Q JEE MAIN 2026

$$ f(x)= \begin{cases}\left|\frac{\sin x}{x}\right| & , x \neq 0 \\ 1 & , x=0\end{cases} $$ The number of critical points...

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Q JEE MAIN 2022

The sum of the maximum and minimum values of the function $f(x)=|5 x-7|+\left[x^2+2 x\right]$ in the interval $\left[\frac{5}{4}, 2\right]$, where...

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Q JEE MAIN 2026

Let $f: \mathrm{i} \rightarrow \mathrm{i}$ be a differentiable function such that $f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)}{3}$ for all $x, y \in \mathrm{i}$, and $f^{\prime}(0)=3$....

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Q JEE MAIN 2026

$\max _{0 \leq x \leq \pi}\left(16 \sin \left(\frac{x}{2}\right) \cos ^3\left(\frac{x}{2}\right)\right)$ is equal to:

JEE Main Mathematics Medium

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Q JEE Main 2026

Let $f(x)$ be a polynomial of degree 5 , and have extrema at $x=1$ and $x=-1$. If $...

JEE Main Mathematics Easy

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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 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

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

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

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

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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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Q JEE MAIN 2024

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 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

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...

JