ -1 & 2\end{array}\right)$. Then the number of...

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Let the points on the plane P be equidistant from the points (.4, 2, 1) and (2, .2, 3). Then...

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The number of ways to distribute 30 identical candies among four children $\mathrm{C}_1, \mathrm{C}_2, \mathrm{C}_3$ and $\mathrm{C}_4$ so that $\mathrm{C}_2$...

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The line $y=x+1$ meets the ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$ at two points $P$ and $Q$. If $r$ is the radius of the...

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If m is the slope of a common tangent to the curves $\frac{x^2}{16}+\frac{y^2}{9}=1$ and $x^2+y^2=12$, then $12 m^2$ is equal...

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Let f(x) be a quadratic polynomial such that f(–2) + f(3) = 0. If one of the roots of f(x)...

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If the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda}$ and $\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5}$ is $\frac{1}{\sqrt{3}}$, then the sum of all possible values of...

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If the equation of the parabola, whose vertex is at $(5,4)$ and the directrix is $...

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If the solution of the differential equation $\frac{d y}{d x}+e^x\left(x^2-2\right) y=\left(x^2-2 x\right)\left(x^2-2\right) e^{2 x}$ satisfies y(0) = 0, then the...

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Let $\mathrm{R}_1=\{(\mathrm{a}, \mathrm{b}) \in \mathrm{N} \times \mathrm{N}:|\mathrm{a}-\mathrm{b}| \leq 13\}$ and $R_2=\{(a, b) \in N \times N:|a-b| \neq 13\}$. Then on...

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The value of $\tan ^{-1}\left(\frac{\cos \left(\frac{15 \pi}{4}\right)-1}{\sin \left(\frac{\pi}{4}\right)}\right)$ is equal to :

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If the solution curve of the differential equation $\left(\left(\tan ^{-1} y\right)-x\right) d y=\left(1+y^2\right) d x$ passes through the point $(1,0)$...

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If $y=y(x)$ is the solution of the differential equation $x \frac{d y}{d x}+2 y=x e^x, y(1)=0$, then the local maximum...

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The number of solutions of the equation cos $\left(x+\frac{\pi}{3}\right) \cos \left(\frac{\pi}{3}-x\right)=\frac{1}{4} \cos ^2 2 x, x \in[-3 \pi, 3 \pi]$...

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The integral $\int_0^1 \frac{1}{7^{\left[\frac{1}{x}\right]}} \mathrm{dx}$, where [.] denotes the greatest integer function is equal to :

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If the line $y=4+k x, k>0$, is the tangent to the parabola $y=x-x^2$ at the point $P$ and $V$ is...

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Let $\mathrm{b}_1 \mathrm{~b}_2 \mathrm{~b}_3 \mathrm{~b}_4$ be a 4 -element permutation with $\mathrm{b}_{\mathrm{i}} \in\{1,2,3, \ldots \ldots \ldots, 100\}$ for $...

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Let $f$ be a differentiable function in $\left(0, \frac{\pi}{2}\right)$. If $\int_{\cos x}^1 t^2 f(t) d t=\sin ^3 x+\cos x$ then...

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The negation of the Boolean expression $((\sim q) \wedge p) \Rightarrow((\sim p) \vee q)$ is logically equivalent to

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Let $P_1: \bar{r} .(2 \hat{i}+\hat{j}-3 \hat{k})=4$ be a plane. Let $P_2$ be another plane which passes through the points $(2,-3,2)(2$,...

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If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $...

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Let $c, k \in R$. If $f(x)=(c+1) x^2+\left(1-c^2\right) x+2 k$ and $f(x+y)=f(x)+f(y)-x y$, for all $x, y \in R$, then...

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If $a_1, a_2, a_3 \ldots$. and $b_1, b_2, b_3 \ldots$. are A.P. and $a_1,=2, a_{10}=3, a_1 b_1=1=a_{10} b_{10}$ then $...

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$50 \tan \left(3 \tan ^{-1}\left(\frac{1}{2}\right)+2 \cos ^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)+4 \sqrt{2} \tan \left(\frac{1}{2} \tan ^{-1}(2 \sqrt{2})\right)$ is equal to $\_\_\_\_$

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Let $S=2+\frac{6}{7}+\frac{12}{7^2}+\frac{20}{7^3}+\frac{30}{7^4}+\ldots$. then $4 S$ is equal to

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Q JEE MAIN 2022

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Q JEE MAIN 2022

If $\int \frac{1}{x} \sqrt{\frac{1-x}{1+x}} d x=g(x)+c, g(1)=0$, then $g\left(\frac{1}{2}\right)$ is equal to :

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Q JEE MAIN 2022

Let $A$ and $B$ be two $3 \times 3$ matrices such that $A B=I$ and $|A|=\frac{1}{8}$ then $|\operatorname{adj}(\operatorname{Badj}(2 A))|$ is...

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Q JEE MAIN 2022

The number of solutions of the equation $2 \theta-\cos ^2 \theta+\sqrt{2}=0$ is $R$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2022

The number of elements in the set $...

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Q JEE MAIN 2022

Let for some real numbers $\alpha$ and $\beta, \mathrm{a}=\alpha-\mathrm{i} \beta$. If the system of equations 4ix $+(1+\mathrm{i}) \mathrm{y}=0$ and $...

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Q JEE MAIN 2022

Let $d$ be the distance between the foot of perpendiculars of the points $P(1,2-1)$ and $Q(2,-1,3)$ on the plane $-x+y+z=1$....

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Q JEE MAIN 2022

The number of distinct real roots of the equation $x^7-7 x-2=0$ is

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Q JEE MAIN 2022

Let $...

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Q JEE MAIN 2022

The area of the region bounded by $y^2=8 x$ and $y^2=16(3-x)$ is equal to:-

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Q JEE MAIN 2022

Let the area of the triangle with vertices $\mathrm{A}(1, \alpha), \mathrm{B}(\alpha, 0)$ and $\mathrm{C}(0, \alpha)$ be 4 sq. units. If...

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Q JEE MAIN 2022

The number of points of intersection of $|z-(4+3 i)|=2$ and $|z|+|z-4|=6, z \in C$ is :

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Q JEE MAIN 2022

Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of...

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Q JEE MAIN 2022

Let the maximum area of the triangle that can be inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{4}=1, a>2$, having one of its...

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Q JEE MAIN 2022

Let $f(x)=\min \{1,1+x \sin x\}, 0 \leq x \leq 2 \pi$. If m is the number of points, where f...

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Q JEE MAIN 2022

Let $S=\{z \in C:|z-2| \leq 1, z(1+i)+\bar{z}(1-i) \leq 2\}$. Let $|z-4 i|$ attains minimum and maximum values, respectively, at $...

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Q JEE MAIN 2022

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to...

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Q JEE MAIN 2022

Let the mean and the variance of 5 observations $x_1, x_2, x_3, x_4, x_5$ be $\frac{24}{5}$ and $\frac{194}{25}$ respectively. If...

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Q JEE MAIN 2022

Let the tangent to the circle $C_1: x^2+y^2=2$ at the point $M(-1,1)$ intersect the circle $C_2:(x-3)^2+(y-2)^2=5$ at two distinct points...

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Q JEE MAIN 2022

$\lim _{n \rightarrow \infty}\left(\frac{n^2}{\left(n^2+1\right)(n+1)}+\frac{n^2}{\left(n^2+4\right)(n+2)}+\frac{n^2}{\left(n^2+9\right)(n+3)}+\ldots+\frac{n^2}{\left(n^2+n^2\right)(n+n)}\right)$ is equal to

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Q JEE MAIN 2022

Let $P Q$ be a focal chord of the parabola $y^2=4 x$ such that it subtends an angle of $\frac{\pi}{2}$...

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Q JEE MAIN 2022

$\lim _{x \rightarrow 0} \frac{\cos (\sin x)-\cos x}{x^4}$ is equal to :

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Q JEE MAIN 2022

$\int_0^5 \cos \left(\pi\left(x-\left[\frac{x}{2}\right]\right)\right) d x$, Where $[t]$ denotes greatest integer less than or equal to $t$, is equal to :

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Q JEE MAIN 2022

The value of the integral $\int_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^x\right)\left(\sin ^6 x+\cos ^6 x\right)}$ is equal to

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Q JEE MAIN 2022

If the constant term in the expansion of $\left(3 x^3-2 x^2+\frac{5}{x^5}\right)^{10}$ is $2^k . l$, where $l$ is an odd...

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Q JEE MAIN 2022

If $A=\sum_{n=1}^{\infty} \frac{1}{\left(3+(-1)^n\right)^n}$ and $B \sum_{n=1}^{\infty} \frac{(-1)^n}{\left(3+(-1)^n\right)^n}$, then $\frac{A}{B}$ is equal to :

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Q JEE MAIN 2022

Let $f(x)=\left\{\begin{array}{ccc}\frac{\sin (x-[x])}{x-[x]}, & x \in(-2,-1) \\ \{\max 2 x, 3[|x|]\}, & |x|

JEE Main Mathematics Medium

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Q JEE MAIN 2022

The domain of the function $\cos ^{-1}\left(\frac{2 \sin ^{-1}\left(\frac{1}{4 x^2-1}\right)}{\pi}\right)$ is

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Q JEE MAIN 2022

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be...

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Q JEE MAIN 2022

If the system of equations $\alpha x+y+z=5, x+2 y+3 z=4, x+3 y+5 z=\beta$, has infinitely many solutions, then the ordered...

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Q JEE MAIN 2022

The distance between the two points $A$ and $A^{\prime}$ which lie on $y=2$ such that both the line segments $...

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Q JEE MAIN 2022

Let $x, y>0$. If $x^3 y^2=2^{15}$, then the least value of $3 x+2 y$ is

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Q JEE MAIN 2022

Let $\left\{a_n\right\}_{n-0}^{\infty}$ be a sequence such that $a_0=a_1=0$ and $a_{n+2}=2 a_{n+1}-a_n+1$ for all $n \geq 0$. Then, $\sum_{n-2}^{\infty} \frac{a_n}{7^n}$ is...

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Q JEE MAIN 2022

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x)=x-1$ and $g: \mathbb{R}-\{1,-1\}$ be defined as $g(x)=\frac{x^2}{x^2-1}$ Then the function fog...

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Q JEE MAIN 2022

Let a set $A=A_1 \cup A_2 \cup \ldots \cup A_{k^{\prime}}$ where $A_i \cap A_j=\phi$ for $...

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Q JEE MAIN 2022

Let $\mathrm{A}=\left[a_{i j}\right]$ be a square matrix of order 3 such that $a_{i j}=2^{i-i}$, for all $\mathrm{i}_{\text {and }} \mathrm{j}=1,2,3$....

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Q JEE MAIN 2022

A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of...

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Q JEE MAIN 2022

The value of $2 \sin \left(12^{\circ}\right)-\sin \left(72^{\circ}\right)$ is :

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Q JEE MAIN 2022

If the angle made by the tangent at the point $\left(x_0, y_0\right)$ on the curve $...

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Q JEE MAIN 2022

If $y=y(x)$ is the solution of the differential equation $2 x^2 \frac{d y}{d x}-2 x y+3 y^2=0$ such that $y(e)=\frac{e}{3}$,...

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Q JEE MAIN 2022

If $b_n=\int_0^{\frac{\pi}{2}} \frac{\cos ^2 n x}{\sin x} d x, n \in N$, then

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Q JEE MAIN 2022

Water is being filled at the rate of $1 \mathrm{~cm}^3 / \mathrm{sec}$ in a right circular conical vessel (vertex downwards)...

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Q JEE MAIN 2022

A circle touches both the y-axis and the line x + y = 0. Then the locus of its center...

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Q JEE MAIN 2022

Let $P$ be the plane passing through the intersection of the planes $\vec{r} \cdot(\hat{i}+3 \hat{j}-\hat{k})=5$ and $\vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=3$, and...

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Q JEE MAIN 2022

The sum $1+2 \cdot 3+3 \cdot 3^2+\ldots \ldots+10 \cdot 3^9$ is equal to

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Q JEE MAIN 2022

The coefficient of $x^{101}$ in the expression $(5+x)^{500}+x(5+x)^{499}+x^2(5+x)^{498}+\ldots . . x^{500} \ldots>0$, is

JEE Main Mathematics Medium

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Q JEE MAIN 2022

The area of the region enclosed between the parabolas $y^2=2 x-1$ and $y^2=4 x-3$ is

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Q JEE MAIN 2022

$\lim _{x \rightarrow \frac{\pi}{2}}\left(\tan ^2 x\left(\left(2 \sin ^2 x+3 \sin x+4\right)^{\frac{1}{2}}-\left(\sin ^2 x+6 \sin x+2\right)^{\frac{1}{2}}\right)\right)$ is equal to

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Q JEE MAIN 2022

Let the system of linear equations $$ \begin{aligned} & x+y+\alpha z=2 \\ & 3 x+y+z=4 \\ & x+2 z=1 \end{aligned}...

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Q JEE MAIN 2022

The system of equations $$ \begin{aligned} & -k x+3 y-14 z=25 \\ & -15 x+4 y-k z=3 \\ & -4...

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Q JEE MAIN 2022

Let $z_1$ and $z_2$ be two complex numbers such that $\bar{z}_1=i \bar{z}_2$ and arg $\left(\frac{z_1}{\bar{z}_2}\right)=\pi$. Then

JEE Main Mathematics Medium

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Q JEE MAIN 2022

The sum of all the real roots of the equation $\left(e^{2 x}-4\right)\left(6 e^{2 x}-5 e^x+1\right)=0$ is

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Q JEE MAIN 2022

Let $a, b \in R$ be such that the equation $a x^2-2 b x+15=0$ has a repeated root $\alpha$. If...

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Q JEE MAIN 2022

Let $\Delta \in\{\wedge, \vee, \Rightarrow, \Leftrightarrow\}$ be such that $(p \wedge q) \Delta((p \vee q) \Rightarrow q)$ is a tautology....

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Q JEE MAIN 2022

Let $A=\{x \in R:|x+1|

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Q JEE MAIN 2022

Let $x * y=x^2+y^3$ and $(x * 1) * 1=x *(1 * 1)$. Then a value of $2 \sin ^{-1}\left(\frac{x^4+x^2-2}{x^4+x^2+2}\right)$

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Q JEE MAIN 2022

Let the lines $y+2 x=\sqrt{11}+7 \sqrt{7}$ and $2 y+x=2 \sqrt{11}+6 \sqrt{7}$ be normal to a circle $C:(x-h)^2+(y-k)^2=r^2$. If the line...

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Q JEE MAIN 2022

The number of elements in the set $\{z=a+i b \in \mathbb{C}: a, b \in \mathbb{Z}$ and $1

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Q JEE MAIN 2022

The number of positive integers $k$ such that the constant term in the binomial expansion of $...

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Q JEE MAIN 2022

Let the mirror image of the point $(a, b, c)$ with respect to the plane $3 x-4 y+12 z+19=0$ be...

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Q JEE MAIN 2022

Let $\mathrm{A}=\left\{1, \mathrm{a}_1, \mathrm{a}_2 \ldots . . \mathrm{a}_{18}, 77\right\}$ be a set of integers with $1

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Q JEE MAIN 2022

A circle of radius 2 unit passes through the vertex and the focus of the parabola $y^2=2 x$ and touches...

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Q JEE MAIN 2022

A rectangle R with end points of the one of its dies as (1, 2) and (3, 6) is inscribed...

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Q JEE MAIN 2022

If the sum of the first ten terms of the series $\frac{1}{5}+\frac{2}{65}+\frac{3}{325}+\frac{4}{1025}+\frac{5}{2501}+\ldots .$. is $\frac{m}{n}$, where $m$ and $n$ are...

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Q JEE MAIN 2022

A ray of light passing through the point $P(2,3)$ reflects on the $x$-axis at point $A$ and the reflected ray...

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Q JEE MAIN 2022

Let $A_1=\left\{(x, y):|x| \leq y^2,|x|+2 y \leq 8\right\}$ and $A_2=\{(x, y):|x|+|y| \leq k\}$. If $27\left(\operatorname{Area} A_1\right)=5\left(\operatorname{Area} A_2\right)$, then $k$ is...

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Q JEE MAIN 2022

If $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}, \vec{b}=3 \hat{i}+3 \hat{j}+\hat{k}$ and $\vec{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$ are coplanar vectors and $...

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Q JEE MAIN 2022

The mean and standard deviation of 15 observations are found to be 8 and 3 respectively. On rechecking it was...

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Q JEE MAIN 2022

If the coefficient of $x^{10}$ in the binomial expansion of $\left(\frac{\sqrt{x}}{5^{\frac{1}{4}}}+\frac{\sqrt{5}}{x^{\frac{1}{3}}}\right)^{60}$ is $5^k l$, where $l, k \in N$ and...

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Q JEE MAIN 2022

The number of real solutions of the equation $\mathrm{e}^{4 \mathrm{x}}+4 \mathrm{e}^{3 \mathrm{x}}-58 \mathrm{e}^{2 \mathrm{x}}+4 \mathrm{e}^{\mathrm{x}}+1=0$ is $\_\_\_\_$ .

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Q JEE MAIN 2022

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in...

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Q JEE MAIN 2022

The positive value of the determinant of the matrix $A$, whose $...

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Q JEE MAIN 2022

Let $R_1$ and $R_2$ be relations on the set $\{1,2, \ldots \ldots, 50\}$ such that $\mathrm{R}_1=\left\{\left(\mathrm{p}, \mathrm{p}^{\mathrm{n}}\right): \mathrm{p}\right.$ is a...

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Q JEE MAIN 2022

If the sum of all the roots of the equation $e^{2 x}-11 e^x-45 e^{-x}+\frac{81}{2}=0$ is $\log _e P$, then $p$...

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Q JEE MAIN 2022

Let p, q, r be three logical statements. Consider the compound statements Then, which of the following is NOT true?

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Q JEE MAIN 2022

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+(2 i-1)=0$. Then, the value of $\left|\alpha^8+\beta^8\right|$ is equal to...

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Q JEE MAIN 2022

Let $f: R \rightarrow R$ be a function defined $f(x)=\frac{2 e^{2 x}}{e^{2 x}+e}$. Then $f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+\ldots . .+f\left(\frac{99}{100}\right)$ is equal to

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Q JEE MAIN 2022

Let $A B$ and $P Q$ be two vertical poles, 160 m apart from each other. Let $C$ be the...

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Q JEE MAIN 2022

The Boolean expression $\left(\sim\left(p^{\wedge} q\right)\right) \vee q$ is equivalent to :

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Q JEE MAIN 2022

The probability, that in a randomly selected 3-digit number at least two digits are odd, is

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Q JEE MAIN 2022

Let the plane $P: \vec{r} \cdot \vec{a}=d$ contain the line of intersection of two planes $\vec{r} \cdot(\hat{i}+3 \hat{j}-\hat{k})=6$ and $...

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Q JEE MAIN 2022

$\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)+\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)+\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$ is equal to :

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Q JEE MAIN 2022

The acute angle between the planes $\mathrm{P}_1$ and $\mathrm{P}_2$, when $\mathrm{P}_1$ and $\mathrm{P}_2$ are the planes passing through the intersection...

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Q JEE MAIN 2022

The value of $\cos \left(\frac{2 \pi}{7}\right)+\cos \left(\frac{4 \pi}{7}\right)+\cos \left(\frac{6 \pi}{7}\right)$ is equal to:

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Q JEE MAIN 2022

Let X be a random variable having binomial distribution B(7, p). If P(X = 3) = 5P(X = 4), then...

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Q JEE MAIN 2022

If two distinct point $Q, R$ lie on the line of intersection of the planes $-x+2 y-z=0$ and $...

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Q JEE MAIN 2022

If the system of linear equations $...

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Q JEE MAIN 2022

Five numbers $x_1, x_2, x_3, x_4, x_5$ are randomly selected from the numbers $1,2,3, \ldots \ldots, 18$ and are arranged...

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Q JEE MAIN 2022

Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$. Then the number of vectors $\overrightarrow{\mathrm{b}}$ such that $\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}$ and $...

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Q JEE MAIN 2022

The area enclosed by $y^2=8 x$ and $y=\sqrt{2} x$ that lies outside the triangle formed by $...

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Q JEE MAIN 2022

If two straight lines whose direction cosines are given by the relations $I+m-n=0,3 I^2+m^2+c n l=0$ are parallel, then the...

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Q JEE MAIN 2022

Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ be $\frac{1}{4}$. If this ellipse passes through the point $...

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Q JEE MAIN 2022

If the tangents drawn at the point $\mathrm{O}(0,0)$ and $\mathrm{P}(1+\sqrt{5}, 2)$ on the circle $x^2+y^2-2 x-4 y=0$ intersect at the...

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Q JEE MAIN 2022

Let $S=(0,2 \pi)-\left\{\frac{\pi}{2}, \frac{3 \pi}{4}, \frac{3 \pi}{2}, \frac{7 \pi}{4}\right\}$. Let $y=y(x), x \in S$, be the solution curve of the...

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Q JEE MAIN 2022

Let $\vec{a}=\alpha \hat{i}+3 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}-\beta \hat{j}+4 \hat{k}$ and $\vec{c}=\hat{i}+2 \hat{j}-2 \hat{k}$ where $\alpha, \beta \in R$, be three vectors....

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Q JEE MAIN 2022

Let the eccentricity of the hyperbola $H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6 \sqrt{2}$, If...

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Q JEE MAIN 2022

Let $\mathrm{A}=\sum_{i=1}^{10} \sum_{j=1}^{10} \min \{\mathrm{i}, \mathrm{j}\}$ and $\mathrm{B}=\sum_{i=1}^{10} \sum_{j=1}^{10} \max \{\mathrm{i}, \beta\}$. Then $\mathrm{A}+\mathrm{B}$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2022

Let $f: R \rightarrow R$ be a function defined by: $...

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Q JEE MAIN 2022

Let $y=y(x)$ be the solution of the differential equation $x\left(1-x^2\right) \frac{d y}{d x}+\left(3 x^2 y-y-4 x^3\right)=0, x>1$, with $y(2)=-2$. Then...

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Q JEE MAIN 2022

Let $A=\{n \in N:$ H.C.F. $(n, 45)=1\}$ and Let $B=\{2 k: k \in\{1,2, \ldots, 100\}\}$. Then the sum of all...

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If $\sin ^2\left(10^{\circ}\right) \sin \left(20^{\circ}\right) \sin \left(40^{\circ}\right) \sin \left(50^{\circ}\right) \sin \left(70^{\circ}\right)=\alpha-\frac{1}{16} \sin \left(10^{\circ}\right)$, then $16+\alpha^{-1}$ is equal to $\_\_\_\_$

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Q JEE MAIN 2022

If the mirror image of the point (2, 4, 7) in the plane 3x – y + 4z = 2...

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Q JEE MAIN 2022

Let the solution curve $y=y(x)$ of the differential equation, $\left[\frac{x}{\sqrt{x^2-y^2}}+e^{\frac{y}{x}}\right] x \frac{d y}{d x}=x+\left[\frac{x}{\sqrt{x^2-y^2}}+e^{\frac{y}{x}}\right] y$ pass through the points $(1,0)$...

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Q JEE MAIN 2022

Let the solution curve $y=y(x)$ of the differential equation $\left(4+x^2\right) d y-2 x\left(x^2+3 y+4\right) d x=0$ pass through the origin....

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Q JEE MAIN 2022

The area of the region $S=\left\{(x, y): y^2 \leq 8 x, y \geq \sqrt{2} x, x \geq 1\right\}$ is

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Q JEE MAIN 2022

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $$ f(x)=\left[\begin{array}{ll} {\left[e^x\right],} & x

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Q JEE MAIN 2022

Let $f(x)=\max \{|x+1|,|x+2|, \ldots,|x+5|\}$. Then $\int_{-6}^0 f(x) d x$ is equal to $\_\_\_\_$ .

JEE Main Mathematics Easy

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Q JEE MAIN 2022

Let the solution curve of the differential equation $x \frac{d y}{d x}-y=\sqrt{y^2+16 x^2}, y(1)=3$ be $y=y(x)$. Then $y(2)$ is equal...

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Q JEE MAIN 2022

In an isosceles triangle $A B C$, the vertex $A$ is $(6,1)$ and the equation of the base $B C$...

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Q JEE MAIN 2022

The probability that a randomly chosen 2 × 2 matrix with all the entries from the set of first 10...

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Q JEE MAIN 2022

Let $[t]$ denote the greatest integer less than or equal to $t$. Then, the value of the integral $...

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Q JEE MAIN 2022

Let the common tangents to the curves $4\left(x^2+y^2\right)=9$ and $y^2=4 x$ intersect at the point $Q$. Let an ellipse, centered...

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Q JEE MAIN 2022

Let $A_1, A_2, A_3, \ldots \ldots$ be an increasing geometric progression of positive real numbers. If $A_1 A_3 A_5 A_7=\frac{1}{1296}$...

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Q JEE MAIN 2022

If $\frac{d y}{d x}+\frac{2^{x-y}\left(2^y-1\right)}{2^x-1}=0, x, y>0, y(1)=1$, then $y(2)$ is equal to :

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Q JEE MAIN 2022

The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which...

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Q JEE MAIN 2022

There are ten boys $\mathrm{B}_1, \mathrm{~B}_2, \ldots ., \mathrm{B}_{10}$ and five girls $\mathrm{G}_1, \mathrm{G}_2, \ldots ., \mathrm{G}_5$ in a class....

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Q JEE MAIN 2022

Let $A$ be a matrix of order $3 \times 3$ and $\operatorname{det}(A)=2$. Then $\operatorname{det}\left(\operatorname{det}(A)\right.$ adj $\left(5 \operatorname{adj}\left(A^3\right)\right)$ ) is equal...

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Q JEE MAIN 2022

If the system of linear equations where $\lambda \in \mathbb{R}$, has no solution, then

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Q JEE MAIN 2022

The sum of the cubes of all the roots of the equation $x^4-3 x^3-2 x^2+3 x+1=10$ is $\_\_\_\_$ .

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Q JEE MAIN 2022

Let a function $f: \mathbb{N} \rightarrow \mathbb{N}$ be defined by $...

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Q JEE MAIN 2022

Let $\Delta, \nabla \in\{\wedge, \vee\}$ be such that $p \nabla q \Rightarrow((p \nabla q) \nabla r)$ is a tautology. Then...

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Q JEE MAIN 2022

The value of the integral $\int_{-2}^2 \frac{\left|x^3+x\right|}{\left(e^{x|x|}+1\right)} d x$ is equal to:

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Q JEE MAIN 2022

If $\sum_{k=1}^{31}\left({ }^{31} C_k\right)\left({ }^{31} C_{k-1}\right)-\sum_{k=1}^{30}\left({ }^{30} C_k\right)\left({ }^{30} C_{k-1}\right)=\frac{\alpha(60!)}{(30!)(31!)}$, Where $\alpha \in R$, then the value of $16 \alpha$...

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Q JEE MAIN 2022

$\int \frac{\left(x^2+1\right) e^x}{(x+1)^2} d x=f(x) e^x+C$, Where $C$ is a constant, then $\frac{d^3 f}{d x^3}$ at $x=1$ is equal to...

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Q JEE MAIN 2022

Let $f(x)=2 \cos ^{-1} x+4 \cot ^{-1} x-3 x^2-2 x+10, x \in[-1,1]$. If $[a, b]$ is the range of the...

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Q JEE MAIN 2022

If $\cos ^{-1}\left(\frac{y}{2}\right)=\log _e\left(\frac{x}{5}\right)^5,|y|

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Q JEE MAIN 2022

The mean of the numbers $a, b, 8,5,10$ is 6 and their variance is 6.8 . If $M$ is the...

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Q JEE MAIN 2022

The lengths of the sides of a triangle are $10+x^2, 10+x^2$ and $20-2 x^2$. If for $x=k$, the area of...

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Q JEE MAIN 2022

The number of distinct real roots of $x^4-4 x+1=0$ is :

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Q JEE MAIN 2022

Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability...

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Q JEE MAIN 2022

Let a be an integer such that $\lim _{x \rightarrow 7} \frac{18-[1-x]}{[x-3 a]}$ exists, where $[t]$ is greatest integer $...

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Q JEE MAIN 2022

The greatest integer less than or equal to the sum of first 100 terms of the sequence $...

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Q JEE MAIN 2022

Let $\frac{d y}{d x}=\frac{a x-b y+a}{b x+c y+a}$, where $a, b, c$ are constants, represent a circle passing through the...

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Q JEE MAIN 2022

If $\vec{a} \cdot \vec{b}=1, \vec{b} \cdot \vec{c}=2$ and $\vec{c} \cdot \vec{a}=3$, then the value of $...

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Q JEE MAIN 2022

Let A be a 3 × 3 matrix having entries from the set {–1, 0, 1}. The number of all...

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Q JEE MAIN 2022

If $x=\sum_{n=0}^{\infty} a^n, y=\sum_{n=0}^{\infty} b^n, z=\sum_{n=0}^{\infty} c^n$, where $a, b, c$ are in A.P. and $|a|

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Q

Let the lines intersect at the point $S$. If a plane $a x+$ by $-z+d=0$ passes through $S$ and is...

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Q JEE MAIN_2022

Let the plane $2 x+3 y+z+20=0$ be rotated through a right angle about its line of intersection with the plane...

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Q JEE MAIN 2022

Let $f: R \rightarrow R$ be a function defined by $f(x)=\left(2\left(1-\frac{x^{25}}{2}\right)\left(2+x^{25}\right)\right)^{\frac{1}{50}}$. If the function $g(x)=f(f(f(x)))+f(f(x))$, the greatest integer less than...

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Q JEE MAIN 2022

Let the system of linear equations $x+2 y+z=2, \alpha x+3 y-z=\alpha,-\alpha x+y+2 z=-\alpha$ be inconsistent. Then $\alpha$ is equal to...

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Q JEE MAIN 2022

For a natural number n , let $\mathrm{a}_{\mathrm{n}}=19^{\mathrm{n}}-12^{\mathrm{n}}$. Then, the value of $\frac{31 \alpha_9-\alpha_{10}}{57 \alpha_8}$ is

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Q JEE MAIN 2022

If the two lines $\ell_1: \frac{x-2}{3}=\frac{y+1}{-2}, z=2$ and $\ell_2: \frac{x-1}{1}=\frac{2 y+3}{\alpha}=\frac{z+5}{2}$ perpendicular, then an angle between the lines $l_2$ and...

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Q JEE MAIN 2022

The number of values of $x$ in the interval $\left(\frac{\pi}{4}, \frac{7 \pi}{4}\right)$ for which $...

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Q JEE MAIN 2022

Let the normal at the point P on the parabola $\mathrm{y}^2=6 \mathrm{x}$ pass through the point $(5,-8)$. If the tangent...

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Q JEE MAIN 2022

Let $C$ be a circle passing through the points $A(2,-1)$ and $B(3,4)$. The line segment $A B$ is not a...

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Q JEE MAIN 2022

Let the abscissae of the two points $P$ and $Q$ be the roots of $2 x^2-r x+p=0$ and the ordinates...

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Q JEE MAIN 2022

The area of the polygon, whose vertices are the non-real roots of the equation $\bar{z}=i z^2$ is:

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Q JEE MAIN 2022

Let $\theta$ be the angle between the vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ where $|\overrightarrow{\mathrm{a}}|=4,|\overrightarrow{\mathrm{~b}}|=3 \theta \in\left(\frac{\pi}{4}, \frac{\pi}{3}\right)$. Then

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Q JEE MAIN 2022

Let $R$ be the point $(3,7)$ and let $P$ and $Q$ be two points on the line $x+y=5$ such that...

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Q JEE MAIN 2022

The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is ______.

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Q JEE-MAIN 2022

The area bounded by the curve $y=\left|x^2-9\right|$ and the line $y=3$ is :

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Q JEE MAIN 2022

If the shortest distance between the line $\vec{r}=(-\hat{i}+3 \hat{k})+\lambda(\hat{i}-a \hat{j})$ and $\vec{r}=(-\hat{j}+2 \hat{k})+\mu(\hat{i}-\hat{j}+\hat{k})$ is $\sqrt{\frac{2}{3}}$ then the integral value of...

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Q JEE MAIN 2022

Let $S$ be the set of all the natural numbers, for which the line $\frac{x}{a}+\frac{y}{b}=2$ is a tangent to the...

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Q JEE MAIN 2022

Let $S$ be the region bounded by the curves $y=x^3$ and $y^2=x$. The curve $y=2|x|$ divides $S$ into two regions...

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Q JEE MAIN 2022

If two tangents drawn from a point ( $\alpha, \beta$ ) lying on the ellipse $25 x^2+4 y^2=1$ to the...

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Q JEE MAIN 2022

The sum of the absolute minimum and the absolute maximum values of the function $f(x)=\left|3 x-x^2+2\right|-x$ in the interval $[-1,2]$...

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Q JEE MAIN 2022

Let $C_r$ denote the binomial coefficient of $x^r$ in the expansion of $(1+x)^{10}$. If $...

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Q JEE MAIN 2022

Let a circle C in complex plane pass although the points $z_1=3+4 i, z_2=4+3 i$ and $z_3=5 i$. If $...

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Q JEE MAIN 2022

Let $\operatorname{Max}_{0 \leq x \leq 2}\left\{\frac{9-x^2}{5-x}\right\}=\alpha$ and $\operatorname{Max}_{0 \leq x \leq 2}\left\{\frac{9-x^2}{5-x}\right\}=\beta$ If $...

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Q JEE MAIN 2022

Let $x=2 t, y=\frac{t^2}{3}$ be a conic. Let $S$ be the focus and $B$ be the point on the axis...

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Q JEE MAIN 2022

If the solution curve $y=y(x)$ of the differential equation $y^2 d x+\left(x^2-x y+y^2\right) d y=0$, which passes through the point...

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Q JEE MAIN 2022

Let $f(\theta)=\sin \theta+\int_{-\pi / 2}^{\pi / 2}(\sin \theta+t \cos \theta) f(t) d t$. Then the value of $...

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Q JEE MAIN 2022

Let $Q$ be the mirror image of the point $P(1,0,1)$ with respect to the plane $S: x+y+z=5$. If a line...

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Q JEE MAIN 2022

If $y=m_1 x+c_1$ and $y=m_2 x+c_2, m_1 \neq m_2$ are two common tangents of circle $x^2+y^2=2$ and parabola $y^2=x$, then...

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Q JEE MAIN 2022

Let a line having direction ratios $1,-4,2$ intersect the lines $\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}$ and $\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}$ at the point $A$ and $B$. Then...

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Q JEE MAIN 2022

Let $A\left(\frac{3}{\sqrt{a}}, \sqrt{a}\right) a>0$ be a fixed point in the $x y$-plane. The image of $A$ in $y$-axis be $B$...

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Q JEE MAIN 2022

Let $y=y(x)$ be the solution of the differential equation $(x+1) y^{\prime}-y=e^{3 x}(x+1)^2$, with $y(0)=\frac{1}{3}$. Then, the point $x =-\frac{4}{3}$ for...

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Q JEE MAIN 2022

$\lim _{x \rightarrow \frac{1}{\sqrt{2}}} \frac{\sin \left(\cos ^{-1} x\right)-x}{1-\tan \left(\cos ^{-1} x\right)}$ is equal to :

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Q JEE MAIN 2022

In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct There...

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Q JEE MAIN 2022

Let $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k} \quad a_i>0, i=1,2,3$ be a vector which makes equal angles with the coordinates axes OX,...

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Q JEE MAIN 2022

The number of one-one function $f:\{a, b, c, d\} \rightarrow\left\{0,1,2_{\text {sui. }} .10\right\}$ such that $2 f(a) . f(b)+3 f(c)+f(d)=0$...

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Q JEE MAIN 2022

The remainder when $(2021)^{2023}$ is divided by 7 is :

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Q JEE MAIN 2022

The number of choices of $\Delta \in\{\wedge, \vee, \Rightarrow, \Leftrightarrow\}$, such that $...

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Q JEE MAIN 2022

Let $\mathrm{S}\left\{\theta \in[-\pi, \pi]-\left\{ \pm \frac{\pi}{2}\right\}: \sin \theta \tan \theta+\tan \theta=\sin 2 \theta\right\}$ If $T=\sum_{\theta \in S} \cos 2 \theta$,...

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Q JEE MAIN 2022

Let $\mathrm{g}:(0, \infty) \rightarrow \mathrm{R}$ be a differentiable function such that $\int\left(\frac{x(\cos x-\sin x)}{e^x+1}+\frac{g(x)\left(e^x+1-x e^x\right)}{\left(e^x+1\right)^2}\right) d x=\frac{x g(x)}{e^x+1}+c$, for all...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

The domain of the function $f(x)=\frac{\cos ^{-1}\left(\frac{x^2-5 x+6}{x^2-9}\right)}{\log _e\left(x^2-3 x+2\right)}$ is

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Q JEE MAIN 2022

The ordered pair ( $a, b$ ), for which the system of linear equations $$ \begin{aligned} & 3 x-2 y+z=b...

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Q JEE MAIN 2022

If a random variable $X$ follows the Binomial distribution $B(33, p)$ such that $3 P(X=0)=P(X=1)$, then the value of $\frac{P(X=15)}{P(X=18)}-\frac{P(X=16)}{P(X=17)}$...

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Q JEE MAIN 2022

Let $A=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]$. If $M$ and $N$ are two matrices given by $M=\sum_{k=1}^{10} A^{2 k}$...

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Q JEE MAIN 2022

Let: $A=\left\{z \in C:\left|\frac{z+1}{z-1}

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Q JEE MAIN 2022

Let $\hat{a}, \hat{b}$ be unit vectors. If $\vec{c}$ be a vector such that the angle between $\hat{a}$ and $\hat{c}$ is...

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Q JEE MAIN 2022

Let $E_1$ and $E_2$ be two events such that the conditional probabilities $P\left(E_1 \mid E_2\right)=\frac{1}{2}, P\left(E_2 \mid E_1\right)=\frac{3}{4}$ and $...

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Q JEE MAIN 2022

Let $\lambda x-2 y=\mu$ be a tangent to the hyperbola $a^2 x^2-y^2=b^2$. Then $\left(\frac{\lambda}{a}\right)^2-\left(\frac{\mu}{b}\right)^2$ is equal to:

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Q JEE MAIN 2022

Let $f(x)$ be a polynomial function such that $f(x)+f^{\prime}(x)+f^{\prime \prime}(x)=x^5+64$. Then, the value of $\lim _{x \rightarrow 1} \frac{f(x)}{x-1}$

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Q JEE MAIN 2022

If $\frac{1}{2 \cdot 3^{10}}+\frac{1}{2^2 \cdot 3^9}+\ldots \cdot \frac{1}{2^{10} \cdot 3}=\frac{\mathrm{K}}{2^{10} \cdot 3^{10}}$, then the remainder when K is divided by...

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Q JEE MAIN 2022

Consider the following two propositions: If the proposition $p \rightarrow((\sim p) \vee q)$ is evaluated as FALSE, then:

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Q JEE MAIN 2022

Let: $f(x)=\frac{x-1}{x+1}, x \in R-\{0,-1,1\}$. If $f^{n+1}(x)=f\left(f^n(x)\right)$ for all $x \in N$, then $f^6(6)+f^7(7)$ is equal to:

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Q JEE MAIN 2022

If $x=x(y)$ is the solution of the differential equation $y \frac{d x}{d y}=2 x+y^3(y+1) e^y, x(1)=0$; then $x(e)$ is equal...

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Q JEE MAIN 2022

If $\left\{a_i\right\}_{i=1}^n$ where $n$ is an even integer, is an arithmetic progression with common difference 1 , and $\sum_{i=1}^n a_i=192$,...

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Q JEE MAIN 2022

The sum of absolute maximum and absolute minimum values of the function $f(x)=\left|2 x^2+3 x-2\right|+\sin x \cos x$ in the...

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Q JEE MAIN 2022

$f(x)=4 \log _e(x-1)-2 x^2+4 x+5, x>1$, which one of the following is NOT correct?

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Q JEE MAIN 2024

Let $A$ be $a \times 3$ real matrix such that $...

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Q JEE MAIN 2022

Let $f: N \rightarrow R$ be a function such that $f(x+y)=2 f(x) f(y)$ for natural numbers $x$ and $y$. If...

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Q JEE MAIN 2022

Let $S=\{\sqrt{n}: 1 \leq n \leq 50$ and $n$ is odd $\}$ Let $a \in S$ and $...

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Q JEE MAIN 2022

Let $a, b$ and $c$ be the length of sides of a triangle $A B C$ such that $\frac{a+b}{7}=\frac{b+c}{8}=\frac{c+a}{9}$. If...

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Q JEE MAIN 2022

The set of all values of $k$ for which $\left(\tan ^{-1} x\right)^3+\left(\cot ^{-1} x\right)^3=k \pi^3, x \in R$, is the...

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Q JEE MAIN 2022

The value of $\int_0^\pi \frac{e^{\cos x} \sin x}{\left(1+\cos ^2 x\right)\left(e^{\cos x}+e^{-\cos x}\right)} d x$ is equal to

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Q JEE MAIN 2022

If the sum of the squares of the reciprocals of the roots $a$ and $b$ of the equation $...

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Q JEE MAIN 2022

Let a circle $C$ touch the lines $L_1: 4 x-3 y+K_1=0$ and $L_2: 4 x-3 y+K_2=0, K_1, K_2 \in R$....

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Q JEE MAIN 2022

The number of values of a for which the system of equations : $...

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Q JEE MAIN 2022

Let $x^2+y^2+A x+B y+C=0$ be a circle passing through $(0,6)$ and touching the parabola $y=x^2$ at $(2,4)$. Then $\mathrm{A}+\mathrm{C}$ is...

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Q JEE MAIN 2022

Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and...

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Q JEE MAIN 2022

The surface area of a balloon of spherical shape being in flated, increases at a constant rate. If initially, the...

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Q JEE MAIN 2022

The remainder when $3^{2022}$ is divided by 5 is

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Q JEE MAIN 2022

Let $A=\{z \in C: 1 \leq|z-(1+i)| \leq 2\}$ and $B=\{z \in A:|z-(1-i)|=1\}$. Then, $B:$

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Q

Let the area enclosed by the lines $x+y=2, y=0, x=0$ and the curve $f(x)=\min \left\{x^2+\frac{3}{4}, 1+[x]\right\}$ where $[x]$ denotes the...

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Q JEE MAIN 2023

Let $R=\{a, b, c, d, e\}$ and $S=\{1,2,3,4\}$. Total number of onto function $f \mid: R \rightarrow S$ such that...

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Q JEE MAIN

Let $P_1$ be the plane $3 x-y-7 z=11$ and $P_2$ be the plane passing through the points $(2,-1,0),(2,0,-1)$, and $(5,1,1)$....

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Q JEE MAIN 2023

The ordinates of the points $P$ and $Q$ on the parabola with focus $(3,0)$ and directrix $x=-3$ are in the...

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Q JEE MAIN

Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2-12 x+[x]+31=0$ and $...

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Q JEE MAIN 2023

If domain of the function $\log _e\left(\frac{6 x^2+5 x+1}{2 x-1}\right)+\cos ^{-1}\left(\frac{2 x^2-3 x+4}{3 x-5}\right)$ is $(\alpha, \beta) \cup(\gamma, \delta]$, then...

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Q JEE MAIN 2023

Let $[t]$ denote the greatest integer function. If $\int_0^{2.4}\left[X^2\right] d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}+\delta \sqrt{5}$, then $\alpha+\beta+\gamma+\delta$ is equal to $\_\_\_\_$...

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Q JEE-Main 2023

Let $A=\{1,2,3,4,5,6,7\}$. Then the relation $R=\{(x, y) \in A \times A: x+y=7\}$ is

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Q JEE-Main 2023

Let $\mathrm{A}(0,1), \mathrm{B}(1,1)$ and $\mathrm{C}(1,0)$ be the mid - points of the sides of a triangle with in Centre at...

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Q

If $\mathrm{A}=\left[\begin{array}{cc}1 & 5 \\ \lambda & 10\end{array}\right], \mathrm{A}^{-1}=\alpha \mathrm{A}+\beta \mathrm{I}$ and $\alpha+\beta=-2$, then $4 \alpha^2+\beta 2+\lambda^2$ is equal to...

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Q [JEE-Main 2023

If $\alpha>\beta>0$ are the roots of the equation $a x^2+b x+1=0$, and $\lim _{x \rightarrow \frac{1}{\alpha}}\left(\frac{1-\cos \left(x^2+b x+a\right)}{2(1-\alpha x)^2}\right)^{\frac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta}-\frac{1}{\alpha}\right)$, then...

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Q JEE-Main 2023

The neqation of $(p \wedge(\sim q)) \vee(\sim p)$ is equivalent to

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Q JEE-Main 2023

Let $O$ be the origin and $O P$ and $O Q$ be the tangents to the circle $x^2+y^2-6 x+4 y+8=0$...

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Q JEE-Main 2023

Let the vectors $\overrightarrow{\mathrm{u}}_1=\hat{\mathrm{i}}+\hat{\mathrm{j}}+a \hat{\mathrm{k}}, \overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+b \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{u}}_3=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ be coplanar. If the vectors $...

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Q JEE MAIN 2023

If the area of the region $\left\{(x, y):\left|x^2-2\right| \leq y \leq x\right\}$ is $A$, then $6 A+16 \sqrt{2}$ is equal...

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Q JEE MAIN 2023

In the figure, $\theta_1+\theta_2=\frac{\pi}{2}$ and $\sqrt{3}(B E)=4(A B)$. If the area of $\triangle C A B$ is $2 \sqrt{3}-3$ unit...

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Q JEE MAIN 2023

The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is...

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Q JEE MAIN_2023

$25^{190}-19^{190}-8^{190}+2^{190}$ is divisible by

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Q JEE MAIN 2023

If the domain of the function $f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)$ is $[\alpha, \beta] \cup[\gamma, \delta]$, then $|3 \alpha+10(\beta+\gamma)+21 \delta|$ is...

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Q JEE MAIN 2023

Suppose $a_1, a_2, 2, a_3, a_4$ be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression...

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Q JEE MAIN 2025

Let the area of the triangle formed by the lines $x+2=y-1=z, \frac{x-3}{5}=\frac{y}{-1}=\frac{z-1}{1}$ and $\frac{x}{-3}=\frac{y-3}{3}=\frac{z-2}{1}$ be $A$. Then $A^2$ is equal...

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Q JEE MAIN 2023

Let the tangent at any point P on a curve passing through the points $(1,1)$ and $\left(\frac{1}{10}, 100\right)$, intersect positive...

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Q JEE-Main 2023

For $\mathrm{a}, \mathrm{b} \in \mathrm{Z}$ and $|\mathrm{a}-\mathrm{b}| \leq 10$, let the angle between the plane $\mathrm{P}: \mathrm{ax}+\mathrm{y}-\mathrm{z}=\mathrm{b}$ and the line...

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Q JEE MAIN 2023

Let quadratic curve passing through the point $(-1,0)$ and touching the line $y=x$ at $(1,1)$ be $y=f(x)$. Then the x-intercept...

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Q JEE MAIN 2023

Let the foot of perpendicular from the point $\mathrm{A}(4,3,1)$ on the plane $\mathrm{P}: \mathrm{x}-\mathrm{y}+2 \mathrm{z}+3=0$ be N . If $...

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Q JEE MAIN 2023

Let S be the set of values of $\lambda$, for which the system of equations $$ \begin{aligned} & 6 \lambda...

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Q JEE MAIN 2025

Let $r$ be the radius of the circle, which touches $x$ - axis at point $(a, 0), a

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Q JEE-Main 2023

The area of the quadrilateral ABCD with vertices A(2, 1, 1), B(1, 2, 5), C(–2, –3, 5) and D(1, –6,...

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Q JEE MAIN 2023

Let the equations of two adjacent sides of a parallelogram ABCD be $2 \mathrm{x}-3 \mathrm{y}=-23$ and $5 \mathrm{x}+4 \mathrm{y}=23$. If...

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Q JEE MAIN 2025

Let the area of the bounded region $\left\{(x, y): 0 \leq 9 x \leq y^2, y \geq 3 x-6\right\}$ be...

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Q JEE MAIN 2023

The remainder, when $7^{103}$ is divided by 17 is $\_\_\_\_$ $-$

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Q JEE MAIN 2023

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Q JEE MAIN 2025

The product of the last two digits of $(1919)^{1919}$ is $\_\_\_\_$

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Q JEE MAIN 2023

The foci of a hyperbola are $( \pm 2,0)$ and its eccentricity is $\frac{3}{2}$. A tangent, perpendicular to the line...

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Q JEE MAIN 2023

If $y=y(x)$ is the solution of the differential equation $\frac{d y}{d x}+\frac{4 x}{\left(x^2-1\right)} y=\frac{x+2}{\left(x^2-1\right)^{5 / 2}}, x>1$ such that $...

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Q JEE MAIN 2025

The number of integral terms in the expansion of $\left(5^{\frac{1}{2}}+7^{\frac{1}{8}}\right)^{1016}$ is

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Q JEE MAIN 2025

Let $\alpha$ be a solution of $x^2+x+1=0$, and for some $a$ and $b$ in $...

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Q JEE MAIN 2023

Let $S=\left\{z=x+i y: \frac{2 z-3 i}{4 z+2 i}\right.$ is a real number $\}$. Then which of the following is NOT...

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Q JEE MAIN 2023

Let the line $=\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}$ intersect the lines $\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}$ and $\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}$ at the points $A$ and $B$ respectively. Then the distance...

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Q JEE MAIN 2023

Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is reciprocal to that of the hyperbola $2 x^2-2 y^2=1$. If the ellipse...

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Q JEE-Main 2023

The value of $36\left(4 \cos ^2 9^{\circ}-1\right)\left(4 \cos ^2 27^{\circ}-1\right)\left(4 \cos ^2 81^{\circ}-1\right)\left(4 \cos ^2 243^{\circ}-1\right)$ is

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Q JEE MAIN 2023

If the coefficients of $x$ and $x^2$ in $(1+x)^p(1-x)^q$ are 4 and -5 respectively, then $2 p+3 q$ is equal...

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Q JEE MAIN 2023

Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is reciprocal to that of the hyperbola $2 x^2-2 y^2=1$. If the ellipse...

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Q JEE MAIN 2023

For $\mathrm{x} \in(-1,1]$, the number of solutions of the equation $\sin ^{-1} \mathrm{x}=2 \tan ^{-1} \mathrm{x}$ is equal to

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Q JEE MAIN 2023

Let $[\alpha]$ denote the greatest integer $\leq \alpha$. Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots .+[\sqrt{120}]$ is equal to $\_\_\_\_$ .

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Q JEE-Main 2023

If the probability that the random variable $X$ takes values $x$ is given by $P(X=x)=k(x+1) 3^{-x}, x=0,1$, $2,3 \ldots \ldots$,...

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Q JEE MAIN 2023

For $\alpha, \beta, z \in C$ and $\lambda>1$, if $\sqrt{\lambda-1}$ is the radius of the circle $|z-\alpha|^2+|z-\beta|^2=2 \lambda$, then $|\alpha-\beta|$...

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Q JEE MAIN 2023

Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2,...

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Q JEE MAIN 2023

Let $f(x)=\sum_{k=1}^{10} k x^k, x \in R$. If $2 f(2)+f^{\prime}(2)=119(2)^n+1$, then $n$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2023

The number of points, where the curve $y=x^5-20 x^3+50 x+2$ crosses the $x$-axis, is $\_\_\_\_$ .

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Q JEE MAIN 2023

Let A= {–4, –3, –2, 0, 1, 3, 4} and R = {(a, b) ∈ A × A : b...

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Q JEE MAIN 2023

If $(20)^{19}+2(21)(20)^{18}+3(21)^2(20)^{17}+\ldots \ldots . .+20(21)^{19}=\mathrm{k}(20)^{19}$, then k is equal to $\_\_\_\_$ .

JEE Main Mathematics Hard

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Q JEE-Main 2023

Let $S$ be the set of all values of $\theta \in[-\pi, \pi]$ for which the system of linear equations $...

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Q JEE MAIN 2023

The mean and standard deviation of the marks of 10 students were found to be 50 and 12 respectively. Later,...

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Q JEE MAIN 2023

If the lines and intersect, then the magnitude of the minimum value of 8is _________.

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Q JEE MAIN 2023

The value of $...

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Q JEE MAIN 2023

The value of tan 9° – tan 27° – tan 63° + tan 81° is _________.

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Q JEE-Main 2023

If the number of words, with or without meaning, which can be made using all the letters of the word...

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Q JEE MAIN 2023

Let for a triangle ABC , $$ \begin{aligned} & \overrightarrow{\mathrm{AB}}=-2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}} ; \\ & \overrightarrow{\mathrm{CB}}=\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}} \\...

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Q JEE MAIN 2023

Let $f(x)=\frac{x}{\left(1+x^n\right)^{\frac{1}{n}}}, x \in R-\{-1\}, n \in N, n>2$. If $f^n(x)=$ (fofof ..... upto $n$ times) ( $x$ ), then...

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Q JEE MAIN 2025

Let $f(x)$ be a positive function and $I_1=\int_{-\frac{1}{2}}^1 2 x f(2 x(1-2 x)) d x$ and $I_2=\int_{-1}^2 f(x(1-x)) d x$....

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Q JEE MAIN 2023

The area of the region $\left\{(x, y): x^2 \leq y \leq\left|x^2-4\right|, y \geq 1\right\}$ is

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Q JEE MAIN 2025

The Value of $\cot ^{-1}\left(\frac{\sqrt{1+\tan ^2(2)}-1}{\tan (2)}\right)-\cot ^{-1}\left(\frac{\sqrt{1+\tan ^2\left(\frac{1}{2}\right)}+1}{\tan \left(\frac{1}{2}\right)}\right)$ is equal to

JEE Main Mathematics Medium

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Q JEE MAIN 2023

If $\operatorname{gcd}(m, n)=1$ and $1^2-2^2+3^2-4^2+\ldots \ldots+(2021)^2-(2022)^2+(2023)^2=1012 m^2 n$, then $m^2-n^2$ is equal to

JEE Main Mathematics Hard

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Q JEE MAIN 2023

Let $S=\left\{Z \in C: \bar{z}=i\left(z^2+\operatorname{Re}(\bar{z})\right)\right\}$. Then $\sum_{z \in S}|z|^2$ is equal to $\_\_\_\_$ .

JEE Main Mathematics Medium

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Q JEE MAIN 2025

The sum of the squares of the roots of $|x-2|^2+|x-2|-2=0$ and the squares of the roots of $x^2-2|x-3|-5=0$, is

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let the vectors $\vec{a}, \vec{b}, \vec{c}$ represent three coterminous edges of a parallelopiped of volume $V$. Then the volume of...

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Q JEE MAIN 2023

Let for $\mathrm{A}=\left[\begin{array}{lll}1 & 2 & 3 \\ \mathrm{a} & 3 & 1 \\ 1 & 1 & 2\end{array}\right],|\mathrm{A}|=2$. If...

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Q JEE MAIN 2023

The random valuable X follows binomial distribution B(n, p) for which the difference of the mean and the variance is...

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Q JEE-Main 2023

The absolute difference of the coefficients of $x^{10}$ and $x^7$ in the expansion of $\left\{2 x^2+\frac{1}{2 x}\right\}^{11}$ is equal to

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Q JEE MAIN 2023

If the tangents at the points P and Q on the circle x2 + y2 – 2x + y =...

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Q JEE MAIN 2023

The statement (p ∧(~ q)∨ ((~ p)∧q)∨((~ p)∧(~ q)) is equivalent to

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Q JEE MAIN 2023

Among the statements $(\mathrm{S} 1):(\mathrm{p} \Rightarrow \mathrm{q}) \vee((\sim \mathrm{p}) \wedge \mathrm{q})$ is a tautology $...

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Q JEE MAIN 2023

The range of $f(x)=4 \sin ^{-1}\left(\frac{x^2}{x^2+1}\right)$

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Q JEE MAIN 2025

Let $A=\{0,1,2,3,4,5\}$. Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $...

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Q JEE MAIN 2023

If the coefficients of $x^7$ in $\left(a x^2+\frac{1}{2 b x}\right)^{11}$ and $x^{-7}$ in $\left(a x-\frac{1}{3 b x^2}\right)^{11}$ are equal, then

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Q JEE MAIN 2023

Let $f(x)$ be a function satisfying $f(x)+f(\pi-x)=\pi^2, \forall x \in R$. Then $\int^\pi f(x) \sin x d x$ is equal...

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Q JEE MAIN_2025

If $\int\left(\frac{1}{x}+\frac{1}{x^3}\right)\left(\sqrt[7]{3 x^{-24}+x^{-26}}\right) \mathrm{d} x=-\frac{\alpha}{3(\alpha+1)}\left(3 x^\beta+x^7\right)^{\frac{\alpha+4}{\alpha}}+C, x>0,(\alpha, \beta, \gamma \in \mathbf{Z})$, where C is the constant of integration, then $\alpha+\beta+\gamma$...

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Q JEE MAIN 2023

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written...

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Q JEE MAIN 2023

Let $S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^2 x}+9^{\tan ^2 x}=10\right\}$ and $\beta=\sum_{x \in s} \tan ^2\left(\frac{x}{3}\right)$, then $\frac{1}{6}(\beta-14)^2$ is equal to

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Q JEE MAIN 2025

There are 12 points in a plane, no three of which are in the same straight line, except 5 points...

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Q JEE MAIN 2023

The statement $\sim[p \vee(\sim(p \wedge q))]$ is equivalent to

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Q JEE MAIN 2023

Let the centre of a circle C be (α, β) and its radius r < 8. Let 3x + 4y...

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Q JEE-Main 2023

Let A $=\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1-i \sin \theta}\right.$ is purely imaginary $\}$. Then the sum of the...

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Q JEE MAIN 2023

Let the line L pass through the point $(0,1,2)$, intersect the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and be parallel to the plane $...

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Q JEE MAIN 2023

Let N be the foot of perpendicular from the point P(1, –2, 3) on the line passing through the points...

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Q JEE MAIN 2023

If the points $P$ and $Q$ are respectively the circumcentre and the orthocentre of a $\triangle A B C$, then...

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Q JEE MAIN_2025

If the function $f(x)=\frac{\tan (\tan x)-\sin (\sin x)}{\tan x-\sin x}$ is continuous at $x=0$, then $f(0)$ is equal to $\_\_\_\_$...

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Q JEE MAIN 2025

The integral $\int_{-1}^{\frac{3}{2}}\left(\left|\pi^2 x \sin (\pi x)\right|\right) d x$ is equal to :

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Q JEE MAIN 2023

All the letters of the word PUBLIC are written in all possible orders and these words are written as in...

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Q JEE MAIN 2023

If $\mathrm{S}_{\mathrm{n}}=4+11+21+34+50+$ $\_\_\_\_$ to n terms, then $\frac{1}{60}\left(\mathrm{~S}_{29}-\mathrm{S}_9\right)$ is equal to

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Q JEE MAIN_2025

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2 a and 2...

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Q JEE MAIN 2023

Let $\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+2 \hat{k}$. Let $\vec{d}$ be a vector which is perpendicular to both...

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Q JEE MAIN 2025

Given below are two statements: Statement I : $\lim _{x \rightarrow 0}\left(\frac{\tan ^{-1} x+\log _e \sqrt{\frac{1+x}{1-x}}-2 x}{x^5}\right)=\frac{2}{5}$ Statement II: $...

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Q JEE MAIN 2023

Let P be a square matrix such that $\mathrm{P}^2=\mathrm{I}-\mathrm{P}$. For $\alpha, \beta, \gamma, \delta \in \mathrm{N}$, if $\mathrm{P}^\alpha+\mathrm{P}^\beta=\gamma \mathrm{I}-29 \mathrm{P}$...

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Q JEE MAIN_2025

For $\mathrm{t}>-1$, let $\alpha_{\mathrm{t}}$ and $\beta$, be the roots of the equation $$ \left((t+2)^{1 / 7}-1\right) x^2+\left((t+2)^{1 / 0}-1\right) x+\left((t+2)^{2...

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Q JEE MAIN 2023

Let $|\vec{a}|=2,|\vec{b}|=3$ and the angle between the vectors $\vec{a}$ and $\vec{b}$ be $\frac{\pi}{4}$. Then $|(\vec{a}+2 \vec{b}) \times(2 \vec{a}-3 \vec{b})|^2$ is...

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Q JEE-Main 2023

Let P be the plane passing through the line $\frac{x-1}{1}=\frac{y-2}{-3}=\frac{z+5}{7}$ and the point (2, 4, –3). If the image of...

JEE Main Mathematics Medium

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Q JEE MAIN_2025

Let $c_1$ and $c_2$ be the eccentricities of the ellipse $\frac{x^2}{b^2}+\frac{y^2}{25}=1$ and the hyperbola $\frac{x^2}{16}-\frac{y^2}{b^2}=1$. respectively. If $\mathrm{b}

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Q JEE MAIN 2025

Let $A=\left\{\theta \in[0,2 \pi]: 1+10 \operatorname{Re}\left(\frac{2 \cos \theta+i \sin \theta}{\cos \theta-3 i \sin \theta}\right)=0\right\}$. Then $\sum_{\theta \in \mathrm{A}} \theta^2$ is...

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Q JEE MAIN 2023

The area bounded by the curves y = |x – 1| + |x – 2| and y = 3 is...

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Q JEE MAIN 2023

Let $(\alpha, \beta)$ be the centroid of the triangle formed by the lines $15 x-y=82,6 x-5 y=-4$ and $...

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Q JEE MAIN 2023

Let $g(x)=f(x)+f(1-x)$ and $f^{\prime \prime}(x)>0, x \in(0,1)$, If $g$ is decreasing in the interval ( $0, \alpha$ ) and increasing...

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Q JEE MAIN 2025

A line passing through the point $P(a, 0)$ makes an acute angle $\alpha$ with the positive $x$-axis. Let this line...

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Q JEE MAIN 2023

For the system of equations $$ \begin{aligned} & x+y+z=6 \\ & x+2 y+\alpha z=10 \end{aligned} $$ $x+3 y+5 z=\beta$, which...

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Q JEE MAIN_2025

A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is...

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Q JEE MAIN 2023

Let $a_1, a_2, a_{3, \ldots . .}$ be a G.P. of increasing positive numbers. Let the sum of its 6th...

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Q JEE MAIN 2023

Let a die be rolled $n$ times. Let the probability of getting odd numbers seven times be equal to the...

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Q JEE MAIN 2025

Let the values of $\lambda$ for which the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-\lambda}{3}=\frac{y-4}{4}=\frac{z-5}{5}$ is $\frac{1}{\sqrt{6}}$ be $\lambda_1$...

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Q JEE MAIN_2025

Let a random variable $X$ take values $0,1,2,3$ with $P(X=0)=P(X=1)=p, P(X=2)=P(X=3)$ and $E\left(X^2\right)=2 E(X)$. Then the value of $8 p-1$...

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Q JEE MAIN 2023

Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{2} x+2=0$. Then $\alpha^{14}+\beta^{14}$ is equal to

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Q JEE MAIN 2023

Let $A$ be the point $(1,2)$ and $B$ be any point on the curve $x^2+y^2=16$. If the centre of the...

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Q JEE MAIN_2025

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a polynomial function of degree four having extreme values at $x=4$ and $x=5$. If...

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Q JEE MAIN 2023

The sum of all values of a, for which the points whose position vectors $...

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Q JEE-Main 08-04-2023

Let an be the nth term of the series 5 + 8 + 14 + 23 + 35 + 50...

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Q JEE MAIN 2025

Let $f(x)=x-1$ and $g(x)=e^x$ for $x \in \mathbb{R}$. If $\frac{d y}{d x}=\left(e^{-2 \sqrt{x}} g(f(f(x)))-\frac{y}{\sqrt{x}}\right), y(0)=0$, then $y(1)$ is

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Q JEE MAIN 2025

If the sum of the first 10 terms of the series $...

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Q JEE MAIN 2023

If $A=\frac{1}{5!6!7!}\left[\begin{array}{lll}5! & 6! & 7! \\ 6! & 7! & 8! \\ 7! & 8! & 9!\end{array}\right]$, then $...

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Q JEE MAIN 2023

The plane, passing through the points (0, –1, 2) and (–1, 2, 1) and parallel to the line passing through...

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Q JEE MAIN_2025

Let $p$ be the number of all triangles that can be formed by joining the vertices of a regular polygon...

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Q JEE MAIN 2023

If the solution curve f(x, y)=0 of the differential equation $...

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Q JEE MAIN 2025

If the set of all $a \in R-\{1\}$, for which the roots of the equation $(1-a) x^2+2(a-3) x+9=0$ are positive...

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Q JEE MAIN 2023

Let $A=\{2,3,4\}$ and $B=\{8,9,12\}$. Then the number of elements in the relation $...

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Q JEE MAIN 2025

Let $\vec{a}=\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-\hat{k}$. Let $\hat{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and...

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Q JEE MAIN 2023

The coefficient of $x^5$ in the expansion of $\left(2 x^3-\frac{1}{3 x^2}\right)^5$

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Q JEE-Main 2023

Let the mean and variance of 12 observations be $\frac{9}{2}$ and 4 respectively. Later on, it was observed that two...

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Q JEE MAIN 2023

Let the image of the point $\mathrm{P}(1,2,6)$ in the plane passing through the points $\mathrm{A}(1,2,0), \mathrm{B}(1,4,1)$ and $\mathrm{C}(0,5,1)$ be $...

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Q JEE MAIN 2025

If $y=\cos \left(\frac{\pi}{3}+\cos ^{-1} \frac{x}{2}\right)$, then $(x-y)^2+3 y^2$ is equal to

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Q JEE MAIN 2023

Let $a \neq b$ be two non-zero real numbers. Then the number of elements in the set $...

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Q JEE MAIN 2025

Let $A(4,-2), B(1,1)$ and $C(9,-3)$ be the vertices of a triangle ABC . Then the maximum area of the parallelogram...

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Q JEE MAIN 2023

If the system of equations $$ \begin{aligned} & 2 x+y-z=5 \\ & 2 x-5 y+\lambda z=\mu \\ & x+2 y-5...

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Q JEE MAIN 2023

For, $\alpha, \beta, \gamma, \delta \in \mathrm{N}$, if $\int\left(\left(\frac{\mathrm{x}}{\mathrm{e}}\right)^{2 \mathrm{x}}+\left(\frac{\mathrm{e}}{\mathrm{x}}\right)^{2 \mathrm{x}}\right) \log _{\mathrm{e}} \mathrm{xdx}=\frac{1}{\alpha}\left(\frac{\mathrm{x}}{\mathrm{e}}\right)^{\beta \mathrm{x}}-\frac{1}{\gamma}\left(\frac{\mathrm{x}}{\mathrm{e}}\right)^{\delta \mathrm{x}}+\mathrm{C}$ Where $e=\sum_{n=0}^{\infty} \frac{1}{n!}$ and...

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Q JEE MAIN 2025

$ \begin{aligned} & \text { If } \frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\ldots \infty=\frac{\pi^4}{90} \\ & \frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+\ldots \infty=\alpha, \\ & \frac{1}{2^4}+\frac{1}{4^4}+\frac{1}{6^4}+\ldots \infty=\beta \end{aligned} $ then...

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Q JEE MAIN 2025

Let $\left(1+x+x^2\right)^{10}=a_0+a_1 x+a_2 x^2+\ldots+a_{20} x^{20}$. If $\left(a_1+a_3+a_5+\ldots+a_{19}\right)-11 a_2=121 k$, then $k$ is equal to $\_\_\_\_$

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Q JEE MAIN 2025

Let $y=y(x)$ be the solution of the differential equation $...

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Q JEE MAIN 2023

Eight persons are to be transported from city A to city B in three cars of different makes. If each...

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Q JEE MAIN 2023

$\operatorname{Lim}_{n \rightarrow \infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right) \cdots \cdots\left(2^{\frac{1}{2}}-2^{\frac{1}{2 n+1}}\right)\right\}$ is equal to

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Q JEE MAIN 2023

Let $f$ be a continuous function satisfying $\int_0^{t^2}\left(f(x)+x^2\right) d x=\frac{4}{3} t^3, \forall t>0$. Then $f\left(\frac{\pi^2}{4}\right)$ is equal to :

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Q JEE MAIN 2025

If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $3 x^2-y^2-\alpha x+\beta y+\gamma=0$, then $\alpha+\beta+\gamma$ is equal...

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Q JEE MAIN 2025

Let $a$ be the length of a side of a square OABC with O being the origin. Its side OA...

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Q JEE MAIN 2025

Let $\vec{a}=2 \hat{i}-3 \hat{j}+\hat{k}, \vec{b}=3 \hat{i}+2 \hat{j}+5 \hat{k}$ and a vector $\vec{c}$ be such that $...

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Q JEE MAIN 2023

Among the statements: (S1): $2023^{2022}-1999^{2022}$ is divisible by 8 . $(\mathrm{S} 2): 13(13)^n-11 \mathrm{n}-13$ is divisible by 144 for infinitely...

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Q JEE MAIN 2025

Let m and $\left.\mathrm{n}_{( } \mathrm{m}

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Q JEE MAIN 2025

Let $\vec{a}=\hat{i}+2 \hat{j}+\hat{k}, \vec{b}=3 \hat{i}-3 \hat{j}+3 \hat{k}, \vec{c}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{d}$ be a vector such that $...

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Q JEE MAIN 2025

Let the ellipse $3 x^2+p y^2=4$ pass through the centre $C$ of the circle $x^2+y^2-2 x-4 y-11=0$ of radius $r$....

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $...

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Q JEE MAIN 2025

Let $...

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If $\lim _{x \rightarrow 0}\left(\frac{\tan x}{x}\right)^{\frac{1}{x^2}}=p$, then $96 \log _e p$ is equal to $\_\_\_\_$

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Let $y=y(x)$ be the solution of the differential equation $\left(x^2+1\right) y^2-2 x y=\left(x^4+2 x^2+1\right) \cos x$. $y(0)=1$. Then $...

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The number of real roots of the equation $x|x-2|+3|x-3|+1=0$ is i

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Let the function $f(x)=\frac{x}{3}+\frac{3}{x}+3, x \neq 0$ be strictly increasing in $\left(-\infty, \alpha_1\right) \cup\left(\alpha_2, \infty\right)$ and strictly decreasing in $...

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If the orthocenter of the triangle formed by the lines $y=x+1, y=4 x-8$ and $y=m x+c$ is at $(3,-1)$ ,...

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If $\int \frac{\left(\sqrt{1+x^2}+x\right)^{10}}{\left(\sqrt{1+x^2}-x\right)^9} \mathrm{~d} x=\frac{1}{\mathrm{~m}}\left(\left(\sqrt{1+x^2}+x\right)^0\left(\pi \sqrt{1+x^2}-x\right)\right)+\mathrm{C}$ where C is the constant of integration and $m, n \in \mathbf{N}$, then $m+n$...

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Q JEE MAIN 2025

Let the area of the triangle formed by a straight line $L: x+b y+c=0$ with co-ordinate axes be 48 square...

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If $A$ and $B$ are two events such that $P(A)=0.7, P(B)=0.4$ and $P(A \cap \bar{B})=0.5$, where $\bar{B}$ denotes the complement...

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Q JEE MAIN 2025

A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and...

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Q JEE MAIN 2025

If $\alpha$ is a root of the equation $x^2+x+1=0$ and $\sum_{k=1}^n\left(\alpha^k+\frac{1}{\alpha^k}\right)^2=20$, then n is equal to $\_\_\_\_$ .

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Q JEE MAIN 2025

$4 \int_0^1\left(\frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right) d x-3 \log _e(\sqrt{3})$ is equal to :

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Let the three sides of a triangle ABC be given by the vectors $2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}$ and $...

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The number of solutions of the equation $(4-\sqrt{3}) \sin x-2 \sqrt{3} \cos ^2 x=-\frac{4}{1+\sqrt{3}}, x \in\left[-2 \pi, \frac{5 \pi}{2}\right]$ is

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The number of solutions of the equation $\cos 2 \theta \cos \frac{\theta}{2}+\cos \frac{5 \theta}{2}=2 \cos ^2 \frac{5 \theta}{2}$ in $...

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Q JEE MAIN_2025

If the locus of $z \in \mathrm{C}$, such that $\operatorname{Re}\left(\frac{z-1}{2 z+i}\right)+\operatorname{Re}\left(\frac{\bar{z}-1}{2 \bar{z}-i}\right)=2$, is a circle of radius r and center...

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Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y$ - and $z$-axes, respectively,...

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Let $\mathrm{a}>0$. If the function $f(x)=6 x^3-45 a x^2+108 a^2 x+1$ attains its local maximum and minimum values at the...

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If the area of the region $\left\{(x, y): 1+x^2 \leqslant y \leqslant \min \{x+7.11-3 x\}\right\}$ is A , then 3...

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If the four distinct points $(4,6),(-1,5),(0,0)$ and $(k, 3 k)$ lie on a circle of radius $r$, then $10 k+r^2$...

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The sum of the infinite series $\cot ^{-1}\left(\frac{7}{4}\right)+\cot ^{-1}\left(\frac{19}{4}\right)+\cot ^{-1}\left(\frac{39}{4}\right)+\cot ^{-1}\left(\frac{67}{4}\right)+\ldots$. is :

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If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the...

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Let the sum of the focal distances of the point $\mathrm{P}(4,3)$ on the hyperbola $\mathrm{H}: \frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ be $8 \sqrt{\frac{5}{3}}$. If...

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Let the Mean and Variance of five observations $x_1=1, x_2=3, x_3=a, x_4=7$ and $x_5=b, a>b$, be 5 and 10 respectively....

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If the range of the function $f(x)=\frac{5-x}{x^2-3 x+2}, x \neq 1,2$, is $(-\infty, \alpha] \cup[\beta, \infty)$, then $\alpha^2+\beta^2$ is equal...

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Q JEE MAIN 2025

Let the domains of the functions $f(x)=\log _4 \log _3 \log _7\left(8-\log _2\left(x^2+4 x+5\right)\right)$ and $\mathrm{g}(x)=\sin ^{-1}\left(\frac{7 x+10}{x-2}\right)$ be $...

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Q JEE MAIN 2025

Let the product of $\omega_1=(8+i) \sin \theta+(7+4 i) \cos \theta$ and $\omega_2=(1+8 i) \sin \theta+(4+7 i) \cos \theta$ be $...

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Q JEE MAIN_2025

Let $a_n$ be the $n^n$ term of an A.P. If $S_n=a_1+a_2+a_3+\ldots+a_n=700, a_0=7$ and $S_7=7$, then $a_n$ is equal to:

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Q JEE MAIN 2025

If the system of equations $$ \begin{aligned} & 2 x+\lambda y+3 z=5 \\ & 3 x+2 y-z=7 \\ & 4...

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The centre of a circle $C$ is at the centre of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$. Let $C$ pass through...

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Let $A=\{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R}:|\alpha-1| \leqslant 4$ and $|\beta-5| \leqslant 6\}$ and $...

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Q JEE MAIN 2025

If $\theta \in\left[-\frac{7 \pi}{6}, \frac{4 \pi}{3}\right]$, then the number of solutions of $\sqrt{3} \operatorname{cosec}^2 \theta-2(\sqrt{3}-1) \operatorname{cosec} \theta-4=0$, is equal tori

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Let $\quad f(x)+2 f\left(\frac{1}{x}\right)=x^2+5 \quad$ and $\quad 2 g(x)-3 g\left(\frac{1}{2}\right)=x, x>0$. If $\quad \alpha=\int_1^2 f(x) \mathrm{d} x, \quad$ and $...

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Q JEE MAIN_2025

If the equation of the line passing through the point $\left(0,-\frac{1}{2}, 0\right)$ and perpendicular to the lines $...

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Let $\mathrm{A}=\{-3,-2,-1,0,1,2,3\}$ and R be a relation on A defined by $x \mathrm{Ry}$ if and only if $2 x-y \in\{0,1\}$....

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Q JEE MAIN 2025

The number of terms of an A.P. is even; the sum of all the odd terms is 24 , the...

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Let $\vec{a}$ and $\vec{b}$ be the vectors of the same magnitude such that $\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|}=\sqrt{2}+1$. Then $\frac{|\vec{a}+\vec{b}|^2}{|\vec{a}|^2}$ is :

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If a curve $y=y(x)$ passes through the point $\left(1, \frac{\pi}{2}\right)$ and satisfies the differential equation $...

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Q JEE MAIN 2025

Let A be a $3 \times 3$ real matrix such that $A^2(A-2 I)-4(A-I)=O$, where I and O are the identity...

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Q JEE MAIN 2025

The sum $1+\frac{1+3}{2!}+\frac{1+3+5}{3!}+\frac{1+3+5+7}{4!}+\ldots$ upto $\infty$ terms, is equal to

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Q JEE MAIN 2025

If $\sum_{r=0}^{10}\left(\frac{10^{r+1}-1}{10^r}\right) \cdot{ }^{11} C_{r+1}=\frac{\alpha^{11}-11^{11}}{10^{10}}$, then $\alpha$ is equal to :

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A line passing through the point $\mathrm{A}(-2,0)$, touches the parabola $\mathrm{P}: y^2=x-2$ at the point $B$ in the first quadrant....

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If $z_1, z_2, z_3 \in \mathbb{C}$ are the vertices of an equilateral triangle, whose centroid is $z_0$, then $\sum_{k=1}^3\left(z_k-z_0\right)^2$ is...

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Q JEE MAIN_2025

Let the length of a latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be 10 . If its eccentricity is the minimum...

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Q JEE MAIN 2025

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the...

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Q JEE MAIN 2025

If the sum of the first 20 terms of the series $$ \frac{4 \cdot 1}{4+3 \cdot 1^2+1^4}+\frac{4 \cdot 2}{4+3 \cdot...

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Q JEE MAIN_2025

Let the system of equations $$ \begin{aligned} & x+5 y-z=1 \\ & 4 x+3 y-3 z=7 \\ & 24 x+y+\lambda...

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Q JEE MAIN 2025

Let . Let R be a relation on defined by if and only if . Let be the number of...

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Let for two distinct values of p the lines $y=x+\mathrm{p}$ touch the ellipse $\mathrm{E}: \frac{x^2}{4^2}+\frac{y^2}{3^2}=1$ at the points A and...

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Let $f:[1, \infty) \rightarrow[2, \infty)$ be a differentiable function. If $10 \int_1^x f(t) d t=5 x f(x)-x^5-9$ for all $...

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Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $\left(k, \frac{k}{2}\right)$ from the line $...

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Q JEE MAIN 2025

Given three identical bags each containing 10 balls, whose colours are as follows : A person chooses a bag at...

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Let $A$ be the point of intersection of the lines $L_1: \frac{x-7}{1}=\frac{y-5}{0}=\frac{z-3}{-1}$ and $L_2: \frac{x-1}{3}=\frac{y+3}{4}=\frac{z+7}{5}$. Let $B$ and $C$ be...

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Q JEE MAIN 2025

The area of the region $\{(x, y):|x-y| \leq y \leq 4 \sqrt{x}\}$ is

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Consider the lines $\mathrm{L}_1: x-1=y-2=\mathrm{z}$ and $\mathrm{L}_2: x-2=y=\mathrm{z}-1$. Let the feet of the perpendiculars from the point $P(5,1,-3)$ on the...

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Q JEE MAIN 2025

Let the mean and the standard deviation of the observation $2,3,3,4,5,7, a, b$ be 4 and $\sqrt{2}$ respectively. Then the...

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Let $A=\{1,2,3, \ldots, 100\}$ and $R$ be a relation on $A$ such that $R=\{(a, b): a=2 b+1\}$. Let $...

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The distance of the point $(7,10,11)$ from the line $\frac{x-4}{1}=\frac{y-4}{0}=\frac{z-2}{3}$ along the line $\frac{x-9}{2}=\frac{y-13}{3}=\frac{z-17}{6}$ is

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Let $f$ be a differentiable function on $\mathbf{R}$ such that $f(2)=1, f^{\prime}(2)=4$. Let $\lim _{x \rightarrow 0}(f(2+x))^{3 / x}=\mathbf{e}^\alpha$. Then...

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Let $(a, b)$ be the point of intersection of the curve $x^2=2 y$ and the straight line $y-2 x-6=0$ in...

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Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function defined by $f(x)=\|x+2|-2| x\|$. If $m$ is the number of points of...

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Q JEE MAIN 2025

If $1^2 \cdot\left({ }^{15} C_1\right)+2^2 \cdot\left({ }^{15} C_2\right)+3^2 \cdot\left({ }^{15} C_3\right)+\ldots+15^2 \cdot\left({ }^{15} C_{15}\right)=2^m \cdot 3^n \cdot 5^k$, where $...

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Q JEE MAIN 2025

Consider the lines $x(3 \lambda+1)+y(7 \lambda+2)=17 \lambda+5, \lambda$ being a parameter, all passing through a point P : One of...

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Q JEE MAIN 2025

Let the point P of the focal chord PQ of the parabola $y^2=16 x$ be $(1,-4)$. If the focus of...

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Line $L_1$ of slope 2 and line $L_2$ of slope $\frac{1}{2}$ intersect at the origin $O$. In the first quadrant,...

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If the domain of the function $f(x)=\log _7\left(1-\log _4\left(x^2-9 x+18\right)\right)$ is $(\alpha, \beta) \cup(\gamma, \delta)$, then $\alpha+\beta+\gamma+\delta$ is equal to

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Q JEE MAIN 2025

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes...

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Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the...

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Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+3\left(\tan ^2 x\right) y+3 y=\sec ^2 x, y(0)=\frac{1}{3}+e^3$. Then...

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If $\lim _{x \rightarrow 0} \frac{\cos (2 x)+a \cos (4 x)-b}{x^4}$ is finite, then $(a+b)$ is equal to :

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Q JEE MAIN 2025

The integral $\int_0^\pi \frac{8 x d x}{4 \cos ^2 x+\sin ^2 x}$ is equal to

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Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$...

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The line $L_1$ is parallel to vector $\vec{a}=-3 \hat{i}+2 \hat{j}+4 \hat{k}$ and passes through the point $(7,6,2)$ and the line...

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Q JEE MAIN 2025

If the probability that the random variable $X$ takes the value $x$ is given by $P(X=x)=k(x+1) 3^{-x}, x=0,1,2,3 \ldots$, where...

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Q JEE MAIN 2025

If the mean and the variance of 6, 4, a, 8, b, 12,10,13 are 9 and 9.25 respectively, then a...

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Let the values of p , for which the shortest distance between the lines $\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}$ and $...

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Let $C$ be the circle of minimum area enclosing the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\frac{1}{2}$ and foci $...

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Q JEE MAIN 2025

The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at...

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Let $f$ be a function such that $f(x)+3 f\left(\frac{24}{x}\right)=4 x, x \neq 0$. Then $f(3)+f(8)$ is equal to.

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Q JEE MAIN 2025

The shortest distance between the curves $y^2=8 x$ and $x^2+y^2+12 y+35=0$ is:

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If the domain of the function $f(x)=\frac{1}{\sqrt{10+3 x-x^2}}+\frac{1}{\sqrt{x+|x|}}$ is $(a, b)$, then $(1+a)^2+b^2$ is equal to:

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If the image of the point $\mathrm{P}(1,0,3)$ in the line joining the points $\mathrm{A}(4,7,1)$ and $\mathrm{B}(3,5,3)$ is $Q(\alpha, \beta, \gamma)$,...

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Let the mean of the data \begin{tabular}{|l|l|l|l|l|l|} \hline$x$ & 1 & 3 & 5 & 7 & 9 \\ \hline...

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Let $\vec{a}=3 \hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2 \hat{i}-3 \hat{j}+3 \hat{k}$. If $\vec{b}$ is a vector such that $\vec{a}=\vec{b} \times \vec{c}$ and $|\vec{b}|^2=50$,...

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Q JEE MAIN 2023

Let the image of the point $\left(\frac{5}{3}, \frac{5}{3}, \frac{8}{3}\right)$ in the plane $x-2 y+z-2=0$ be $P$. If the distance of...

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Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $P(4,1)$ to the hyperbola $H: \frac{y^2}{25}-\frac{x^2}{16}=1$....

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Q JEE MAIN 2023

The number of seven digit positive integers formed using the digits 1,2,3 and 4 only and sum of the digits...

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The sum to 20 terms of the series $2.2^2-3^2+2.4^2-5^2+2.6^2-\ldots \ldots$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2023

Let for $\mathrm{x} \in \mathbb{R}, \mathrm{So}(\mathrm{x})=\mathrm{x}$, $S_k(x)=C_k x+k \int_0^x S_{k-1}(t) d t$, where $...

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Let $\omega=\mathrm{z} \overline{\mathrm{z}}+\mathrm{k} 1 \mathrm{z}+\mathrm{k} 2 \mathrm{z}+\lambda(1+\mathrm{j}), \mathrm{k}_1, \mathrm{k} 2 \in \mathbb{R}$. Let $\operatorname{Re}(\omega)=0$ be the circle C of radius...

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If $S=\left\{x \in \mathbb{R}: \sin ^{-1}\left(\frac{x+1}{\sqrt{x^2+2 x+2}}\right)-\sin ^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right)=\frac{\pi}{4}\right\}$, then $\sum_{x \in \mathbb{R}}\left(\sin \left(x^2+x+5\right) \frac{\pi}{2}\right)-\cos \left(\left(x^2+x+5\right) \pi\right)$ is equal to $\_\_\_\_$...

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Q JEE MAIN 2023

Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{6}{x^{\frac{3}{2}}}\right)^n, n \leq 15$. If the sum of the...

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Let $f(x)=\int \frac{d x}{\left(3+4 x^2\right) \sqrt{4-3 x^2}},|x|0$, then $\alpha^2+\beta^2$ is equal to. $\_\_\_\_$ .

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Q JEE MAIN 2023

Let the equation of plane passing through the line of intersection of the planes $x+2 y+a z=2$ and $x-y +z=3$...

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Consider the triangles with vertices $A(2,1) B(0,0)$ and $C(t, 4), t \in[0,4]$. It the maximum and the minimum perimeters of...

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If the area bounded by the curve $2 y^2=3 x$, lines $x+y=3, y=0$ and outside the circle $(x-3)^2+y^2=2$ is $A$,...

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Let $y=y_1(x)$ and $y=y_2(x)$ be the solution curves of the differential equation $\frac{d y}{d x}=y+7$ with initial conditions $y_1(0)=y_2(0)=1$ respectively....

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Q JEE MAIN 2023

If the line $x=y=z$ intersects the line $...

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Let $s 1, s 2, s s_{3000-u s 10}$ respectively be the sum to 12 terms of 10 A.P.s whose...

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The number of elements in the set $\left\{n \in N: 10 \leq n \leq 100\right.$ and $3^n-3$ is a multiple...

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Let A = {1, 2, 3, 4} and R be a relation on the set A × A defined by...

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Let an ellipse with centre $(1,0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along $x$-axis. If its...

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Let the plane $P$ contain the line $2 x+y-z-3=0=5 x-3 y+4 z+9$ and be parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$. Then...

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Q JEE MAIN 2023

A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different,...

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Q JEE MAIN 2023

If the sum of the series

JEE Main Mathematics Medium

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If the domain of the function Then $36|\alpha+\beta|$ is equal to-

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Let S be the set of all $(\lambda, \mu)$ for which the vectors $\lambda \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\mu \hat{\mathrm{k}}$ and $...

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$\int_0^1 \frac{1}{\left(5+2 x-2 x^2\right)\left(1+e^{(2-4 x)}\right)} d x=\frac{1}{\alpha} \log _e\left(\frac{\alpha+1}{\beta}\right), \alpha, \beta>0$, then $\alpha^4-\beta^4$ is equal to:

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A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal...

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Let the foot of perpendicular of the point $\mathrm{P}(3,-2-9)$ on the plane passing through the points $(-1,-2,-3),(9,3,4),(9,-2,1)$ be $...

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If $(\alpha, \beta)$ is the orthocentre of the triangle ABC with vertices $\mathrm{A}(3,-7), \mathrm{B}(-1,2)$ and $\mathrm{C}(4,5)$, then $9 \alpha-6 \beta+60$...

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Q JEE MAIN 2023

Let the system of linear equations has a unique solution $x=\alpha, y=\beta, z=\gamma$. Then the distance of the point $...

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The shortest distance between the lines $\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}$ and $\frac{x-4}{1}=\frac{y-1}{-2}=\frac{z+3}{0}$ is

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Q JEE MAIN 2023

Negation of $p \wedge(q \wedge \sim(p \wedge q))$ is

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Q JEE MAIN 2023

$$ \text { The number of common tangents, to the circles } x^2+y^2-18 x-15 y+131=0 \text { and } x^2+y^2-6...

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$$ \text { If the set }\left\{\operatorname{Re}\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right): z \in \mathbb{C}, \operatorname{Re}(z)=3\right\} \text { is equal to the...

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Q JEE MAIN 2023

Let $B=\left[\begin{array}{ccc}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & \alpha\end{array}\right], \alpha>2$ be...

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Q JEE MAIN 2023

Let $\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$ and $\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}$. If a vector $\vec{d}$ satisfies $...

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Q JEE MAIN 2023

Fractional part of the number $\frac{4^{2022}}{15}$ is equal to

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For $x \in R$, two real valued functions $f(x)$ and $g(x)$ are such that, $g(x)=\sqrt{x}+1$ and $f g(x)=x+3-\sqrt{x}$. Then $f(0)$...

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Q JEE MAIN 2023

Let P Q be a focal chord of the parabola $y^2=36 x$ of length 100 , making an acute angle...

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Q JEE MAIN 2023

Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2, G_3$ be three geometric means of two distinct positive...

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Q JEE MAIN 2023

Let the determinant of a square matrix A of order m be m – n, where m and n satisfy...

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Q JEE MAIN 2023

Let $\left(a+b x+c x^2\right)^{10}=\sum_{i=0}^{20} p_i x^i, a, b, c \in \mathbb{N}$. If $p_1=20$ and $p_2=210$, then $2(a+b+c)$ is equal to

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Q JEE MAIN 2023

The number of symmetric matrices of order 3, with all the entries from the set {0, 1, 2, 3, 4,...

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Q JEE MAIN 2023

The area of the region enclosed by the curve $f(x)=\max \{\sin x, \cos x\}-\pi \leq x \leq \pi$ and the...

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Q JEE MAIN 2023

The number of real roots of the equation $x|x|-5|x+2|+6=0$, is

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Q JEE MAIN 2023

Let the tangent and normal at the point $(3 \sqrt{3}, 1)$ on the ellipse $\frac{x^2}{36}+\frac{y^2}{4}=1$ meet the $y$-axis at the...

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Q JEE MAIN 2023

Let $[x]$ denote the greatest integer function and $f(x)=\max \{1+x+[x], 2+x, x+2[x]\}, 0 \leq x \leq 2$. Let $m$ be...

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Q JEE MAIN 2023

Let $x=x(y)$ be the solution of the differential equation $2(y+2) \log _e(y+2) d x+\left(x+4-2 \log _e(y+2)\right) d y=0, y>-1$ with...

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Q JEE MAIN 2023

For the differentiable function $f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$, let $3 f(x)+2 f\left(\frac{1}{x}\right)=\frac{1}{x}-10$, then $\left|f(3)+f^{\prime}\left(\frac{1}{4}\right)\right|$ is equal to

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Q JEE-Main 2023

The sum of all the roots of the equation is :$\left|x^2-8 x+15\right|-2 x+7=0$ is :

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Q JEE MAIN 223

For the system of linear equations $$ \begin{aligned} & 2 x+4 y+2 a z=b \\ & x+2 y+3 z=4 \\...

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Q JEE-Main 2023

Statement $(P \Rightarrow Q) \wedge(R \Rightarrow Q)$ is logically equivalent to

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Q JEE MAIN 2023

Let $A B C D$ be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $...

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Q JEE-Main 2023

If the equation of the plane passing through the line of intersection of the planes 2x - y + z...

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Q JEE MAIN 2023

The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one...

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Let $I(x)=\int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x$. If $I(0)=0$ the $I\left(\frac{\pi}{4}\right)$ is equal to

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Q JEE MAIN 2023

Let S be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac{x-\lambda}{0}=\frac{y-3}{4}=\frac{x+6}{1}$ and...

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Q JEE MAIN 2023

A coin is biased so that the head is 3 times as likely to occur as tail. This coin is...

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Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{i}}\right] 2 \times 2$ where $\mathrm{a}_{\mathrm{i}} \neq 0$ for all $\mathrm{i}, \mathrm{j}$ and $\mathrm{A}^2=\mathrm{I}$. Let a be the sum...

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Q JEE MAIN 2023

The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8,...

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Q JEE MAIN 2023

The distance of the point $(-1,2,3)$ from the plane $\vec{r} \cdot(\hat{i}-2 \hat{j}+3 \hat{k})=10$ parallel to the line of the shortest...

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Q JEE-Main 2023

The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of...

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Q JEE MAIN 2023

Let $A=\left[\begin{array}{lll}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{array}\right]$, where $...

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Q JEE-Main 2023

The sum of the first 20 terms of the series 5 +11 + 19 + 29 + 41 + ......

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Q JEE MAIN 2023

The negation of the statement $((A \wedge(B \vee C)) \Rightarrow(A \vee B)) \Rightarrow A$ is

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Q JEE MAIN 2023

If a and b are the roots of equation $x^2-7 x-1=0$, then the value of $\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is equal to

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Q JEE MAIN 2023

If $\int_{-0.15}^{0.15}\left|100 \mathrm{x}^2-1\right| \mathrm{dx}=\frac{\mathrm{k}}{3000}$, then k is equal to $\_\_\_\_$ .

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Q JEE-Main 2023

From the top A of a vertical wall AB of height 30 m, the angles of depression of the top...

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Q JEE MAIN 2023

Let $H_n=\frac{x^2}{1+n}-\frac{y^2}{3+n}=1, n \in N$. Let $k$ be the smallest even value of $n$ such that the eccentricity of $H_k$...

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Q JEE MAIN 2023

The number of relations, on the set {1, 2, 3} containing (1, 2) and (2,3), which are reflexive and transitive...

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Q JEE MAIN 2023

The set of all $a \in \mathbb{R}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root is :

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Q JEE MAIN 2023

The number of ordered triplets of the truth values of $p, q$ and $r$ such that the truth value of...

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Q JEE-Main 2023

The straight lines $I_1$ and $I_2$ pass through the origin and trisect the line segment of the line L: 9x...

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Q JEE MAIN 2023

Let the positive numbers $a_1, a_2, a_3, a_4$ and $a_5$ be in a G.P. Let their mean and variance be...

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Q JEE MAIN 2023

The number of integral terms in the expansion of $\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680}$ is equal to

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Q JEE MAIN 2023

$\max _{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=$

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Q JEE-Main 2023

Let $\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}-2 \hat{k}$ and $\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$. If $\vec{d}$ is a vector perpendicular to...

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Q JEE MAIN 2023

Let a line $I$ pass through the origin and be perpendicular to the lines $...

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Q JEE MAIN 2023

Let $D_k=\left|\begin{array}{ccc}1 & 2 k & 2 k-1 \\ n & n^2+n+2 & n^2 \\ n & n^2+n & n^2+n+2\end{array}\right|$....

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Q JEE MAIN 2023

In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in...

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Q JEE MAIN 2023

$\int_0^{\infty} \frac{6}{e^{3 x}+6 e^{2 x}+1}$

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Q JEE-Main

If the ratio of the fifth term from the begining to the fifth term from the end in the expansion...

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Q JEE MAIN 2023

For $m, n>0$, let $\alpha(m, n)=\int_0^2 t^m(1+3 t)^n d t$. If $11 \alpha(10,6)+18 \alpha(11,5)=p(14)^6$, then $p$ is equal to

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Q JEE MAIN 2023

Let $I(x)=\int \sqrt{\frac{x+7}{x}} d x$ and $I(9)=12+7 \log _e 7$. If $I(1)=\alpha+7 \log _e(1+2 \sqrt{2})$, then $\alpha^4$ is equal to...

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Q JEE MAIN 2023

Let $S=109+\frac{108}{5}+\frac{107}{5^2}+\ldots \ldots \ldots+\frac{2}{5^{107}}+\frac{1}{5^{108}}$. Then the value of $\left(16 S-(25)^{-54}\right)$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2023

Let the plane $x+3 y-2 z+6=0$ meet the co-ordinate axes at the points A,B,C. It the orthocentre of the triangle...

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Q JEE-Main_2023

If ${ }^{2 n} C_3 \cdot{ }^n C_3=10: 1$, then the ratio $\left(n^2+3 n\right):\left(n^2-3 n+4\right)$ is

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Q JEE MAIN 2023

The mean of the coefficients of $x, x^2$ $\_\_\_\_$ $x^7$ in the binomial expansion of $(2+x)^9$ is $\_\_\_\_$ .

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Q JEE MAIN 2023

Let $[x]$ be the greatest integer $\leq x$. Then the number of points in the interval $(-2,1)$, where the function...

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Q JEE-Main 2023

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If...

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Q JEE MAIN 2023

Let the mean and variance of 8 numbers x, y, 10, 12, 6, 12, 4, 8, be 9 and 9.25...

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Q JEE MAIN 2023

Let $x_1, x_2$ $\_\_\_\_$ $x_{100}$ be in an arithmetic progression, with $x_1=2$ and their mean equal to 200 . If...

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Q JEE MAIN 2023

Let the digits a, b, c be in A.P. Nine-digit numbers are to be formed using each of these three...

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Q JEE MAIN 2023

Let $\lambda_1, \lambda_2$ be the values of $\lambda$ for which the points $\left(\frac{5}{2}, 1, \lambda\right)$ and $(-2,0,1)$ are at equal...

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Q JEE MAIN 2023

The number of elements in the set $S=\left\{\theta \in[0,2 \pi]: 3 \cos ^4 \theta-5 \cos ^2 \theta-2 \sin ^2 \theta+2=0\right\}$...

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Q JEE MAIN 2023

A fair n (n > 1) faces die is rolled repeatedly until a number less than n appears. If the...

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Q JEE MAIN 2023

If the solution curve of the differential equation $\left(y-2 \log _e x\right) d x+\left(x \log _e x^2\right) d y=0, x>1$...

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Q JEE MAIN 2023

Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j}, \hat{i}+\hat{k}$...

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Q JEE MAIN 2023

In a triangle $A B C$, if $\cos A+2 \cos B+\cos C=2$ and the lengths of the sides opposite to...

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Q JEE MAIN 2023

Consider a circle $C_1: x^2+y^2-4 x-2 y=\alpha-5$. Let its mirror image in the line $y=2 x+1$ be another circle $...

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Q JEE MAIN 2023

Let $(\alpha, \beta, \gamma)$ be the image of the point $\mathrm{P}(2,3,5)$ in the plane $2 \mathrm{x}+\mathrm{y}-3 \mathrm{z}=6$. Then $\alpha+\beta+\gamma$ is...

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Q JEE MAIN 2023

Two dice A and B are rolled, Let the numbers obtained on A and B be $\alpha$ and $\beta$ respectively....

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Q JEE MAIN 2023

Let $A$ be a $2 \times 2$ matrix with real entries such that $A^{\prime}=\alpha A+I$, where $\alpha \in \mathbb{R}-\{-1,1\}$. If...

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Q JEE MAIN 2023

Let A = {0, 3, 4, 6, 7, 8, 9, 10} and R be the relation defined on A such...

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Q JEE MAIN 2023

If the point $\left(\alpha, \frac{7 \sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments of the...

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Q JEE MAIN 2023

If $a_n$ is the greatest term in the sequence $a_n=\frac{n^3}{n^4+147}, n=1,2,3 \ldots \ldots$, then $a$ is equal to $\_\_\_\_$ .

JEE Main Mathematics Hard

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Q JEE MAIN 2023

Let $y=y(x)$ be a solution curve of the differential equation, $\left(1-x^2 y^2\right) d x=y d x+x d y$. If the...

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Q JEE-Main 2023

Let $5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3, x>0$. Then $18 \int_1^2 f(x) d x$ is equal to :

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Q JEE MAIN 2023

The largest natural number $n$ such that $3^n$ divides $66!$ is $\_\_\_\_$

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Q JEE MAIN 2023

Let $f:[2,4] \rightarrow \mathbb{R}$ be a differentiable function such that $...

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Q JEE MAIN 2023

Let $\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}$, and $\vec{c}$ be vectors such that $...

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Q JEE MAIN 2023

Let C be the circle in the complex plane with centre $z_0=\frac{1}{2}(1+3 i)$ and radius $r=1$. Let $z_1=1+i$ and the...

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Q JEE MAIN 2023

Let $[t]$ denotes the greatest integer $\leq t$. If the constant term in the expansion of $\left(3 x^2-\frac{1}{2 x^5}\right)^7$ is...

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Q JEE MAIN 2023

Let $[t]$ denotes the greatest integer $\leq t$. Then $...

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Q JEE MAIN 2023

Let the plane $P: 4 x-y+z=10$ be rotated by an angle $\frac{\pi}{2}$ about its line of intersection with the plane...

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Q JEE MAIN 2023

Let $A=\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]$. If $...

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Q JEE MAIN 2023

If for $z=\alpha+i \beta=|z+2|=z+4(1+i)$, then $\alpha+\beta$ and $\alpha \beta$ are the roots of the equation

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Q JEE MAIN 2023

Area of the region $\left\{(x, y): x^2+(y-2)^2 \leq 4, x^2 \geq 2 y\right\}$ is

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Q JEE MAIN 2023

The area of the region enclosed by the curve $y=x^3$ and its tangent at the point $(-1,-1)$ is

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Q JEE MAIN 2023

$\lim _{x \rightarrow 0}\left(\left(\frac{1-\cos ^2(3 x)}{\cos ^3(4 x)}\right)\left(\frac{\sin ^3(4 x)}{\left(\log _e(2 x+1)\right)^5}\right)\right)$ is equal to $\_\_\_\_$

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Q JEE MAIN 2023

The number of integral solutions $x$ of $\log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^2 \geq 0$ is

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Q JEE MAIN 2023

The sum, of the coefficients of the first 50 terms in the binomial expansion of $(1-x)^{100}$, is equal to

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Q JEE MAIN 2023

Let the number of elements in sets A and B be five and two respectively. Then the number of subsets...

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Q JEE MAIN 2023

The number of elements in the set $\left\{n \in \mathbb{Z}:\left|n^2-10 n+19\right|

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Q JEE MAIN 2023

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played...

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Q JEE MAIN 2023

Let $\lambda \in Z, \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}$. Let $\vec{c}$ be a vector such that $...

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Q JEE MAIN 2023

The area of the region $\left\{(x, y): x^2 \leq y \leq 8-x^2, y \leq 7\right\}$ is -

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Consider ellipses $E_k: k x^2+k^2 y^2=1, k=1,2, \ldots \ldots, 20$. Let $C_k$ be the circle which touches the four chords...

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Q JEE MAIN 2023

If the mean of the frequency distribution then its variance is ______.

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Q JEE MAIN 2023

Among the two statements $(\mathrm{S} 1) \sim(\mathrm{p} \Rightarrow \mathrm{q}) \wedge(\mathrm{q} \wedge(\sim \mathrm{q}))$ is a contradiction and $$ (S 2) \sim(p...

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Q JEE-Main_2023

The negation of the statement $((A \wedge(B \vee C)) \Rightarrow(A \vee B)) \Rightarrow A$ is

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Q JEE MAIN 2023

The shortest distance between the lines $\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3}$ and $\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2}$ is -

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Q JEE MAIN 2023

Let ij $\mathrm{S}=\left\{\mathrm{M}=\left[a_{i j}\right], a_{i j} \in\{0,1,2\}, 1 \leq \mathrm{i}_k \mathrm{j} \leq 2\right\}$ be a sample space and $...

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Q JEE MAIN 2023

Let $y=p(x)$ be the parabola passing through the points $(-1,0),(0,1)$ and $(1,0)$. If the area of the region $...

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Q JEE MAIN 2023

An organization awarded 48 medals in event ‘A’, 25 in event ‘B’ and 18 in event ‘C’. If these medals...

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Let $y=y(x), y>0$, be a solution curve of the differential equation $\left(1+x^2\right) d y=y(x-y) d x$. If $y(0)=1$ and $...

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Q JEE MAIN 2023

Negation of $(p \Rightarrow q) \Rightarrow(q \Rightarrow p)$ is

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Q JEE MAIN 2023

Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be three distinct positive real numbers such that $(2 \mathrm{a})^{\log _{\mathrm{a}} \mathrm{a}}=(\mathrm{bc})^{\log _{\mathrm{a}} \mathrm{b}}$ and $...

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Q JEE MAIN 2023

Let w1 be the point obtained by the rotation of z1 = 5 + 4i about the origin through a...

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Q JEE MAIN 2023

The number of permutations, of the digits 1, 2, 3, …..7 without repetition, which neither contain the string 153 nor...

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Q JEE MAIN 2023

https://competishun.com/let-d-be-the-domain-of-the-function-fxsin-1leftlog-_3-xleftfrac62-log-_3-x-5-xrightright-if-the-range-of-the-function-g_i-d-rightarrow-r-defined-by-gxx-xleftxright-is-the-greatest-integer-function-is-alpha-beta-then-alpha2frac5beta-is-equal-to/

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Q JEE-Main_2023

The set of all $a \in \mathbb{R}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root is :

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Q JEE MAIN 2023

Let $A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right]$. If |adj...

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Q JEE MAIN 2023

Let a common tangent to the curves $y^2=4 x$ and $(x-4)^2+y^2=16$ touch the curves at the points $P$ and $Q$....

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Q JEE MAIN 2023

For any vector $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$, with $10\left|a_i\right|

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Q JEE MAIN 2023

If the local maximum value of the function $f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^2 x}, x \in\left(0, \frac{\pi}{2}\right)$, is $\frac{k}{e}$, then...

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Q JEE MAIN 2023

Let $S_k=\frac{1+2+\ldots \ldots+K}{K}$ and $\sum_{j=1}^n S_j^2=\frac{n}{A}\left(B n^2+C n+D\right)$, where $A, B, C, D \in N$ and $A$ has least value....

JEE Main Mathematics Hard

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Q JEE-Main_2023

$\max _{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}$

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Q JEE MAIN 2023

The sum of all those terms, of the arithmetic progression 3, 8, 13,…… 373, which are not divisible by 3,...

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Q JEE MAIN 2023

Let $a, b, c$ be three distinct real numbers, none equal to one. If the vectors $a \hat{i}+\hat{j}+\hat{k}, \hat{i}+b \hat{j}+\hat{k}$...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

If the coefficients of the three consecutive terms in the expansion of $(1+x)^n$ are in the ratio $1: 5: 20$,...

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Q JEE MAIN 2023

The number of triplets (x, y, z). where x, y, z are distinct non negative integers satisfying x + y...

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Q JEE MAIN 2023

Let $P\left(\frac{2 \sqrt{3}}{\sqrt{7}}, \frac{6}{\sqrt{7}}\right), Q, R$ and $S$ be four points on the ellipse $9 x^2+4 y^2=36$. Let $P Q$...

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Q JEE-Main_2023

$\int_0^{\infty} \frac{6}{e^{3 x}+6 e^{2 x}+11 e^x+6} d x$

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Q JEE MAIN 2023

The coefficient of $x^7$ in $\left(1-x+2 x^3\right)^{10}$ is $\_\_\_\_$

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Q JEE MAIN 2023

Let $f(x)=\left[x^2-x\right]+|-x+[x]|$, where $x \in \mathbb{R}$ and $[t]$ denotes the greatest integer less than or equal to $t$. Then, f...

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Q JEE MAIN 2023

If the points with vectors $\alpha \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+13 \hat{\mathrm{k}}, 6 \hat{\mathrm{i}}+11 \hat{\mathrm{j}}+11 \hat{\mathrm{k}}, \frac{9}{2} \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}-8 \hat{\mathrm{k}}$ are collinear, then...

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Q JEE MAIN 2023

An are PQ of a circle subtends a right angle at its centre O . The mid point of the...

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Q JEE MAIN 2023

Let sets A and B have 5 elements each. Let the mean of the elements in sets A and B...

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Q JEE MAIN 2023

Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum...

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Let $\alpha, \beta$ be the roots of the quadratic equation $x^2+\sqrt{6} x+3=0$. Then $\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}$ is

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Q JEE MAIN 2023

Let $f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}, x \in[0, \pi]-\left\{\frac{\pi}{4}\right\}$. Then $f\left(\frac{7 \pi}{12}\right) f^{\prime \prime}\left(\frac{7 \pi}{12}\right)$ is equal to -

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Let the tangent to the curve $x^2+2 x-4 y+9=0$ at the point P(1, 3) on it meet the y-axis at...

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If $I(x)=\int e^{\sin 2} x(\cos x \sin 2 x-\sin x) d x$ and $I(0)=1$, then $I\left(\frac{\pi}{3}\right)$ is equal to

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Q JEE MAIN 2023

Let R be a rectangle given by the lines $x=0, x=2, y=0$ and $y=5$. Let $A(\alpha, 0)$ and $...

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Q JEE MAIN 2023

The number of five digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits...

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Let the image of the point P(1, 2, 3) in the plane 2x – y + z = 9 be...

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Q JEE MAIN 2023

The number of arrangements of the letter of the word “INDEPENDENCE” in which all the vowels always occur together is-

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Q JEE MAIN 2023

If equation of the plane that contains the point $(-2,3,5)$ and is perpendicular to each of the planes $...

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Let A={1, 2, 3, 4,.....10} and B = {0, 1, 2, 3, 4}. The number of elements in the relation...

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Let two vertices of triangle ABC be $(2,4,6)$ and $(0,-2,-5)$, and its centroid be $(2,1,-1)$. If the image of third...

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Q JEE MAIN 2023

In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. of...

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The coefficient of $x^{18}$ in the expansion of $\left(x^4-\frac{1}{x^3}\right)^{15}$ is $\_\_\_\_$

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The value of the integral $\int_{-\log _e 2}^{\log _e 2} \mathrm{e}^{\mathrm{x}}\left(\log _{\mathrm{e}}\left(\mathrm{e}^{\mathrm{x}}+\sqrt{1+\mathrm{e}^{2 \mathrm{x}}}\right)\right) \mathrm{dx}$ is equal to

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Q JEE MAIN 2023

If the coefficient of $x^7$ in $\left(a x-\frac{1}{b x^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(a x+\frac{1}{b x^2}\right)^{13}$ are equal,...

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Q JEE MAIN 2023

The number of ways, in which 5 girls and 7 boys can be seated at a round table so that...

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Let y = y(x) be a solution of the differential equation (xcos x)dy + (xysinx + ycos x – l)dx...

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Q JEE MAIN 2023

The negation of the statement $(p \vee q)^{\wedge}(q \vee(\sim r))$ is

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Let $P$ be the point of intersection of the line $\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}$ and the plane $x+y+z=2$. If the distance of the...

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Let $\alpha, \beta, \gamma$ be the three roots of the equation $x^3+b x+c=0$. If $\beta \gamma=1=-\alpha$, then $...

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Let the point $(p, p+1)$ lie inside the region $...

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Q JEE MAIN 2023

For the system of linear equations 2x - y + 3z = 5 3x + 2y – z = 7...

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Q JEE MAIN 2023

Let $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=P Q P^{\top}$. If...

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If the area of the region $\mathrm{S}=\left\{(\mathrm{x}, \mathrm{y}) ; 2 \mathrm{y}-\mathrm{y}^2 \leq \mathrm{x}^2 \leq 2 \mathrm{y}, \mathrm{x} \geq \mathrm{y}\right\}$ is...

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Q JEE MAIN 2023

The slope of tangent at any point $(x, y)$ on a curve $y=y(x)$ is $\frac{x^2+y^2}{2 x y}, x>0$. If $y(2)=0$,...

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The number of ways of giving 20 distinct oranges to 3 children such that each child gets atleast one orange...

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Q JEE MAIM 2023

Let $C(\alpha, \beta)$ be the circumcenter of the triangle formed by the lines $$ \begin{aligned} & 4 x+3 y=69 \\...

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Q JEE-Main 2023

A circle passing through the point P($\alpha, \beta$) in the first quadrant touches the two coordinate axes at the points...

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Q JEE MAIN 2023

If $A$ is a $3 \times 3$ matrix and $|A|=2$, then $\left|3 \operatorname{adj}\left(|3 A| A^2\right)\right|$ is equal to

JEE Main Mathematics Easy

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Q JEE MAIN 2023

$96 \cos \frac{\pi}{33} \cos \frac{2 \pi}{33} \cos \frac{4 \pi}{33} \cos \frac{8 \pi}{33} \cos \frac{16 \pi}{33}$ is equal to

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Q JEE-Main 2023

Let $a \in Z$ and [t] be the greatest integer $\leq$ t. Then the number of points, where the function...

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Q JEE MAIN 2023

Let R be the focus of the parabola $\mathrm{y}^2=20 \mathrm{x}$ and the line $\mathrm{y}=m \mathrm{x}+c$ intersect the parabola at two...

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Q JEE MAIN 2023

Let the complex number $z=x+i y$ be such that $\frac{2 z-3 i}{2 z+i}$ is purely imaginary. If $x+y^2=0$, then $y^4+y^2-y$...

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Q JEE-Main 2023

Let $A=\{x \in R ;[x+3]+[x+4] \leq 3\},\left\{x \in R: 3^x\left(\sum_{t=1}^{\infty} \frac{3}{10^t}\right)^{x-3}

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Q JEE MAIN 2023

A line segment AB of length l moves such that the points A and B remain on the periphery of...

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Q JEE MAIN 2023

If the equation of the plane containing the line $x+2 y+3 z-4=0=2 x+y-z+5$ and perpendicular to the plane $...

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Q JEE-Main 2023

Let the position vectors of the points A, B, C and D be $...

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Let $f$ be a differentiable function such that $x^2 f(x)-x=4 \int_0^x t f(t) d t, f(1)=\frac{2}{3}$. Then $18 f(3)$ is...

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Q JEE MAIN 2023

Let $I(x)=\int \frac{(x+1)}{x\left(1+x e^x\right)^2} d x, x>0$, If $\lim _{x \rightarrow \infty} I(x)=0$, then $I(1)$ is equal to

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Let the ellipse $E_{\text {in }} x^2+9 y^2=9$ intersect the positive $x$ - and $y$-axes at the points $A$ and...

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Q JEE-Main 2023

One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y...

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Q JEE MAIN 2023

A square piece of tin of side 30 cm is to be made into a box without top by cutting...

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Q JEE-Main 2023

If $2 x^y+3 y^x=20$, then $\frac{d y}{d x}$ at $(2,2)$ is equal to

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Q JEE MAIN 2023

If $f(x)=\frac{\left(\tan 1^0\right) x+\log _e(123)}{x \log _e(1234)-\left(\tan 1^0\right)}, x>0$, then the least value of $f(f(x))+f\left(f\left(\frac{4}{x}\right)\right)$ is

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Q JEE-Main 2023

If the system of equations x + y + az = b 2x + 5y + 2z = 6 x...

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Q JEE-Main 2023

Let a1, a2, a3 ...... an be n positive consecutive terms of an arithmetic progression. If d > 0 is...

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Let $O$ be the origin and the position vector of the point $P$ be $-\hat{i}-2 \hat{j}+3 \hat{k}$. If the position...

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Q JEE-Main 2023

The sum of all the roots of the equation $\left|x^2-8 x+15\right|-2 x+7=0$ is :

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Q JEE-Main 2023

If the equation of the plane passing through the line of intersection of the planes 2x - y + z...

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$\operatorname{Let} I(x)=\int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x$. If $I(0)=0$ the $I\left(\frac{\pi}{4}\right)$ is equal to

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Let $\mathrm{A}=\left[a_{i j}\right]_{2 \times 2}$ where $a_{i j} \neq 0$ for all $\mathrm{j}_i \mathrm{j}$ and $\mathrm{A}^2=\mathrm{I}$. Let a be the...

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Q JEE-Main 2023

The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of...

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Q JEE-Main

The sum of the first 20 terms of the series 5 +11 + 19 + 29 + 41 + ......

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Q JEE-Main 2023

The straight lines l1 and l2 pass through the origin and trisect the line segment of the line L: 9x...

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Let $\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}-2 \hat{k}$ and $\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$. If $\vec{d}$ is a vector perpendicular to...

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Q JEE-Main 2023

If the ratio of the fifth term from the begining to the fifth term from the end in the expansion...

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If ${ }^{2 n} C_3 \cdot{ }^n C_3=10: 1$, then the ratio $\left(n^2+3 n\right):\left(n^2-3 n+4\right)$ is

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Q JEE-Main 2023

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If...

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Let $5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3, x>0$. Then $18 \int_1^2 f(x) d x$ is equal to :

JEE Main Mathematics Easy

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Q JEE MAIN 2024

If $\sin x = - \frac{3}{5}$, where $\pi < x < \frac{{3\pi }}{2}$, then $...

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Q JEE MAIN

Let f:(0,π)→R be a function given by Where a,b∈Z. If f is continuous at $\mathrm{x}=\frac{\pi}{2}$, then $\mathrm{a}^2+\mathrm{b}^2$ is equal to

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Let A={2,3,6,7} and B={4,5,6,8}. Let R be a relation defined on A×B by $(a_1,b_1 )R(a_2,b_2 ) $ is and only...

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If a function f satisfies f(m+n)=f(m)+f(n) for all m,n∈N and f(1)=1, then the largest natural number λ such that $...

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Let the centre of a circle, passing through the point (0,0),(1,0) and touching the circle $x^2+y^2$ =9, be (h,k). Then...

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Q JEE MAIN

Let A be a non-singular matrix of order 3. If $\operatorname{det}(3 \operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A)))=3^{-13} \cdot 2^{-10}$ and det $...

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Q JEE MAIN

The remainder when $428^{2024}$ is divided by 21 is

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Let $...

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Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $\left(1+x\left(\lambda^2-x^2\right)\right)$...

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The sum of the square of the modulus of the elements in the set {z=a+ib:a,b∈Z,z∈C,|z-1|≤1,|z-5|≤|z-5i|} is

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Let a,b and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are...

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Let $f(x)=x^2+9, g(x)=\frac{x}{x-9}$ and $a=\operatorname{fog}(10), b=\operatorname{gof}(3)$. If e and 1 denote the eccentricity and the length of the latus rectum...

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Let $\overrightarrow{\mathrm{a}}=2 \hat{\imath}-3 \hat{\jmath}+4 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=3 \hat{\imath}+4 \hat{\jmath}-5 \hat{\mathrm{k}}$, and a vector $\vec{c}$ be such that $...

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Let α,β be the roots of the equation $x^2+2 \sqrt{2} x-1=0$. . The quadratic equation, whose roots are $\alpha^4+\beta^4$ and...

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Q JEE MAIN

Let three vectors $\vec{a}=\alpha \hat{\imath}+4 \hat{\jmath}+2 \hat{k}, \vec{b}=5 \hat{\imath}+3 \hat{\jmath}+4 \hat{k}, \vec{c}=x \hat{\imath}+y \hat{\jmath}+z \hat{k}$ from a triangle such that...

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Let the first term of a series be $\mathrm{T}_1=6$ and its $\mathrm{r}^{\text {th }}$ term $T_r=3 T_{r-1}+6^r, r=2,3, \ldots, n$....

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If the second, third and fourth terms in the expansion of $(x+y)^n$ are 135,30 and $\frac{10}{3}$, respectively, then $6\left(n^3+x^2+y\right)$ is...

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Let $L_1, L_2$ be the lines passing through the point $P(0,1)$ and touching the parabola $9 x^2+12 x+18 y- 14=0$....

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The solution of the differential equation $\left(x^2+y^2\right) d x-5 x y d y=0, y(1)=0$, is :

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Let $\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4$ be the solution of the equation $4 x^4+8 x^3-17 x^2-12 x+9=0$ and $\left(4+x_1^2\right)\left(4+x_2^2\right)(4+ \left.x_3^2\right)\left(4+x_4^2\right)=\frac{125}{16} m$....

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The frequency distribution of the age of students in a class of 40 students is given below. If the mean...

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Q JEE MAIN 2024

The value of $\lim _{x \rightarrow 0} 2\left(\frac{1-\cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \ldots \ldots 10}{x^2}\right)$ is

JEE Main Mathematics Medium

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The shortest distance between the line $\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5}$ and $\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}$ is :

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Q JEE MAIN 2024

Let the area of the region enclosed by the curve $y=\min \{\sin x, \cos x\}$ and the $x$-axis between $x=-\pi$...

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Let $r_k=\frac{\int_0^1\left(1-x^7\right)^k d x}{\int_0^1\left(1-x^7\right)^{k+1} d x}, k \in N$. Then the value of $\sum_{\mathrm{k}=1}^{10} \frac{1}{7\left(\mathrm{r}_{\mathrm{k}}-1\right)}$ is equal to $\_\_\_\_$

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Q JEE MAIN 2024

Suppose $A B$ is a focal chord of the parabola $y^2=12 x$ of length $l$ and slope $m

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Let a circle passing through (2,0) have its centre at the point (h,k). Let $\left(x_c, y_c\right)$ be the point of...

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Q JEE MAIN 2024

The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is

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Q JEE MAIN 2024

If $\int_0^{\frac{\pi}{4}} \frac{\sin ^2 x}{1+\sin x \cos x} d x=\frac{1}{a} \log _e\left(\frac{a}{3}\right)+\frac{\pi}{b \sqrt{3}}$, where $a, \mathrm{~b} \in \mathrm{~N}$, then $\mathrm{a}+\mathrm{b}$...

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Let a conic C pass through the point (4,-2) and P(x,y),x≥3, be any point on C. Let the slope of...

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Let $\vec{a}=9 \hat{\imath}-13 \hat{\jmath}+25 k, b=3 \hat{\imath}+7 \hat{\jmath}-13 k$ and $\overrightarrow{\mathrm{c}}=17 \hat{\imath}-2 \hat{\jmath}+\mathrm{k}$ be three given vectros. If $\overrightarrow{\mathrm{r}}$ is...

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Q JEE MAIN 2024

Let $\vec{a}=\hat{i}-3 \hat{j}+7 k \hat{k}=2 \hat{i}-\hat{j}+k$ and $\vec{c}$ be a vector such that $(\vec{a}+2 \vec{b}) \times \vec{c}=3(\vec{c} \times \vec{a})$. If...

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Let $f(x)=a x^3+b x^2+e x+41$ be such that $f(1)=40, f^{\prime}(1)=2$ and $f^{\prime \prime}(1)=4$. Then $a^2+b^2+c^2$ is equal to :

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Q JEE MAIN 2024

Let ABC be a triangle of area $15 \sqrt{2}$ and the vectors $\overrightarrow{\mathrm{AB}}=\hat{\imath}+2 \hat{\jmath}-7 \hat{k}, \overrightarrow{\mathrm{BC}}=a \hat{\imath}+b \hat{\jmath}+c \hat{k}$ and...

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Let αβγ=45;α,β,γ∈R. If x(α,1,2)+y(1,β,2) +z(2,3,γ)=(0,0,0) for some x,y,z∈R,xyz≠ 0, then 6α+4β+γ is equal to_____.

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Let $a_1, a_2, a_3, \ldots$ be in an arithmetic progression of positive terms. Let $...

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Let the length of the focal chord $P Q$ of the parabola $\mathrm{y}^2=12 \mathrm{x}$ be 15 units. If the distance...

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If the sum of series $\frac{1}{1 \cdot(1+d)}+\frac{1}{(1+d)(1+2 d)}+\cdots \ldots+\frac{1}{(1+9 d)(1+10 d)}$ is equal to 5 , then 50 d is...

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Q JEE MAIN 2024

Let $f$ be a differentiable function in the interval $(0, \infty)$ such that $f(1)=1$ and $...

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Let $\mathrm{f}:(-\infty, \infty)-\{0\} \rightarrow \mathrm{R}$ be a differentiable function such that $f^{\prime}(1)=\lim _{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)$. Then $...

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If the domain of the function $f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)$ is $R-(\alpha, \beta)$ then $12 \alpha \beta$ is equal to :

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Q JEE MAIN 2024

Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $...

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Q JEE MAIN 2024

For the function $f(x)=(\cos x)-x+1, x \in \mathbb{R}$, between the following two statements (S1) $f(x)=0$ for only one value of...

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If $S=\{a \in R:|2 a-1|=3[a]+2\{a\}\}$, where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents...

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Q JEE MAIN 2024

Let a $$ \begin{aligned} t a=1+\frac{{ }^2 C_2}{3!}+\frac{{ }^3 C_2}{4!}+\frac{{ }^4 C_2}{5!}+\cdots, & \\ & \quad b=1+\frac{{ }^1 C_0+{ }^1...

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Let $\alpha=\sum_{\mathrm{r}=0}^{\mathrm{n}}\left(4 \mathrm{r}^2+2 \mathrm{r}+1\right)^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}$ and $\beta=\left(\sum_{r=0}^n \frac{{ }^n C_r}{r+1}\right)+\frac{1}{n+1}$. If $140

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Let $\overrightarrow{\mathrm{OA}}=2 \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{OB}}=6 \overrightarrow{\mathrm{a}}+5 \overrightarrow{\mathrm{~b}}$ and $\overrightarrow{\mathrm{OC}}=3 \overrightarrow{\mathrm{~b}}$, where O is the origin. If the area of the parallelogram...

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If the range of $f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}$ is $[\alpha, \beta]$,...

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Q JEE MAIN 2024

The number of ways of getting a sum 16 on throwing a dice four times is

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Let $|\cos \theta \cos (60-\theta) \cos (60-\theta)| \leq \frac{1}{8}, \theta \in[0,2 \pi]$ Then, the sum of all $\theta \in[0,2 \pi]$,...

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Q JEE MAIN 2024

Let the positive integers be written in the form : If the k^"th " row contains exactly k numbers for...

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Let $A$ be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is...

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Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $...

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Q JEE MAIN 2024

The area of the region enclosed by the parabolas $y=x^2-5 x$ and $y=7 x-x^2$ is

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Q JEE MAIN 2024

If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3 \sqrt{5}} \mathrm{k}$ and $\int_0^{\mathrm{k}}\left[\mathrm{x}^2\right] \mathrm{dx}=\alpha- \sqrt{\alpha}$, where $[\mathrm{x}]$...

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A variable line L passes through the point (3,5) and intersects the positive coordinate axes at the points A and...

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Q JEE MAIN 2024

The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits...

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Let $\int \frac{2-\tan x}{3+\tan x} d x=\frac{1}{2}\left(\alpha x+\log _e|\beta \sin x+\gamma \cos x|\right)+C$, where C is the constant of integration....

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Q JEE-Main 2024

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\left(1+x^2\right) \frac{d y}{d x}+y=e^{\tan ^{-1} x}, \mathrm{y}(1)=0$. Then $\mathrm{y}(0)$ is $\_\_\_\_$...

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Q JEE MAIN 2024

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables...

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Q JEE MAIN 2024

Let the solution $y=y(x)$ of the differential equation $\frac{d y}{d x}-y=1+4 \sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+$ 10 is equal...

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Q JEE MAIN 2024

If the constant term in the expansion of (1+2x-3x^3 ) (3/2 x^2-1/3x)^9 is p, then 108p is equal to

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Q JEE MAIN 2024

If the orthocentre of the triangle formed by the lines 2x+3y-1=0,x+2y-1=0 and ax + by -1=0, is the centroid of...

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Q JEE MAIN 2024

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at...

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Q JEE MAIN

The coefficient of $x^{70}$ in $x^2(1+x)^{98}+x^3(1+x)^{97}+\mathrm{x}^4(1+\mathrm{x})^{96}++x^{54}(1+x)^{46}$ is ${ }^{99} \mathrm{C}_{\mathrm{p}}-{ }^{46} \mathrm{C}_{\mathrm{q}}$. Then a possible value to p+q is :

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Q JEE MAIN 2024

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random....

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Q JEE MAIN 2024

Let $A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]$. If the sum of the diagonal elements of $A^{13}$ is $3^n$, then...

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Q JEE-Main

A circle in inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any...

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Q JEE MAIN 2024

If $\lim _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{m \sqrt{5}}{n(2 n)^{2 / 3}}$,...

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Q JEE MAIN

Let $\lambda, \mu \in R$. If the system of equations $3 x+5 y+\lambda z=3$ 7x+11y-9z=2 $97 x+155 y-189 z=\mu$ has...

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Q JEE MAIN 2024

If the line (2-x)/3=(3y-2)/(4λ+1)=4-z makes a right angle with the line (x+3)/3μ=(1-2y)/6=(5-z)/7, then 4λ+9μ is equal to :

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Q JEE MAIN 2024

Let $\mathrm{H}: \frac{-\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4 \sqrt{3}$....

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Q JEE MAIN 2024

Let $\alpha \in(0, \infty)$ and $...

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Q JEE MAIN

A ray of light coming from the point P(1,2) gets reflected from the point Q on the x-axis and then...

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Q JEE MAIN 2024

Let $y=y(x)$ be the solution of the differential equation $\left(1+y^2\right) e^{\tan x} d x+\cos ^2 x\left(1+e^{2 \tan x}\right) d y=0$,...

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Q JEE MAIN 2024

The value of ∫_(-π)^π (2y(1+sin⁡y))/(1+〖cos〗^2⁡y) dy is :

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Q JEE MAIN 2024

Let $f(\mathrm{x})=\mathrm{x}^5+2 \mathrm{e}^{\mathrm{x} / 4}$ for all $\mathrm{x} \in \mathrm{R}$. Consider a function $g(x)$ such that (gof) $(x)=x$ for all...

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Q JEE MAIN 2024

The set of all $\alpha$, for which the vector $\vec{a}=\alpha t \hat{\imath}+6 \hat{\jmath}-3 \hat{k}$ and $...

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Q JEE MAIN 2024

If A(1,-1,2),B(5,7,-6),C(3,4,-10) and D(-1,-4,-2) are the vertices of a quadrilateral ABCD, then its area is

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Q JEE MAIN

The solution curve, of the differential equation $2 y \frac{d y}{d x}+3=5 \frac{d y}{d x}$ , passing through the point...

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Q JEE MAIN 2024

If the domain of the function $\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _e\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)$ is $(\alpha, \beta]$, then $...

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Q JEE MAIN 2024

Let f(x)=x^5+2x^3+3x+1,x∈R, and g(x) be a function such that g(f(x))=x for all x∈R. Then (g(7))/(g^' (7)) is equal to :

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Q JEE-Main 2024

Let the relations $R_1$ and $R_2$ on the set X={1,2,3,…,20} be given by $R_1$={(x,y):2x-3y=2} and $R_2$={(x,y):-5x+4y=0}. If M and N...

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Q JEE MAIN

The parabola $y^2=4 x$ divides the area of the circle $x^2+y^2=5$ in two parts. The area of the smaller part...

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Q JEE MAIN 2024

A square is inscribed in the circle $x^2+y^2-10 x-6 y+30=0$. One side of this square is parallel to $y= x+3$....

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Q JEE MAIN 2024

If the set $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}) ; \mathrm{a}+5 \mathrm{~b}=42, \mathrm{a}, \mathrm{b} \in \mathbb{N}\}$ has $m$ elements and $\sum_{n=1}^m\left(1+i^{n!}\right)=x+i y$, where $\mathrm{I}=\sqrt{-1}$,...

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Q JEE MAIN 2024

If 1/(√1+√2)+1/(√2+√3)+⋯+1/(√99+√100)=m and 1/(1⋅2)+1/(2⋅3)+⋯+1/(99⋅100)=n, then the point (m,n) lies on the line

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Q JEE-Main 2024

Let, $\alpha, \beta$ be the distinct roots of the equation $\mathrm{x}^2-\left(\mathrm{t}^2-5 \mathrm{t}+6\right) \mathrm{x}+1=0, \mathrm{t} \in \mathrm{R}$ and $\mathrm{a}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}$ Then the...

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Q JEE MAIN 2024

Let $z$ be a complex number such that $|z+2|=1$ and $\operatorname{lm}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\operatorname{Re}(\overline{z+2})|$ is :

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Q JEE MAIN 2024

Let the point, on the line passing through the points $\mathrm{P}(1,-2,3)$ and $\mathrm{Q}(5,-4,7)$, farther from the origin and at a...

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Q JEE MAIN

Let the line L intersect the lines x-2=-y=z-1,2(x+1)=2(y-1)=z+1 and be parallel to the line $\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}$. Then which of the following...

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Q JEE MAIN 2024

Suppose θ∈[0,π/4] is a solution of 4cos⁡θ-3sin⁡θ=1. Then cos⁡θ is equal to :

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Q JEE-Main 2024

A company has two plants A and B to manufacture motorcycles. 60% motorcycles are manufactured at plant A and the...

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Q JEE MAIN 2024

Let $[t]$ be the greatest integer less than or equal to t. Let $A$ be the set of al prime...

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Q JEE MAIN 2024

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $...

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Q JEE MAIN 2024

Consider the following two statements Statement I: For any two non-zero complex numbers $z_1, z_2\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)$ and Statement II:...

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Q JEE MAIN 2024

The equations of two sides AB and AC of a triangle $A B C$ are $4 x+y=14$ and $...

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Q JEE MAIN 2024

If 2 and 6 are the roots of the equation $\mathrm{ax}^2+\mathrm{bx}+1=0$, then the quadratic equation, whose roots are $\frac{1}{2 a+b}$...

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Q JEE-Main 2024

The shortest distance between the lines $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ is $\_\_\_\_$

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Q JEE MAIN 2024

Let $I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x$. If $I(0)=3$, then $I\left(\frac{\pi}{12}\right)$ is equal to :

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Q JEE MAIN 2024

Let a,b∈R. Let the mean and the variance of 6 observations -3,4,7,-6,a,b be 2 and 23 , respectively. The mean...

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Q JEE-Main 2024

For α,β∈R and a natural number n, let

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Q JEE MAIN 2024

Let the line 2x+3y-k=0,k>0, intersect the x-axis and y-axis at the points A and B, respectively. If the equation of...

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Q JEE MAIN 2024

Let the first three terms $2, p$ and $q$, with $q \neq 2$, of a G.P. be respectively the $...

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Q JEE MAIN 2024

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the...

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Q JEE MAIN 2024

Let a unit vector which makes an angle of $60^{\circ}$ with $2 \hat{\imath}+2 \hat{\jmath}-\hat{k}$ and an angle of $45^{\circ}$ with...

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Q JEE-Main 2024

Let C be the circle of minimum area touching the parabola $y=6-x^2$ and the lines y=√3|x|. Then, which one of...

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Q JEE MAIN 2024

Let the sum of two positive integers be 24 . If the probability, that their product is not less than...

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Q JEE MAIN 2024

The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$ is equal to :

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Q JEE-Main 2024

Let A={n∈[100,700]∩N:n is neither a multiple of 3 nor a multiple of 4}. Then the number of elements in A...

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Q JEE MAIN 2024

If the shortest distance between the lines L_1:r ⃗=(2+λ)i ˆ+(1-3λ)j ˆ+(3+4λ)k ˆ,λ∈R L_2:r ⃗=2(1+μ)i ˆ+3(1+μ)j ˆ+(5+μ)k ˆ,μ∈R is m/√n, where...

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Q JEE MAIN 2024

Let $A$ and $B$ be two square matrices of order 3 such that $|A|=3$ and $|B|=2$. Then $...

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Q JEE MAIN 2024

Let $A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$. If $...

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Q JEE-Main 2024

The function $f(x)=\frac{x^2+2 x-15}{x^2-4 x+9}, x \in R$ is $\_\_\_\_$ .

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Q JEE MAIN 2024

Let λ,μ∈R. If the system of equations 3x+5y+λz=3 7x+11y-9z=2 97x+155y-189z=μ has infinitely many solutions, then μ+2λ is equal to :

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Q JEE MAIN 2024

Let $f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10$. The number of points of local maxima of $f$ in interval...

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Q JEE MAIN 2024

Let $f(x)$ be a positive function such that the area bounded by $y=f(x), y=0$ from $x=0$ to $x=a>0$ is $...

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Q JEE MAIN 2024

The integral $\int_0^{\frac{\pi}{4}} \frac{136 \sin x}{3 \sin x+5 \cos x} d x$ is equal to :

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Q JEE-Main 2024

The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On respectively, it was...

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Q JEE MAIN 2024

The number of critical points of the function $f(x)=(x-2)^{2 / 3}(2 x+1)$ is:

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Q JEE-Main 2024

$\int_0^{\pi / 4} \frac{\cos ^2 x \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x$ is equal to $\_\_\_\_$

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Q JEE MAIN 2024

If the function $f(x)=\frac{\sin 3 x+\alpha \sin x-\beta \cos 3 x}{x^3}, x \in R$, is continuous at $x=0$, then $f(0)$...

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Q JEE MAIN 2024

Let $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ be a point in the first octant, whose projection in the xy -plane is the point...

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Q JEE MAIN 2024

Let $A=\{1,3,7,9,11\}$ and $B=\{2,4,5,7,8,10,12\}$. Then the total number of one-one maps $f: A \rightarrow B$, such that $f(1)+f(3)=14$, is :

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Q JEE-Main 2024

If $\mathrm{A}(3,1,-1), \mathrm{B}\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), \mathrm{C}(2,2,1)$ and $\mathrm{D}\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)$ are the vertices of a quadrilateral ABCD , then its...

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Q JEE MAIN 2024

Let the circles $C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2$ and $C_2:(x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2$ touch each other externally at the point $(6,6)$. If the point $(6,6)$ divides the...

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Q JEE MAIN 2024

The sum of all the solutions of the equation $(8)^{2 x}-16 \cdot(8)^x+48=0$ is is:

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Q JEE MAIN 2024N

If the system of equations has infinitely many solutions, then $\lambda^4-\mu$ is equal to:

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Q JEE MAIN 2024

The value of $k \in \mathbb{N}$ for which the integral $I_n=\int_0^1\left(1-x^k\right)^n d x, n \in \mathbb{N}$, satisfies $147 I_{20}=148 I_{21}$...

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Q JEE-Main 2024

If $f(x)=\left\{\begin{array}{cc}x^3 \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x=0\end{array}\right.$, then

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Q JEE MAIN 2024

For the function f(x)=sin⁡x+3x-2x2+x, where x0,2, consider the following two statements : (I) f is increasing in 0,2. (II) f'...

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Q JEE MAIN 2024

There are 5 points $\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3, \mathrm{P}_4, \mathrm{P}_5$ on the side AB , excluding A and B , of...

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Q JEE MAIN

Let $\vec{a}, \vec{b}$ and $\overrightarrow{\mathrm{c}}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear if $\vec{a}+5 \vec{b}$is collinear...

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Q JEE MAIN 2024

If the system of equations $$ \begin{aligned} & x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0 \\ & \mathrm{x}+(\cos \alpha) \mathrm{y}+(\sin...

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Q JEE MAIN 2024

If y=y(x) is the solution of the differential equation dydx+2y=sin⁡(2x),y(0)=34, then y8 is equal to :

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Q JEE MAIN 2024

One of the points of intersection of the curves $y=1+3 x-2 x^2$ and $y=\frac{1}{x}$ is $\left(\frac{1}{2}, 2\right)$. Let the area...

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Q JEE MAIN 2024

Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$ be $\frac{m}{n}$,...

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Q JEE MAIN 2024

If the solution $y=y(x)$ of the differential equation $\left(x^4+2 x^3+3 x^2+2 x+2\right) d y-\left(2 x^2+2 x+3\right) d x=0$ satisfies $y(-1)=-\frac{\pi}{4}$,...

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Q

Let two straight lines drawn from the origin O intersect the line 3x+4y=12 at the points P and Q such...

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Q JEE MAIN 2024

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of...

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Q JEE MAIN 2024

The vertices of a triangle are A(-1,3),B(-2,2) and C(3,-1). A new triangle is formed by shifting the sides of the...

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Q JEE MAIN 2024

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black...

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Q JEE MAIN 2024

Let d be the distance of the point of intersection of the lines x+63=y2=z+11 and x-74=y-93=z-42 from the point (7,8,9)....

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Q JEE MAIN 2024

Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function given by $$ f(x)=\left\{\begin{array}{cl} \frac{1-\cos 2 x}{x^2} & , x0 \end{array}\right. $$...

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Q JEE MAIN 2024

Let $\alpha=1^2+4^2+8^2+13^2+19^2+26^2+\cdots \ldots$ upto 10 terms and $\beta=\sum_{n=1}^{10} n^4$. If $4 \alpha-\beta=55 k+40$, then k is equal to

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Q JEE MAIN 2024

If the function $f(x)=\left\{\begin{array}{cl}\frac{1}{|x|} & , \\ a x^2+2 b, & |x| \geq 2\end{array}\right.$ is differentiable on R , then...

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Q JEE MAIN 2024

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $$ \begin{aligned} & f(x)=\frac{4^x}{4^x+2} \text { and } \\ &...

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Q JEE MAIN 2024

Let $\alpha, \beta \in N$ be roots of equation $x^2-70 x+\lambda=0$, where $\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathrm{~N}$. If $\lambda$ assumes the...

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Q JEE-Main 2024

Let $A=\{1,2,3, \ldots 20\}$. Let $R_1$ and $R_2$ two relation on A such that $\mathrm{R}_1=\{(\mathrm{a}, \mathrm{b}): \mathrm{b}$ is divisible by...

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Q JEE MAIN 2024

Let A={1,2,3,4} and R={(1,2),(2,3),(1,4)} be a relation on A. Let S be the equivalence relation on A such that R⊂S...

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Q JEE MAIN 2024

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$...

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Q JEE-Main 2024

Let the line of the shortest distance between the lines $\mathrm{L}_1: \vec{r}=(\hat{\imath}+2 \hat{\jmath}+3 \hat{k})+\lambda(\hat{\imath}-\hat{\jmath}+\hat{k})$ and $...

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Q JEE MAIN 2024

Let $y=y(x)$ be the solution of the differential equation $\left(1-x^2\right) d y=\left[x y+\left(x^3+2\right) \sqrt{3\left(1-x^2\right)}\right] d x,-1< x

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Q JEE-Main 2024

If $\int_{-\pi / 2}^{\pi / 2} \frac{8 \sqrt{2} \cos x d x}{\left(1+e^{\sin x}\right)\left(1+\sin ^4 x\right)}=\alpha \pi+\beta \log _e(3+2 \sqrt{2})$, where...

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Q JEE MAIN 2024

Let the foci and length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ be $( \pm 5,0)$ and $\sqrt{50}$,...

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Q JEE-Main 2024

Let $\mathrm{P}=\{\mathrm{z} \in \mathbb{C}:|\mathrm{z}+2-3 \mathrm{i}| \leq 1\}$ and $\mathrm{Q}=\{\mathrm{z} \in \mathbb{C}: \mathrm{z}(1+\mathrm{i})+\overline{\mathrm{z}}(1-\mathrm{i}) \leq-8\}$. Let in $\mathrm{P} \cap \mathrm{Q},|\mathrm{z}-3+2 \mathrm{i}|$ be...

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Q JEE MAIN 2024

If $\alpha$ denotes the number of solutions of $|1-\mathrm{i}|^{\mathrm{x}}=2^{\mathrm{x}}$ and $\beta=\left(\frac{|z|}{\arg (z)}\right)$, where $\mathrm{z}=\frac{\pi}{4}(1+\mathrm{i})^4\left(\frac{1-\sqrt{\pi} \mathrm{i}}{\sqrt{\pi}+\mathrm{i}}+\right. \left.\frac{\sqrt{\pi}-\mathrm{i}}{1+\sqrt{\pi} \mathrm{i}}\right), \mathrm{i}=\sqrt{-1}$, then the...

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Q JEE MAIN 2024

The value 9∫_0^9 [√(10x/(x+1))]dx, where [t] denotes the greatest integer less than or equal to t, is

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Q JEE-Main 2024

Let the line $\mathrm{L}: \sqrt{2} \mathrm{x}+\mathrm{y}=\alpha$ pass through the point of the intersection P (in the first quadrant) of the...

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Q JEE-Main 2024

In the expansion of $(1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0$, the sum of the coefficient of $x^3$ and $x^{-13}$ is equal to

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Q JEE-Main 2024

Let $\{x\}$ denote the fractional part of $x$ and $f(x)=\frac{\cos ^{-1}\left(1-\{x\}^2\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^3}, x \neq 0$. If $L$ and R...

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Q JEE MAIN 2024

Let $Q$ and $R$ be the feet of perpendiculars from the point $P(a, a, a)$ on the lines $x=y, z=1$...

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Q JEE MAIN 2024

If d_1 is the shortest distance between the lines x+1=2y=-12z,x=y+2=6z-6 and d_2 is the shortest distance between the lines (x-1)/2=(y+8)/(-7)=(z-4)/5,(x-1)/2=(y-2)/1=(z-6)/(-3),...

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Q JEE-Main 01.02.24_(S1)

Let 3,7,11,15,….,403 and 2,5,8,11,…,404 be two arithmetic progressions. Then the sum, of the common terms in them, is equal to

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Q JEE MAIN 2024

The total number of words (with or without meaning) that can be formed out of the letters of the word...

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Q JEE MAIN 2024

Let $S=(-1, \infty)$ and $f: S \rightarrow \mathbb{R}$ be defined as $f(x)=\int_{-1}^x\left(e^t-1\right)^{11}(2 t-1)^5(t-2)^7(t-3)^{12}(2 t-10)^{61} d t$ Let $p=$ Sum of...

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Q JEE MAIN 2024

A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics & Chemistry. It was found...

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Q JEE-Main 2024

If the Coefficient of $x^{30}$ in the expansion of $\left(1+\frac{1}{x}\right)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 ; x \neq 0$ is $\alpha$, then $|\alpha|$ equals

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If the integral $525 \int_0^{\frac{\pi}{2}} \sin 2 x \cos ^{\frac{11}{2}} x\left(1+\cos ^{\frac{5}{2}} x\right)^{\frac{1}{2}} d x$ is equal to $(n \sqrt{2}-64)$,...

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Let $\mathrm{f}:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$, If the $...

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Let $S$ be the set of positive integral values of a for which $\frac{\mathrm{ax}^2+2(\mathrm{a}+1) \mathrm{x}+9 \mathrm{a}+4}{\mathrm{x}^2-8 \mathrm{x}+32}

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If $2 \sin ^3 x+\sin 2 x \cos x+4 \sin x-4=0$ has exactly 3 solutions in the interval $...

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The number of elements in the set S={(x,y,z):x,y,z∈Z,x+2y+3z=42,x,y,z ≥0} equals

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Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable x to be the number of...

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$f(x)=\left|\begin{array}{ccc}3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2\end{array}\right|$ for all $x \in \mathbb{R}$, then $...

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If $x=x(t)$ is the solution of the differential equation $(t+1) d x=\left(2 x+(t+1)^4\right) d t, x(0)=2$, then, $x(1)$ equals

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A line with direction ratios 2,1,2 meets the lines x=y+2=z and x+2=2y=2z respectively at the point P and Q. if...

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If the shortest distance between the lines $\frac{x-\lambda}{-2}=\frac{y-2}{1}=\frac{z-1}{1}$ and $\frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1}$ is 1 , then the sum of all possible values...

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Q JEE MAIN 2024

If $...

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If $\frac{{ }^{11} \mathrm{C}_1}{2}+\frac{{ }^{11} \mathrm{C}_2}{3}+\cdots \cdot+\frac{{ }^{11} \mathrm{C}_9}{10}=\frac{\mathrm{n}}{\mathrm{m}}$ with $\operatorname{gcd}(\mathrm{n}, \mathrm{m})=1$, then n+m is equal to

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Let $g(x)$ be a linear function and $f(x)=\left\{\begin{array}{ll}g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x>0\end{array}\right.$, is continuous...

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If $5 f(x)+4 f\left(\frac{1}{x}\right)=x^2-2, \forall x \neq 0$ and $y=9 x^2 f(x)$, then y is strictly increasing in :

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The area (in sq. units) of the part of circle $x^2+y^2=169$ which is below the line 5x-y=13 is $...

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If the mean and variance of the data 65,68,58,44, 48, 45, 60, α,β,60 where α>β are 56 and 66.2 respectively,...

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Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles,...

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Let $C: x^2+y^2=4$ and $C^{\prime}: x^2+y^2-4 \lambda x+9=0$ be two circles. If the set of all values of $\lambda$ so...

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Q JEE MAIN 2024

Let M denote the median of the following frequency distribution. Then 20 M is equal to :

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If the solution curve y=y(x) of the differential equation $\left(1+y^2\right)\left(1+\log _e x\right) d x+x d y=0, x>0$ passes through the...

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For $\alpha, \beta, \gamma \neq 0$. If $\sin ^{-1} \alpha+\sin ^{-1} \beta+\sin ^{-1} \gamma=\pi$ and $(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3 \alpha \beta$, then $\gamma$...

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If the points of intersection of two distinct conics $x^2+y^2=4 b$ and $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ lie on the curve $y^2=3 x^2$, then...

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The distance of the point $Q(0,2,-2)$ form the line passing through the point $P(5,-4,3)$ and perpendicular to the lines $...

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Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $\sqrt{14}$. Then...

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Let α,β be the roots of the equation$x^2-x+2=0$ with Im⁡(α)>Im⁡(β). Then $\alpha^6+\alpha^4+\beta^4-5 \alpha^2$ is equal to

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The sum of the series $\frac{1}{1-3 \cdot 1^2+1^4}+\frac{2}{1-3 \cdot 2^2+2^4}+\frac{3}{1-3 \cdot 3^2+3^4}+\cdots$. up to 10 terms is

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Let f:R→R be defined as $f(x)=\left\{\begin{array}{ccc}\frac{a-b \cos 2 x}{x^2} & ; & x1\end{array}\right.$ If f is continuous everywhere in R...

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All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words...

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Q JEE MAIN 2024

Consider the system of linear equation x+y+z= 4μ,x+2y+2λz=10μ,x+3y+4λ^2 z=μ^2+15, where λ,μ∈R. Which one of the following statements is NOT correct?

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Q JEE MAIN 2024

Let $\vec{a}=3 \hat{\imath}+\hat{\jmath}-2 \hat{k}, \vec{b}=4 \hat{\imath}+\hat{\jmath}+7 \hat{k}$ and $\overrightarrow{\mathrm{c}}=\hat{\imath}-3 \hat{\jmath}+4 \hat{k}$ be three vectors. If a vectors $\overrightarrow{\mathrm{p}}$ satisfies $...

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Equation of two diameters of a circle are 2x-3y=5 and 3x-4y=7. The line joining the points $\left(-\frac{22}{7},-4\right)$ and $\left(-\frac{1}{7}, 3\right)$...

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Q JEE MAIN 2024

If the domain of the function f(x)=cos^(-1)⁡((2-|x|)/4)+(log_e⁡(3-x))^(-1) is [-α,β)-{y}, then α+β+γ is equal to :

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For $0

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Let A be a square matrix such that $A A^T=I$. Then $\frac{1}{2} A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]$ is equal to

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Let $y=y(x)$ be the solution of the differential equation $...

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If the system of equations 2x+3y-z=5 x+αy+3z=-4 3x-y+βz=7 has infinitely many solutions, then 13αβ is equal to

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Suppose $f(x)=\frac{\left(2^x+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^2-x+1\right)}}{\left(7 x^2+3 x+1\right)^3}$ Then the value of f^' (0) is equal to

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Let f:R→R and g:R→R be defined as $$f(x)=\left\{\begin{array}{lll} \log _e x & , & x>0 \\ e^{-x} & , &...

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If the value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{2023}}}\right) d x=\frac{\pi}{4}(\pi+a)-2$ then the value of a is

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Let $\alpha, \beta, \gamma, \delta \in \mathrm{Z}$ and let $\mathrm{A}(\alpha, \beta), \mathrm{B}(1,0), \mathrm{C}(\gamma, \delta)$ and $D(1,2)$ be the vertices of...

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Two integers x and y are chosen with replacement from the set {0,1,2,3, 10}. Then the probability that |x-y|>5 is

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The area enclosed by the curves xy+4y=16 and x+y=6 is equal to :

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Let R be a relation on Z×Z defined by (a,b)R(c,d) if and only if ad-bc is divisible by 5 ....

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Let S={x∈R:(√3+√2 )^x+(√3-√2 )^x=10}. Then the number of elements in S is :

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Let PQR be a triangle with R(-1,4,2). Suppose M(2,1,2) is the mid point of PQ. The distance of the centroid...

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The solution curve of the differential equation $y \frac{d x}{d y}=x\left(\log _e x-\log _e y+1\right), x>0, y>0$ passing through the...

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Q JEE MAIN 2024

Let (α,β,γ) be the foot of perpendicular from the point (1,2,3) on the line (x+3)/5=(y-1)/2=(z+4)/3. then 19(α+β+γ) is equal to...

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Let $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha\end{array}\right]$ and $...

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Q JEE MAIN 2024

Let y=y(x) be the solution of the differential equation sec⁡xdy+{2(1-x)tan⁡x+x(2-x)} dx=0 such that y(0)=2. Then y(2) is equal to :

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Consider the function $\mathrm{f}:\left[\frac{1}{2}, 1\right] \rightarrow \mathrm{R}$ defined by $f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1$. Consider the statements (I) The curve...

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Let $g: R \rightarrow R$ be a non constant twice differentiable such that $g^{\prime}\left(\frac{1}{2}\right)=g^{\prime}\left(\frac{3}{2}\right)$. If a real valued function $f$...

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Let O be the origin and the position vector of A and B be $2 \hat{\imath}+2 \hat{\jmath}+\hat{k}$ and $...

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Q JEE MAIN 2024

The area (in square units) of the region bounded by the parabola $y^2=4(x-2)$ and the line $y=2 x-8$

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If n is the number of ways five different employees can sit into four indistinguishable offices where any office may...

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A function y=f(x) satisfies $f(x) \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0$ with condition $f(0)=0$. Then $f\left(\frac{\pi}{2}\right)$ is equal to

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The value of $\lim _{n \rightarrow \infty} \sum_{k=1}^n \frac{n^3}{\left(n^2+k^2\right)\left(n^2+3 k^2\right)}$ is:

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If the system of linear equations $$ \begin{aligned} & x-2 y+z=-4 \\ & 2 x+\alpha y+3 z=5 \\ & 3...

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If $\tan \mathrm{A}=\frac{1}{\sqrt{x\left(x^2+x+1\right)}}, \tan B=\frac{\sqrt{x}}{\sqrt{x^2+x+1}}$ and $\tan C=\left(x^{-3}+x^{-2}+x^{-1}\right)^{\frac{1}{2}}, 0

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If $\alpha,-\frac{\pi}{2}

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For $\mathrm{x} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, if $y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x$ and $...

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Q JEE MAIN 2024

$\lim _{x \rightarrow 0} \frac{\mathrm{e}^{2|\sin x|}-2|\sin x|-1}{x^2}$

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If $\mathrm{A}=\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], \mathrm{B}=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], \mathrm{C}=\mathrm{ABA}^{\mathrm{T}}$ and $\mathrm{X}=\mathrm{A}^{\mathrm{T}} \mathrm{C}^2 \mathrm{~A}$,...

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The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the...

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Let $\left(5, \frac{a}{4}\right)$, be the circumcenter of a triangle with vertices $A(a,-2), B(a, 6)$ and $C\left(\frac{a}{4},-2\right)$. Let $\alpha$ denote the...

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If $f(x)=\frac{4 x+3}{6 x-4}, x \neq \frac{2}{3}$ and (fof) $(x)=g(x)$, where $\mathrm{g}: \mathbb{R}-\left\{\frac{2}{3}\right\} \rightarrow \mathbb{R}-\left\{\frac{2}{3}\right\}$, then (gogog) (4) is equal...

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Q JEE-Main 2024

The value of the integral $\int_0^{\frac{\pi}{4}} \frac{x d x}{\sin ^4(2 x)+\cos ^4(2 x)}$ equals :

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Q JEE MAIN 2024

The area of the region $\left\{(x, y): y^2 \leq 4 x, x0, x \neq 3\right\}$ is

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Let $\vec{a}=a_i \hat{\imath}+a_2 \hat{\jmath}+a_3 \hat{k}$ and $\vec{b}=b_1 \hat{\imath}+b_2 \hat{\jmath}+b_3 \hat{k}$ be two vectors such that $|\vec{a}|=1 ; \vec{a} \cdot \vec{b}=2$...

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If one of the diameters of the circle $x^2+y^2-10 x+4 y+13=0$ is a chord of another circle $C$, whose center...

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In a △ABC, suppose y=x is the equation of the bisector of the angle B and the equation of the...

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If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola...

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$\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{1}{\left(x-\frac{\pi}{2}\right)^2} \int_{x^3}^{\left(\frac{\pi}{2}\right)^3} \cos \left(\frac{1}{t^3}\right) d t\right)$ is equal to

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If z=x+iy,xy≠0, satisfies the equation z^2+iz ‾=0, then |z^2 | is equal to :

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If $\mathrm{z}=\frac{1}{2}-2 \mathrm{i}$, is such that $|\mathrm{z}+1|=\alpha \mathrm{z}+\beta(1+\mathrm{i}), \mathrm{i}=\sqrt{-1}$ and $\alpha, \beta \in \mathrm{R}$, then $\alpha+\beta$ is equal to

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Let $a$ be the sum of all coefficients in the expansion of $\left(1-2 x+2 x^2\right)^{2023}\left(3-4 x^2+2 x^3\right)^{2024}$ and $...

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let $S_a$ denote the sum of first n terms an arithmetic progression. If S_20=790 and S_10=145, then S_15- S_5 is...

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Q JEE MAIN 2024

For 0

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A line passing through the point  makes an angle of  with the positive direction of -axis. If this line is...

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In an A.P., the sixth terms a_6=2. If the a_1 a_4 a_5 is the greatest, then the common difference of...

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If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the...

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Q JEE MAIN 2025

If the domain of the function $\log _{5}\left(18 x-x^{2}-77\right)$ is ( $\alpha, \beta$ ) and the domain of the function...

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Q JEE MAIN 2024

If $\alpha$ satisfies the equation $x^2+x+1=0$ and $(1+\alpha)^7=\mathrm{A}+\mathrm{B} \alpha+\mathrm{C} \alpha^2, \mathrm{~A}, \mathrm{~B}, \mathrm{C} \geq 0$, then $5(3 \mathrm{~A}-2 \mathrm{~B}-\mathrm{C})$ is...

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Q JEE MAIN 2024

Let $...

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Let $f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime}(3), x \in R$. Then $f^{\prime}(10)$ is equal to

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Let the set of all $a \in R$ such that the equation $\cos 2 x+a \sin x=2 a-7$ has a...

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A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and...

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If $8=3+\frac{1}{4}(3+p)+\frac{1}{4^2}(3+2 p)+\frac{1}{4^3}(3+3 p)+\cdots \infty$, then the value of $p$ is

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Let the area of the region $...

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If the solution of the differential equation $(2 x+3 y-2) d x+(4 x+6 y-7) d y=0, y(0)=3$, is $...

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Let for a differentiable function $f:(0, \infty) \rightarrow R$, $f(x)-f(y) \geq \log _{\varepsilon}\left(\frac{x}{y}\right)+x-y, \forall x, y \in(0, \infty).$ Then $...

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Q JEE MAIN 2024

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{\imath}-2 \hat{\jmath}+2 k$ and $...

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The function $\mathrm{f}: \mathrm{N}-\{1\} \rightarrow \mathrm{N}$; defined by $\mathrm{f}(\mathrm{n})=$ the highest prime factor of $n$, is :

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Q JEE MAIN 2024

Consider the matrix $...

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If $a=\lim _{x \rightarrow 0} \frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4}$ and $b=\lim _{x \rightarrow 0} \frac{\sin ^2 x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $...

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Q JEE MAIN 2024

Let $\overrightarrow{\mathrm{a}}=\hat{\imath}+2 \hat{\jmath}+\mathrm{k}, \overrightarrow{\mathrm{b}}=3(\hat{\imath}-\hat{\jmath}+\mathrm{k})$. Let $\vec{c}$ be the vector such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$. Then $...

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The portion of the line $4 x+5 y=20$ in the first quadrant is trisected by the lines $\mathrm{L}_1$ and $\mathrm{L}_2$...

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Q JEE MAIN 2024

The length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid point is $\left(1, \frac{2}{5}\right)$, is equal to :

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Let $a_1, a_2, \ldots . . a_{10}$ be 10 observations such that $\sum_{\mathrm{k}=1}^{10} \mathrm{a}_{\mathrm{k}}=50$ and $\sum_{\forall \mathrm{k}

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Consider the function $f(x)=\left\{\begin{array}{cl} \frac{a\left(7 x-12-x^2\right)}{b\left|x^2-7 x+12\right|} & , x3 \\ b & , x=3 \end{array}\right.$ Where $[\mathrm{x}]$ denotes the...

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Q JEE MAIN 2024

Four distinct points (2k,3k),(1,0),(0,1) and (0,0) lie on a circle for k equal to :

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If $S=\{z \in C:|z-i|=|z+i|=|z-1|\}$, then, $n(S)$ is:

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Let $S=\{1,2,3, \ldots, 10\}$. Suppose M is the set of all the subsets of S , then the relation $...

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If $\int_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}} d x=a+b \sqrt{2}+c \sqrt{3}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are rational numbers, then $2 \mathrm{a}+3 \mathrm{~b}-4 \mathrm{c}$ is...

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If the shortest distance between the lines $\frac{x-4}{1}=\frac{y+1}{2}=\frac{z}{-3}$ and $\frac{x-\lambda}{2}=\frac{y+1}{4}=\frac{z-2}{-5}$ is $\frac{6}{\sqrt{5}}$, then the sum of all possible values of...

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If the shortest distance of the parabola $y^2=4 x$ from the centre of the circle $x^2+y^2-4 x-16 y+64=0$ is d...

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Q JEE MAIN 2024

The number of common terms in the progressions 4,9,14,19,....., up to 25th term and 3,6,9,12, up to 37th term is...

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Q JEE MAIN 2024

If A denotes the sum of all the coefficients in the expansion of $\left(1-3 x+10 x^2\right)^n$ and B denotes the...

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If $(a, b)$ be the orthocentre of the triangle whose vertices are $(1,2),(2,3)$ and $(3,1)$, and $...

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Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\frac{\mathrm{dx}}{\mathrm{dt}}+\mathrm{ax}=0$ and $\frac{\mathrm{dy}}{\mathrm{dt}}+\mathrm{by}=0$ respectively, $\mathrm{a}, \mathrm{b} \in \mathrm{R}$. Given that...

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The distance, of the point (7,-2,11) from the line $\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3}$ along the line $\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6}$, is :

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Q JEE MAIN 2024

${ }^{n-1} C_r=\left(k^2-8\right)^n C_{r+1}$ if and only if :

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Q JEE MAIN 2025

If $24 \int_{0}^{\frac{\pi}{4}}\left(\sin \left|4 x-\frac{\pi}{12}\right|+[2 \sin x]\right) \mathrm{d} x=2 \pi+\alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is...

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Q JEE MAIN 2025

If $\lim _{\mathrm{t} \rightarrow 0}\left(\int_{0}^{1}(3 x+5)^{\mathrm{t}} \mathrm{d} x\right)^{\frac{1}{\mathrm{t}}}=\frac{\alpha}{5 \mathrm{e}}\left(\frac{8}{5}\right)^{\frac{2}{3}}$, then $\alpha$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2025

Let integers $a, b \in[-3,3]$ be such that $a+b \neq 0$. Then the number of all possible ordered pairs (a,...

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Q JEE MAIN 2025

Let $y^{2}=12 x$ be the parabola and $S$ be its focus. Let PQ be a focal chord of the parabola...

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Q JEE MAIN 2025

Let $a_{1}, a_{2}, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_{1}+\left(a_{5}+a_{10}+a_{15}+\ldots+a_{2020}\right)+a_{2024}=2233$. Then $a_{1}+a_{2}+a_{3}+\ldots+a_{2024}$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2025

If $\alpha x+\beta y=109$ is the equation of the chord of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, whose mid point is $\left(\frac{5}{2}, \frac{1}{2}\right)$,...

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Q JEE MAIN 2025

If the set of all $a \in \mathbf{R}$, for which the equation $2 x^2+(a-5) x+15=3$ a has no real root,...

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Q JEE MAIN 2025

If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as...

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Q JEE MAIN 2025

Let $\hat{\mathrm{a}}$ be a unit vector perpendicular to the vectors $\overrightarrow{\mathrm{b}}=\hat{i}-2 \hat{j}+3 \hat{k}$ and $\overrightarrow{\mathrm{c}}=2 \hat{i}+3 \hat{j}-\hat{k}$, and makes an...

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Q JEE MAIN 2025

Let $A=\left[a_{i j}\right]$ be a matrix of order $3 \times 3$, with $\mathrm{a}_{i j}=(\sqrt{2})^{i+j}$. If the sum of all the...

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Q JEE MAIN 2025

If for the solution curve $y=f(x)$ of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+(\tan x) y=\frac{2+\sec x}{(1+2 \sec x)^{2}}$, $...

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Q JEE MAIN 2025

Let a circle $C$ pass through the points $(4,2)$ and $(0,2)$, and its centre lie on $3 x+2 y+2=0$. Then...

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Q JEE MAIN 2025

Let the area enclosed between the curves $|y|=1-x^{2}$ and $x^{2}+y^{2}=1$ be $\alpha$. If $9 \alpha=\beta \pi+\gamma ; \beta, \gamma$ are...

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Q JEE MAIN 2025

If $\sin x+\sin ^{2} x=1, x \in\left(0, \frac{\pi}{2}\right)$, then $...

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Q JEE MAIN 2025

Let $\alpha, \beta(\alpha \neq \beta)$ be the values of $m$, for which the equations $x+y+z=1 ; x+2 y+4 z=m$ and...

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Q JEE MAIN 2025

The remainder, when $7^{103}$ is divided by 23 , is equal to :

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Q JEE MAIN 2025

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black...

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Q JEE MAIN 2025

Let $S=\mathbf{N} \cup\{0\}$. Define a relation $R$ from $S$ to $\mathbf{R}$ by: $...

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Let P be the foot of the perpendicular from the point $(1,2,2)$ on the line $\mathrm{L}: \frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}$. Let the line...

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Let a straight line $L$ pass through the point $P(2,-1,3)$ and be perpendicular to the lines $\frac{x-1}{2}=\frac{y+1}{1}=\frac{z-3}{-2}$ and $\frac{x-3}{1}=\frac{y-2}{3}=\frac{z+2}{4}$. If...

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Let $\mathrm{A}=\left[\mathrm{a}_{i j}\right]$ be a $2 \times 2$ matrix such that $\mathrm{a}_{i j} \in\{0,1\}$ for all $i$ and $j$. Let...

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Q JEE MAIN 2025

Let $f(x)=\int_{0}^{x} \mathrm{t}\left(\mathrm{t}^{2}-9 \mathrm{t}+20\right) \mathrm{dt}, 1 \leq x \leq 5$. If the range of $f$ is $[\alpha, \beta]$, then $4(\alpha+\beta)$...

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Q JEE MAIN 2025

Let the function $f(x)=\left(x^{2}-1\right)\left|x^{2}-\mathrm{a} x+2\right|+\cos |x|$ be not differentiable at the two points $x=\alpha=2$ and $x=\beta$. Then the distance of...

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Q JEE MAIN 2025

Let the line $x+y=1$ meet the axes of $x$ and $y$ at $A$ and $B$, respectively. A right angled triangle...

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Q JEE MAIN 2025

The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15 ,...

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Q JEE MAIN 2025

If $y=y(x)$ is the solution of the differential equation, $...

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Q

The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^{\circ}$. If the largest...

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Q JEE MAIN 2025

Let $A$ and $B$ be the two points of intersection of the line $y+5=0$ and the mirror image of the...

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Q JEE MAIN 2025

Let $f(x)=\lim _{\mathrm{n} \rightarrow \infty} \sum_{\mathrm{r}=0}^{\mathrm{n}}\left(\frac{\tan \left(x / 2^{r+1}\right)+\tan ^{3}\left(x / 2^{r+1}\right)}{1-\tan ^{2}\left(x / 2^{r+1}\right)}\right)$. Then $...

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Q JEE MAIN 2025

Let $[x]$ denote the greatest integer less than or equal to $x$. Then the domain of $f(x)=\sec ^{-1}(2[x]+1)$ is :

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Bag $B_{1}$ contains 6 white and 4 blue balls, Bag $B_{2}$ contains 4 white and 6 blue balls, and Bag...

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Q JEE MAIN 2025

If the midpoint of a chord of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ is $(\sqrt{2}, 4 / 3)$, and the length of the...

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Q JEE MAIN 2025

If $f(x)=\int \frac{1}{x^{1 / 4}\left(1+x^{1 / 4}\right)} \mathrm{d} x, f(0)=-6$, then $f(1)$ is equal to :

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Q JEE MAIN 2025

Let $f: \mathbf{R}-\{0\} \rightarrow(-\infty, 1)$ be a polynomial of degree 2 , satisfying $f(x) f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$. If $f(\mathrm{~K})=-2 \mathrm{~K}$, then the...

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Q JEE MAIN 2025

The area of the region bounded by the curves $x\left(1+y^{2}\right)=1$ and $y^{2}=2 x$ is:

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Q JEE MAIN 2025

If $\sum_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{r \pi}{6}\right)}\right\}=a \sqrt{3}+b, a, b \in Z$, then $a^{2}+b^{2}$ is equal to :

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Q JEE MAIN 2025

If $\alpha+i \beta$ and $\gamma+i \delta$ are the roots of $x^{2}-(3-2 i) x-(2 i-2)=0, i=\sqrt{-1}$, then $\alpha \gamma+\beta \delta$ is...

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Q JEE MAIN 2025

For $n \ge 2$, let ${S_n}$ denote the set of all subsets of $\{ 1,2, \ldots ,n\} $ with no...

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Q JEE MAIN 2025

Let $A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 0 & 1\end{array}\right]$ and $\mathrm{P}=\left[\begin{array}{cc}\cos \theta & -\sin \theta \sin \theta & \cos \theta\end{array}\right], \theta>0$. If...

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Q JEE MAIN 2025

If the components of $\overrightarrow{\mathbf{a}}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ along and perpendicular to $\overrightarrow{\mathbf{b}}=3 \hat{i}+\hat{j}-\hat{k}$ respectively, are $\frac{16}{11}(3 \hat{i}+\hat{j}-\hat{k})$ and $...

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Q JEE MAIN 2025

The number of points of discontinuity of the function $f(x) = \left[ {\frac{{{x^2}}}{2}} \right] - [\sqrt x ],x \in [0,4]$,...

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Q JEE MAIN 2025

The number of singular matrices of order 2, whose elements are from the set {2,3,6,9}, is ______.

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Q JEE MAIN 2025

Consider the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ having one of its focus at ${\rm{P}}( - 3,0)$. If the latus...

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Q JEE MAIN 2025

The number of relations on the set ${\rm{A}} = \{ 1,2,3\} $, containing at most 6 elements including (1,2), which...

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Q JEE MAIN 2025

If the shortest distance between the lines $\frac{{x - 1}}{2} = \frac{{y - 2}}{3} = \frac{{z - 3}}{4}$ and $...

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Q JEE MAIN 2025

Let the system of equations: $2x + 3y + 5z = 9$ $7x + 3y - 2z = 8$ $...

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Q JEE MAIN 2025

Let A be a $3 \times 3$ matrix such that $|{\mathop{\rm adj}\nolimits} ({\mathop{\rm adj}\nolimits} ({\mathop{\rm adj}\nolimits} {\rm{A}}))| = 81$. If...

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Q JEE MAIN 2025

Let $\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $f(2)=1$. If $\mathrm{F}(x)=x f(x)$ for all $...

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Q JEE MAIN 2025

Let the angle $\theta ,0 < \theta < \frac{\pi }{2}$ between two-unit vectors $\hat a$ and $\hat b$ be $...

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Let $\mathrm{A}, \mathrm{B}, \mathrm{C}$ be three points in $\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $\mathrm{a} \hat{i}+(1-\mathrm{a}) \hat{j}$ respectively with respect to...

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Q JEE MAIN 2025

$...

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Q JEE MAIN 2025

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int_{0}^{x} t f(\mathrm{t}) \mathrm{dt}$....

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Q JEE MAIN 2025

The integral $\int_0^\pi {\frac{{(x + 3)\sin x}}{{1 + 3{{\cos }^2}x}}} dx$ is equal to

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Q JEE MAIN 2025

Let $f:[0,3] \rightarrow$ A be defined by $f(x)=2 x^{3}-15 x^{2}+36 x+7$ and $g:[0, \infty) \rightarrow B$ be defined by $\mathrm{g}(x)=\frac{x^{2025}}{x^{2025}+1}$....

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Q JEE MAIN 2025

Let P be the parabola, whose focus is (-2, 1) and directrix is $2x + y + 2 = 0$....

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Q JEE MAIN 2025

The square of the distance of the point $\left(\frac{15}{7}, \frac{32}{7}, 7\right)$ from the line $\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}$ in the direction of the...

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Q JEE MAIN 2025

The remainder when ${\left( {{{(64)}^{(64)}}} \right)^{(64)}}$ is divided by 7 is equal to

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Q JEE MAIN 2025

Two equal sides of an isosceles triangle are along $-x+2 y=4$ and $x+y=4$. If m is the slope of its...

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Q JEE MAIN 2025

Let $x = - 1$ and $x = 2$ be the critical points of the function $...

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Q JEE MAIN 2025

Let the line L pass through (1,1,1) and intersect the lines $...

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For positive integers $n$, if $4 a_{n}=\left(n^{2}+5 n+6\right)$ and $S_{n}=\sum_{k=1}^{n}\left(\frac{1}{a_{k}}\right)$, then the value of $507 S_{2025}$ is:

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If A and B are the points of intersection of the circle $x^{2}+y^{2}-8 x=0$ and the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ and a...

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Q JEE MAIN 2025

Let ABC be the triangle such that the equations of lines AB and AC be $3y - x = 2$...

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Q JEE MAIN 2025

Let $S$ be the set of all the words that can be formed by arranging all the letters of the...

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Q JEE MAIN 2025

The mean and standard deviation of 100 observations are 40 and 5.1, respectively. By mistake one observation is taken as...

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From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should...

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Let ${x_1},{x_2},{x_3},{x_4}$ be in a geometric progression. If 2,7,9,5 are subtracted respectively from ${x_1},{x_2},{x_3},{x_4}$, then the resulting numbers are in...

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Q JEE MAIN 2025

Let ${C_1}$ be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let ${C_2}$ be...

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Q JEE MAIN 2025

Let $y = y(x)$ be the solution curve of the differential equation $...

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Q JEE MAIN 2025

Let the set of all values of $p \in $ $\mathbb{R}$, for which both the roots of the equation $...

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Q JEE MAIN 2025

If for $\theta \in \left[ { - \frac{\pi }{3},0} \right]$, the points $...

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Q JEE MAIN 2025

Among the statements (S1) : The set $\left\{ {z \in - \{ - i\} :|z| = 1} \right.$ and $...

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Q JEE MAIN 2025

If the area of the region bounded by the curves $y = 4 - \frac{{{x^2}}}{4}$ and $...

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Q JEE MAIN 2025

Let C be the circle ${x^2} + {(y - 1)^2} = 2,{E_1}$ and ${E_2}$ be two ellipses whose centres lie...

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Q JEE MAIN 2025

Let $...

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Let $A = \left[ {\begin{array}{*{20}{c}} {\cos \theta }&0&{ - \sin \theta }\\ 0&1&0\\ {\sin \theta }&0&{\cos \theta } \end{array}} \right]$....

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Q JEE MAIN 2025

If the area of the region $\{ (x,y):|x - 5| \le y \le 4\sqrt x \} $ is A ,...

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Q JEE MAIN 2025

Let $m$ and $n$ be the number of points at which the function $...

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The length of the latus-rectum of the ellipse, whose foci are (2,5) and (2, –3) and eccentricity is $\frac{4}{5}$ ,...

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Q JEE MAIN 2025

In the expansion of ${\left( {\sqrt[3]{2} + \frac{1}{{\sqrt[3]{3}}}} \right)^n},n \in \;{\rm{N}}$ , if the ratio of ${15^{{\rm{th }}}}$ term from...

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Q JEE MAIN 2025

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and...

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Q JEE MAIN 2025

Let the three sides of a triangle are on the lines $...

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The value of $...

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Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $f(0) = 1$ and $f(2x) - f(x) = x$ for...

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Q JEE MAIN 2025

If $...

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Q JEE MAIN 2025

Considering the principal values of the inverse trigonometric functions, $...

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Q JEE MAIN 2025

Consider the sets $A=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^2+y^2=25\right\}, B=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^2+9 y^2=144\right\}$, $...

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Q JEE MAIN 2025

Consider the equation ${x^2} + 4x - n = 0$, where $n \in [20,100]$ is a natural number. Then the...

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Q JEE MAIN 2025

Let $A = \{ 1,\,\,6,\,\,11,\,\,16, \ldots \} $ and $B = \{ 9,16,23,30, \ldots \} $ be the sets consisting...

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Q JEE MAIN 2025

For an integer $n \ge 2$, if the arithmetic mean of all coefficients in the binomial expansion of $...

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Q JEE MAIN 2025

Let the shortest distance between the lines $...

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Q JEE MAIN 2025

Let A and B be two distinct points on the line $...

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Q JEE MAIN 2025

Let $f,g:(1,\infty ) \to $ $\mathbb{R}$ be defined as $f(x) = \frac{{2x + 3}}{{5x + 2}}$ and $...

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Q JEE MAIN 2025

If $10{\sin ^4}\theta + 15{\cos ^4}\theta = 6$, then the value of $...

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Q JEE MAIN 2025

Consider two vectors $\vec u = 3\hat i - \hat j$ and $...

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Q JEE MAIN 2025

The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3...

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Q JEE MAIN 2025

Let $f:[0,\infty)\to$ $\mathbb{R}$ be a differentiable function such that $...

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Q JEE MAIN 2025

$1 + 3 + {5^2} + 7 + {9^2} + .....$ upto 40 terms is equal to

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Q JEE MAIN 2025

The area of the region bounded by the curve $y = \max \{ |x|,x|x - 2|\} $, the x-axis and...

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Q JEE MAIN 2025

Let $...

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Q JEE MAIN 2025

If the number of seven-digit numbers, such that the sum of their digits is even, is $...

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Q JEE MAIN 2025

Let the product of the focal distances of the point $P(4,2\sqrt 3)$ on the hyperbola ${\rm{H}}:\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$...

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Q JEE MAIN 2025

Let \[f(x) = \left\{ {\begin{array}{*{20}{l}} {{{(1 + ax)}^{1/x}}}&{,x < 0}\\ {1 + b}&{,x = 0}\\ {\frac{{{{(x + 4)}^{1/2}} - 2}}{{{{(x...

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Q JEE MAIN 2025

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $...

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Q JEE MAIN 2025

The sum 1 + 3 + 11 + 25 + 45 + 71 + ……… upto 20 terms, is equal...

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Q JEE MAIN 2025

Let the domain of the function $f(x) = {\log _2}{\log _4}{\log _6}\left( {3 + 4x - {x^2}} \right)$ be (a,...

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Q JEE MAIN 2025

The sum of all rational terms in the expansion of ${(2 + \sqrt 3 )^8}$ is

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Q JEE MAIN 2025

Let $f(x) = \int {{x^3}} \sqrt {3 - {x^2}} dx$. If $5f(\sqrt 2 ) = - 4$, then $f(1)$ is...

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Q JEE MAIN 2025

The number of solutions of the equation $...

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Q JEE MAIN 2025

Let $z \in \mathbb{C}$ be such that $\frac{{{z^2} + 3i}}{{z - 2 + i}} = 2 + 3i$. Then the...

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Q JEE MAIN 2025

If $y(x) = \left| {\begin{array}{*{20}{c}} {\sin x}&{\cos x}&{\sin x + \cos x + 1}\\ {27}&{28}&{27}\\ 1&1&1 \end{array}} \right|$, $...

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Q JEE MAIN 2025

Let A be a matrix of order $3 \times 3$ and $|A| = 5$. If $...

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Q JEE MAIN 2025

If the domain of the function $...

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Q JEE MAIN 2025

Let g be a differentiable function such that $\int_0^x g (t)dt = x - \int_0^x t g(t)dt,x \ge 0$ and...

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Q JEE MAIN 2025

Line ${L_1}$ passes through the point (1,2,3) and is parallel to z-axis. Line ${L_2}$ passes through the point $(\lambda ,5,6)$...

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Q JEE MAIN 2025

A line passing through the point $P(\sqrt 5 ,\sqrt 5 )$ intersects the ellipse $\frac{{{x^2}}}{{36}} + \frac{{{y^2}}}{{25}} = 1$ at...

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Q JEE MAIN 2025

Let a line passing through the point (4, 1, 0) intersect the line $...

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Q JEE MAIN 2025

If $...

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Q JEE MAIN 2025

Let ${a_1},{a_2},{a_3}, \ldots $ be a G.P. of increasing positive numbers. If ${a_3}{a_5} = 729$ and $...

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Q JEE MAIN 2025

The radius of the smallest circle which touches the parabolas $...

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Q JEE MAIN 2025

Let $\alpha$ and $\beta $ be the roots of ${x^2} + \sqrt 3 x - 16 = 0$, and $...

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Q JEE MAIN 2025

Three distinct numbers are selected randomly from the set $\{ 1,2,3, \ldots ,40\}$. If the probability, that the selected numbers...

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Q JEE MAIN 2025

Let [.] denote the greatest integer function. If $\int_0^{{e^3}} {\left[ {\frac{1}{{{e^{x - 1}}}}} \right]} dx = \alpha - {\log _e}2$,...

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Q JEE MAIN 2025

Let $f:R \to R$ be a thrice differentiable odd function satisfying $...

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Q JEE MAIN 2025

The absolute difference between the squares of the radii of the two circles passing through the point (-9, 4) and...

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Q JEE MAIN 2025

If the area of the region $...

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Q JEE MAIN 2025

Let $y=y(x)$ be the solution of the differential equation $...

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Q JEE MAIN 2025

Let $\mathrm{H}_{1}: \frac{x^{2}}{\mathrm{a}^{2}}-\frac{y^{2}}{\mathrm{~b}^{2}}=1$ and $\mathrm{H}_{2}:-\frac{x^{2}}{\mathrm{~A}^{2}}+\frac{y^{2}}{\mathrm{~B}^{2}}=1$ be two hyperbolas having length of latus rectums $15 \sqrt{2}$ and $12 \sqrt{5}$ respectively. Let...

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Q JEE MAIN 2025

Number of functions $f:\{1,2, \ldots, 100\} \rightarrow\{0,1\}$, that assign 1 to exactly one of the positive integers less than or...

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Q JEE MAIN 2025

If $\int \frac{2 x^{2}+5 x+9}{\sqrt{x^{2}+x+1}} \mathrm{~d} x=x \sqrt{x^{2}+x+1}+\alpha \sqrt{x^{2}+x+1}+\beta \log _{\mathrm{e}}\left|x+\frac{1}{2}+\sqrt{x^{2}+x+1}\right|+\mathrm{C}$, where $C$ is the constant of integration, then $...

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Q JEE MAIN 2025

Let $P$ be the image of the point $Q(7,-2,5)$ in the line $L: \frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4}$ and $R(5, p, q)$ be a...

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Q

Let $f:(0, \infty) \rightarrow \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the...

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Q JEE MAIN 2025

Let the points $\left(\frac{11}{2}, \alpha\right)$ lie on or inside the triangle with sides $x+y=11, x+2 y=16$ and $2 x+3 y=29$....

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Q JEE MAIN 2025

In an arithmetic progression, if $S_{40}=1030$ and $S_{12}=57$, then $S_{30}-S_{10}$ is equal to:

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Q JEE MAIN 2025

If $\alpha>\beta>\gamma>0$, then the expression $\cot ^{-1}\left\{\beta+\frac{\left(1+\beta^{2}\right)}{(\alpha-\beta)}\right\}+\cot ^{-1}\left\{\gamma+\frac{\left(1+\gamma^{2}\right)}{(\beta-\gamma)}\right\}+\cot ^{-1}\left\{\alpha+\frac{\left(1+\alpha^{2}\right)}{(\gamma-\alpha)}\right\}$ is equal to :

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Let $(2,3)$ be the largest open interval in which the function $f(x)=2 \log _{\mathrm{e}}(x-2)-x^{2}+a x+1$ is strictly increasing and $...

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Q JEE MAIN 2025

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The...

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Q JEE MAIN 2025

If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x+2 y=0$ is $...

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Q JEE MAIN 2025

The function $f:(-\infty, \infty) \rightarrow(-\infty, 1)$, defined by $f(x)=\frac{2^{x}-2^{-x}}{2^{x}+2^{-x}}$ is :

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Q JEE MAIN 2025

Let $\overrightarrow{\mathrm{a}}=3 \hat{i}-\hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=\overrightarrow{\mathrm{a}} \times(\hat{i}-2 \hat{k})$ and $\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}} \times \hat{k}$. Then the projection of $\overrightarrow{\mathrm{c}}-2 \hat{j}$ on $\vec{a}$ is:

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Let $\mathrm{A}=\left\{x \in(0, \pi)-\left\{\frac{\pi}{2}\right\}: \log _{(2 / \pi)}|\sin x|+\log _{(2 / \pi)}|\cos x|=2\right\}$ and $B=\{x \geq 0: \sqrt{x}(\sqrt{x}-4)-3|\sqrt{x}-2|+6=0\}$. Then $...

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Q JEE MAIN 2025

Let $\mathrm{A}=\left[a_{i j}\right]$ be a square matrix of order 2 with entries either 0 or 1 . Let E be...

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Q JEE MAIN 2025

The number of real solution(s) of the equation $x^{2}+3 x+2=\min \{|x-3|,|x+2|\}$ is :

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Q JEE MAIN 2025

For some a, b, let $$ f(x)=\left|\begin{array}{ccc} a+\frac{\sin x}{x} & 1 & \mathbf{b} \\ \mathbf{a} & 1+\frac{\sin x}{x} & \mathbf{b}...

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Q JEE MAIN 2025

If $7=5+\frac{1}{7}(5+\alpha)+\frac{1}{7^{2}}(5+2 \alpha)+\frac{1}{7^{3}}(5+3 \alpha)+\ldots \ldots \ldots$, then the value of $\alpha$ is :

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Q JEE MAIN 2025

The area of the region enclosed by the curves $y=\mathbf{e}^{x}, y=\left|\mathbf{e}^{x}-1\right|$ and $y$-axis is :

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Q JEE MAIN 2025

If the system of equations $x+2 y-3 z=2$ $2 x+\lambda y+5 z=5$ $14 x+3 y+\mu z=33$ has infinitely many solutions,...

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Q

The equation of the chord, of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$, whose mid-point is $(3,1)$ is :

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Q JEE MAIN 2025

Suppose $A$ and $B$ are the coefficients of $30^{\text {th }}$ and $12^{\text {th }}$ terms respectively in the binomial...

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Q JEE MAIN 2025

Let $[\mathrm{x}]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where...

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Q JEE MAIN 2025

If the system of linear equations $...

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Q JEE MAIN 2025

Let one focus of the hyperbola $H:\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ be at $(\sqrt {10} ,0)$ and the corresponding directrix...

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Q JEE MAIN 2025

Let $a \in R$ and A be a matrix of order $3 \times 3$ such that $...

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Q JEE MAIN 2025

For $\alpha ,\beta ,\gamma \in R$ , if $...

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Q JEE MAIN 2025

Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of...

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Q JEE MAIN 2025

The largest $n \in N$ such that ${3^n}$ divides 50 ! is :

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Q JEE MAIN 2025

Let $A = \left[ {\begin{array}{*{20}{c}} \alpha &{ - 1}\\ 6&\beta \end{array}} \right],\alpha > 0$, such that $\det (A) = 0$...

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Q JEE MAIN 2025

The number of sequences of ten terms, whose terms are either 0 or 1 or 2 , that contain exactly...

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Q JEE MAIN 2025

If S and S' are the foci of the ellipse $\frac{{{x^2}}}{{18}} + \frac{{{y^2}}}{9} = 1$ and P be a point...

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Q JEE MAIN 2025

If the function $f(x) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1$, where $a > 0$ , attains its local...

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Q JEE MAIN 2025

If $\theta \in [ - 2\pi ,2\pi ]$, then the number of solutions of $...

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Q JEE MAIN 2025

If $\vec a$ is a nonzero vector such that its projections on the vectors $...

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Q JEE MAIN 2025

The term independent of x in the expansion of $...

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Q JEE MAIN 2025

Let the vertices Q and R of the triangle PQR lie on the line $...

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Q JEE MAIN 2025

Let the focal chord PQ of the parabola ${y^2} = 4x$ make an angle of ${60^\circ}$ with the positive axis,...

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Q JEE MAIN 2025

Let ${a_1},{a_2},{a_3}, \ldots $. be in an A.P. such that $...

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Q JEE MAIN 2025

Let ${P_n} = {\alpha ^n} + {\beta ^n},n \in N$ . If ${P_{10}} = 123,{P_9} = 76,{P_8} = 47$ and...

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Q JEE MAIN 2025

Let $f:R \to R$ be a twice differentiable function such that $...

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Q JEE MAIN 2025

Let z be a complex number such that $|z| = 1$. If $...

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Q JEE MAIN 2025

Let A be the set of all functions $f:Z \to Z$ and R be a relation on A such that...

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Q JEE MAIN 2025

Let $\alpha, \beta$ be the roots of the equation $x^{2}-\mathrm{ax}-\mathrm{b}=0$ with $\operatorname{Im}(\alpha)

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Q JEE MAIN 2025

The roots of the quadratic equation $3 x^{2}-p x+q=0$ are $10^{\text {th }}$ and $11^{\text {th }}$ terms of an...

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Q

The variance of the numbers 8, 21, 34, 47, .............., 320 is._______.

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Q JEE MAIN 2025

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys...

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Q JEE MAIN 2025

The focus of the parabola $y^{2}=4 x+16$ is the centre of the circle $C$ of radius 5 . If the...

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Q JEE MAIN 2025

If $\mathrm{I}=\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} \mathrm{~d} x$, then $...

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Q JEE MAIN 2025

Let $\int \mathrm{x}^{3} \sin x \mathrm{~d} x=\mathrm{g}(x)+\mathrm{C}$, where C is the constant of integration. If $...

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Q JEE MAIN 2025

The distance of the line $\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}$ from the point $(1,4,0)$ along the line $\frac{x}{1}=\frac{y-2}{2}=\frac{z+3}{3}$ is :

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Q JEE MAIN 2025

$\lim _{x \rightarrow \infty} \frac{\left(2 x^{2}-3 x+5\right)(3 x-1)^{\frac{x}{2}}}{\left(3 x^{2}+5 x+4\right) \sqrt{(3 x+2)^{x}}}$ is equal to :

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Q JEE MAIN 2025

If the area of the region $\left\{(x, y):-1 \leq x \leq 1,0 \leq y \leq \mathrm{a}+\mathrm{e}^{|x|}-\mathrm{e}^{-x}, \mathrm{a}>0\right\}$ is $\frac{\mathrm{e}^{2}+8 \mathrm{e}+1}{\mathrm{e}}$,...

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Q JEE MAIN 2025

A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x-y+2=0$...

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Q JEE MAIN 2025

The number of complex numbers $z$, satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$, is :

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Q JEE MAIN 2025

Let the shortest distance from ( $\mathrm{a}, 0$ ), $\mathrm{a}>0$, to the parabola $y^{2}=4 x$ be 4 . Then the...

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Q JEE MAIN 2025

If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5}$ is $\frac{m}{n}$, where $m, n$ are coprime...

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Q JEE MAIN 2025

If in the expansion of $(1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}$, the coefficients of x and $x^{2}$ are 1 and -2 , respectively, then $p^{2}+q^{2}$...

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Q JEE MAIN 2025

Let $X=\mathbf{R} \times \mathbf{R}$. Define a relation R on X as : $\left(a_{1}, b_{1}\right) R\left(a_{2}, b_{2}\right) \Leftrightarrow b_{1}=b_{2}$. Statement I:...

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Q JEE MAIN 2025

Let the point A divide the line segment joining the points $P(-1,-1,2)$ and $Q(5,5,10)$ internally in the ratio $r: 1(r>0)$....

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Q JEE MAIN 2025

The system of equations $x+y+z=6$, $x+2 y+5 z=9$, $x+5 y+\lambda z=\mu$, has no solution if

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Q JEE MAIN 2025

Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y \frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right), y>0$ and $x(1)=\frac{\pi}{2}$ Then $...

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Q JEE MAIN 2025

The length of the chord of the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$, whose mid-point is $\left(1, \frac{1}{2}\right)$, is :

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Q JEE MAIN 2025

A board has 16 square as shown in the figures : Out of these 16 squares, two squares are chosen...

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Q JEE MAIN 2025

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream...

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Let $S=\left\{m \in \mathbf{Z}: A^{m^2}+A^m=3 I-A^{-6}\right\}$, where $A=\left[\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}\right]$. Then $n(S)$ is equal to ______.

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Q JEE MAIN 2025

Let $\mathrm{S}=\left\{x: \cos ^{-1} x=\pi+\sin ^{-1} x+\sin ^{-1}(2 x+1)\right\}$. Then $\sum_{x \in \mathrm{~S}}(2 x-1)^2$ is equal to ______.

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Q JEE MAIN 2025

Let $f:(0,\infty ) \to {\bf{R}}$ be a twice differentiable function. If for some $...

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Q JEE MAIN 2025

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS...

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Q JEE MAIN 2025

Let [t] be the greatest integer less than or equal to t. Then the least value of $p\in \mathbb{N}$ for...

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Q JEE MAIN 2025

Let ${{\rm{L}}_1}:\frac{{x - 1}}{1} = \frac{{y - 2}}{{ - 1}} = \frac{{z - 1}}{2}$ and $...

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Q JEE MAIN 2025

Let $\overrightarrow {\rm{a}} = \hat i + 2\hat j + \hat k$ and $...

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Q JEE MAIN 2025

Let $...

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Q JEE MAIN 2025

Let $A = \left[ {{a_{ij}}} \right] = \left[ {\begin{array}{*{20}{c}} {{{\log }_5}128}&{{{\log }_4}5}\\ {{{\log }_5}8}&{{{\log }_4}25} \end{array}} \right]$. If ${A_{ij}}$ is...

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Q JEE MAIN 2025

Let ${x_1},{x_2}, \ldots .,{x_{10}}$ be ten observations such that $...

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Q JEE MAIN 2025

Let ABC be a triangle formed by the lines $...

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Q JEE MAIN 2025

Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these...

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Q JEE MAIN 2025

Let $\left| {{z_1} - 8 - 2i} \right| \le 1$ and $...

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Q JEE MAIN 2025

Let M and N respectively be the maximum and the minimum values of $...

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Q JEE MAIN 2025

The value of $...

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Q JEE MAIN 2025

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of...

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Q JEE MAIN 2025

Let the line $x + y = 1$ meet the circle ${x^2} + {y^2} = 4$ at the points A...

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Q JEE MAIN 2025

The integral $80\int_0^{\frac{\pi }{4}} {\left( {\frac{{\sin \theta + \cos \theta }}{{9 + 16\sin 2\theta }}} \right)} d\theta $ is equal...

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Q JEE MAIN 2025

The number of solutions of the equation $...

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Q JEE MAIN 2025

Let $y = y(x)$ be the solution of the differential equation $...

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Q JEE MAIN 2025

Let the ellipse ${{\rm{E}}_1}:\frac{{{x^2}}}{{{{\rm{a}}^2}}} + \frac{{{y^2}}}{{\;{{\rm{b}}^2}}} = 1,{\rm{a}} > {\rm{b}}$ and ${{\rm{E}}_2}:\frac{{{x^2}}}{{\;{{\rm{A}}^2}}} + \frac{{{y^2}}}{{\;{{\rm{B}}^2}}} = 1,\;{\rm{A}} < {\rm{B}}$ have same...

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Q JEE MAIN 2025

Define a relation R on the interval $\left[ {0,\frac{\pi }{2}} \right)$ by if and only if $...

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Q JEE MAIN 2025

The least value of n for which the number of integral terms in the Binomial expansion of ${(\sqrt[3]{7} + \sqrt[{12}]{{11}})^n}$...

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Q JEE MAIN 2025

Let P be the set of seven digit numbers with sum of their digits equal to 11 . If the...

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Q JEE MAIN 2025

Let the area of the region $...

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Q JEE MAIN 2025

If $\alpha = 1 + \sum\limits_{r = 1}^6 {{{( - 3)}^{r - 1}}} {\quad ^{12}}{{\rm{C}}_{2r - 1}}$, then the distance...

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Q JEE MAIN 2025

Let $...

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Q JEE MAIN 2025

Let M denote the set of all real matrices of order $3 \times 3$ and let $...

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Q JEE MAIN 2025

Let ${{\rm{E}}_1}:\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$ be an ellipse. Ellipses ${{\rm{E}}_1}$'s are constructed such that their centres and eccentricities are...

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Q JEE MAIN 2025

Let $...

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Q JEE MAIN 2025

If $...

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Q JEE MAIN 2025

Let ${{\rm{T}}_{\rm{r}}}$ be the ${{\rm{r}}^{{\rm{th }}}}$ term of an A.P. If for some ${\rm{m}},{{\rm{T}}_{\rm{m}}} = \frac{1}{{25}},\;{{\rm{T}}_{25}} = \frac{1}{{20}}$ , and...

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Q JEE MAIN 2025

The sum, of the squares of all the roots of the equation $...

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Q JEE MAIN 2025

Let ABCD be a trapezium whose vertices lie on the parabola ${{\rm{y}}^2} = 4{\rm{x}}$. Let the sides AD and BC...

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Q JEE MAIN 2025

If $f(x) = \frac{{{2^x}}}{{{2^x} + \sqrt 2 }},{\rm{x}} \in $, then $\sum\limits_{{\rm{k}} = 1}^{81} f \left( {\frac{{\rm{k}}}{{82}}} \right)$ is equal...

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Q JEE MAIN 2025

Let O be the origin, the point A be ${z_1} = \sqrt 3 + 2\sqrt 2 i$, the point $...

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Q JEE MAIN 2025

Let the equation of the circle, which touches x-axis at the point $(a,0),a > 0$ and cuts off an intercept...

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Q JEE MAIN 2025

The area (in sq. units) of the region $...

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Q JEE MAIN 2025

Let $^{\rm{n}}{{\rm{C}}_{{\rm{r}} - 1}} = 28{,^{\rm{n}}}{{\rm{C}}_{\rm{r}}} = 56$ and $^{\rm{n}}{{\rm{C}}_{{\rm{r}} + 1}} = 70$. Let $...

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Q JEE MAIN 2025

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $...

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Q JEE MAIN 2025

$\cos \left( {{{\sin }^{ - 1}}\frac{3}{5} + {{\sin }^{ - 1}}\frac{5}{{13}} + {{\sin }^{ - 1}}\frac{{33}}{{65}}} \right)$ is equal to:

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Q JEE MAIN 2025

The relation $R = \{ (x,y):x,y \in $ and $x + y$ is even is:

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Let ${\rm{A}}(x,y,z)$ be a point in xy-plane, which is equidistant from three points (0,3,2),(2,0,3) and (0,0,1). Let $...

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Let $\left\langle {{a_{\rm{n}}}} \right\rangle $ be a sequence such that ${a_0} = 0,{a_1} = \frac{1}{2}$ and $...

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Q JEE MAIN 2025

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0 , 1,2,3,4,5,6,7,...

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Q JEE MAIN 2025

The sum of all local minimum values of the function $...

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Let for some function ${\rm{y}} = f(x),\int_0^x t f(t)dt = {x^2}f(x),x > 0$ and $f(2) = 3$. Then $f(6)$ is...

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Q JEE MAIN 2025

Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to...

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Q JEE MAIN 2025

If the image of the point (4,4,3) in the line $...

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Q JEE MAIN 2025

Two number ${k_1}$ and ${k_2}$ are randomly chosen from the set of natural numbers. Then, the probability that the value...

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Q JEE MAIN 2025

Let f be a differentiable function such that $...

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Q JEE MAIN 2025

Let $S = \left\{ {{p_1},{p_2} \ldots ,{p_{10}}} \right\}$ be the set of first ten prime numbers. Let $...

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Q JEE MAIN 2025

If for some $\alpha ,\beta ;\alpha \le \beta ,\alpha + \beta = 8$ and $...

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Q JEE MAIN 2025

The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is

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Q JEE MAIN 2025

Let A be a $3 \times 3$ matrix such that ${X^T}AX = O$ for all nonzero $3 \times 1$ matrices...

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Q JEE MAIN 2025

If $I(m,n) = \int_0^1 {{x^{m - 1}}} {(1 - x)^{n - 1}}dx,m,n > 0$, then $I(9,14) + I(10,13)$ is

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Q JEE MAIN 2025

A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B...

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Q JEE MAIN 2025

$...

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Q JEE MAIN 2025

For some ${\rm{n}} \ne 10$, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of...

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Q JEE MAIN 2025

The product of all the rational roots of the equation $...

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Q JEE MAIN 2025

Let in a $\Delta ABC$, the length of the side AC be 6, the vertex B be (1, 2, 3)...

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Q JEE MAIN 2025

Let the lines $3x - 4y - \alpha = 0,8x - 11y - 33 = 0$, and $...

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Q JEE MAIN 2025

Let $f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$ be a function such that $f(x) - 6f\left( {\frac{1}{x}} \right) = \frac{{35}}{{3x}} - \frac{5}{2}$ ....

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Q JEE MAIN 2025

Let the product of the focal distances of the point $\left( {\sqrt 3 ,\frac{1}{2}} \right)$ on the ellipse $...

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Q JEE MAIN 2025

For a statistical data ${{\rm{x}}_1},{{\rm{x}}_2}, \ldots ,{{\rm{x}}_{10}}$ of 10 values, a student obtained the mean as 5.5 and $...

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Q JEE MAIN 2025

Let $f(x) = \frac{{{2^{x + 2}} + 16}}{{{2^{2x + 1}} + {2^{x + 4}} + 32}}$. Then the value of...

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Q JEE MAIN 2025

Let y = y(x) be the solution of the differential equation $...

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Q JEE MAIN 2025

If $\alpha $ and $\beta $ are the roots of the equation $2{z^2} - 3z - 2i = 0$, where...

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Q JEE MAIN 2025

If the system of equations $2x - y + z = 4$ $5x + \lambda y + 3z = 12$...

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Q JEE MAIN 2025

Let $...

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Q JEE MAIN 2025

Let the line passing through the points (-1, 2, 1) and parallel to the line $...

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Let ${S_n} = \frac{1}{2} + \frac{1}{6} + \frac{1}{{12}} + \frac{1}{{20}} + \ldots $ upto $n$ terms. If the sum of...

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Q JEE MAIN 2025

Consider the region $R = \left\{ {(x,y):x \le y \le 9 - \frac{{11}}{3}{x^2},x \ge 0} \right\}$. The area, of the...

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Q JEE MAIN 2025

The area of the region $\left\{ {(x,y):{x^2} + 4x + 2 \le y \le |x + 2|} \right\}$ is equal...

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Q JEE MAIN 2025

Let circle C be the image of ${x^2} + {y^2} - 2x + 4y - 4 - 0$ in the...

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Q JEE MAIN 2025

If the set of all values of a, for which the equation $5{x^3} - 15x - a = 0$ has...

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Q JEE MAIN 2025

If the area of the larger portion bounded between the curves ${x^2} + {y^2} = 25$ and $...

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Q JEE MAIN 2025

Let the circle C touch the line $x - y + 1 = 0$ , have the centre on the...

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Q JEE MAIN 2025

The sum of all rational terms in the expansion of ${\left( {1 + {2^{1/3}} + {3^{1/2}}} \right)^6}$ is equal to...

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Q JEE MAIN 2025

If the equation $a(b - c){x^2} + b(c - a)x + c(a - b) = 0$ has equal roots, where...

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Q JEE MAIN 2025

One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4....

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If A,B, and $\left( {{\mathop{\rm adj}\nolimits} \left( {{{\rm{A}}^{ - 1}}} \right) + {\mathop{\rm adj}\nolimits} \left( {{{\rm{B}}^{ - 1}}} \right)} \right)$...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $\left| {\frac{{\bar z - i}}{{2\bar z + i}}} \right| = \frac{1}{3},z \in C$ , be the equation of a...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let ${\rm{I}}(x) = \int {\frac{{dx}}{{{{(x - 11)}^{\frac{{11}}{{13}}}}{{(x + 15)}^{\frac{{15}}{{13}}}}}}} \cdot $. If $...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let P be the foot of the perpendicular from the point $Q(10, - 3, - 1)$ on the line $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the area of a $\Delta PQR$ with vertices $P(5,4),Q( - 2,4)$ and $R(a,b)$ be 35 square units. If its...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the position vectors of the vertices A, B and C of a tetrahedron A, B, C, D be $...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

Marks obtains by all the students of class 12 are presented in a freqency distribution with classes of equal width....

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let a curve $y = f(x)$ pass through the points (0,5) and $\left( {{{\log }_e}2,k} \right)$. If the curve satisfies...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If the first term of an A.P. is 3 and the sum of its first four terms is equal to...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If the function $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the arc AC of a circle subtend a right angle at the centre O . If the point B...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

The value of $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If the line $3x-2y+12=0$ intersects the parabola $4y=3{{x}^{2}}$ at the points A and B, then at the vertex of the...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

The value of $\left( {\sin {{{70}}^{{}^\circ }}} \right)\left( {\cot {{{10}}^{{}^\circ }}\cot {{{70}}^{{}^\circ }}-1} \right)$ is

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If the system of equations $(\lambda -1)\text{x}+(\lambda -4)\text{y}+\lambda \text{z}=5$ $\lambda \text{x}+(\lambda -1)\text{y}+(\lambda -4)\text{z}=7$ $(\lambda +1)\text{x}+(\lambda +2)\text{y}-(\lambda +2)\text{z}=9$ has infinitely many...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $R=\{(1,2),(2,3),(3,3)\}$ be a relation defined on the set $\{1,2,3,4\}$. Then the minimum number of elements, needed to be added...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

If $\frac{\pi }{2}\le x\le \frac{{3\pi }}{4}$ , then ${{\cos }^{{-1}}}\left( {\frac{{12}}{{13}}\cos x+\frac{5}{{13}}\sin x} \right)$ is equal to

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $f(x)={{\log }_{e}}x$ and $ g(x)=\frac{{{{x}^{4}}-2{{x}^{3}}+3{{x}^{2}}-2x+2}}{{2{{x}^{2}}-2x+1}}$ . Then the domain of $fog$ is

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $A(6,8), B(10 \cos \alpha,-10 \sin \alpha)$ and $C(-10 \sin \alpha, 10 \cos \alpha)$, be the vertices of a triangle....

JEE Main Mathematics Medium

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Q

Let \( y = f(x) \) be the solution of the differential equation \[\frac{dy}{dx} = \frac{y - x^2 + 4x}{\sqrt{1...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

If $\sum_{r=1}^{30} \frac{r^2\left({ }^{30} C_r\right)^2}{{ }^{30} C_{r-1}}=\alpha \times 2^{29}$, then $\alpha$ is equal to

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

The perpendicular distance, of the line $\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z+3}{2}$ from the point $\mathrm{P}(2,-10,1)$, is :

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let the curve $z(1+i)+\bar{z}(1-i)=4, z \in \mathbf{C}$, divide the region $|z-3| \leq 1$ into two parts of areas $\alpha$ and...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let for $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black...

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Q JEE MAIN 2025

The product of all solutions of the equation $\mathrm{e}^{5\left(\log _{\mathrm{c}} x\right)^2+3}=x^8, x>0$, is :

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $A=\{1,2,3,4\}$ and $B=\{1,4,9,16\}$. Then the number of many-one functions $f: A \rightarrow B$ such that $1 \in f(\mathrm{~A})$ is...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

If $\sum_{r=1}^n T_r=\frac{(2 n-1)(2 n+1)(2 n+3)(2 n+5)}{64}, then {\lim _{n \rightarrow \infty} \sum_{r=1}^n\left(\frac{1}{T_r}\right)}$ is equal to :

JEE Main Mathematics Medium

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Q JEE MAIN 2025

A circle $C$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $r$ be...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the foci of a hyperbola be (1,14) and (1,-12). If it passes through the point(1, 6), then the length...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Suppose that the number of terms in an A.P. is $2 k, k \in \mathbf{N}$. If the sum of all...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $f(x+y)=f(x) f(y)$ for all $x, y \in \mathbf{R}$....

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2 x^2+(\cos \theta) x-1=0, \theta \in(0,2 \pi)$. If m and M...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $A=\{1,2,3, \ldots, 10\}$ and $B = \left\{ {\frac{m}{n}:m,n \in A,m < n} \right.$ and $\left. {gcd(m,n) = 1} \right\}$....

JEE Main Mathematics Hard

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Q JEE MAIN 2025

The area of the region, inside the circle ${(x - 2\sqrt 3 )^2} + {y^2} = 12$ and outside the...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $x=x(y)$ be the solution of the differential equation ${{y}^{2}}~\text{d}x+\left( {x-\frac{1}{y}} \right)\text{d}y=0$. If $x(1)=1$, then $x\left( {\frac{1}{2}} \right)$ is :

JEE Main Chemistry Medium

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Q JEE MAIN 2025

Let ${{z}_{1}},{{z}_{2}}$ and ${{z}_{3}}$ be three complex numbers on the circle $|z|=1$ with $...

JEE Main Physics Easy

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Q JEE MAIN 2025

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x){{f}^{\prime }}(y)+{{f}^{\prime }}(x)f(y)$ for all $x,y\in \mathbf{R}$. Then $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let ${{a}_{1}},{{a}_{2}},{{a}_{3}},\ldots $ be a G.P. of increasing positive terms. If ${{a}_{1}}{{a}_{5}}=28$ and ${{a}_{2}}+{{a}_{4}}=29$, then ${{a}_{6}}$ is equal to :

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If the system of linear equations : $$ \begin{aligned} & x+y+2 z=6 \\ & 2 x+3 y+a z=a+1 \\ &...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways,...

JEE Main Mathematics Easy

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Q JEE MAIN

Let $\mathrm{L}_1: \frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0}$ and $\mathrm{L}_2: \frac{x-2}{2}=\frac{y}{0}=\frac{z+4}{\alpha}, \alpha \in \mathbf{R}$, be two lines, which intersect at the point $B$. If $P$...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

The area of the region enclosed by the curves $y=x^2-4 x+4$ and $y^2=16-8 x$ is:

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let A be a square matrix of order 3 such that $\det (A)=-2$ and $\det (3\operatorname{adj}(-6\operatorname{adj}(3A)))={{2}^{{m+n}}}\cdot {{3}^{{mn}}},m>n$. Then $4m+2n$ is...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If $\sum_{r=0}^5 \frac{{ }^{11} C_{2 r+1}}{2 r+2}=\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}-\mathrm{n}$ is equal to ..........

JEE Main Mathematics Hard

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Q JEE MAIN 2025

Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda>0$, on the vector $\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}$. If $|\vec{a}+\vec{c}|=7$, then...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways,...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If $\int \mathrm{e}^x\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}}+\frac{x}{1-x^2}\right) \mathrm{d} x=\mathrm{g}(x)+\mathrm{C}$, where C is the constant of integration, then...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let the parabola $y={{x}^{2}}+\text{p}x-3$ , meet the coordinate axes at the points P,Q and R. If the circle C with...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

The sum of all values of $\theta \in[0,2 \pi]$ satisfying $2 \sin ^2 \theta=\cos 2 \theta$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let the triangle PQR be the image of the triangle with vertices (1,3)(3,1) and (2,4) in the line $x+2y=2$. If...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a}+2 \vec{b}$...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $\mathrm{E}: \frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}$ and $\mathrm{H}: \frac{x^2}{\mathrm{~A}^2}-\frac{y^2}{\mathrm{~B}^2}=1$. Let the distance between the foci of E and the foci of $H$...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let ${{\text{L}}_{1}}:\frac{{x-1}}{2}=\frac{{y-2}}{3}=\frac{{z-3}}{4}$ and ${{\text{L}}_{2}}:\frac{{x-2}}{3}=\frac{{y-4}}{4}=\frac{{z-5}}{5}$ be two lines. Then which of the following points lies on the line of the shortest...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $f(x)=\int_0^{x^2} \frac{\mathrm{t}^2-8 \mathrm{t}+15}{\mathrm{e}^{\mathrm{t}}} \mathrm{dt}, x \in \mathbf{R}$. Then the numbers of local maximum and local minimum points of $f$,...

JEE Main Mathematics Easy

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Q

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $\mathrm{P}(4,4 \sqrt{3})$ be a point on the parabola $y^2=4 \mathrm{ax}$ and PQ be a focal chord of the parabola....

JEE Main Mathematics Medium

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Q JEE MAIN 2025

The number of non-empty equivalence relations on the set {1,2,3} is :

JEE Main Mathematics Medium

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Q JEE MAIN 2025

If $\lim _{x \rightarrow \infty}\left(\left(\frac{\mathrm{e}}{1-\mathrm{e}}\right)\left(\frac{1}{\mathrm{e}}-\frac{x}{1+x}\right)\right)^x=\alpha$, then the value of $\frac{\log _{\mathrm{e}} \alpha}{1+\log _{\mathrm{e}} \alpha}$ equals :

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If $A$ and $B$ are two events such that $P(A \cap B)=0.1$, and $P(A \mid B)$ and $P(B \mid A)$...

JEE Main Mathematics Easy

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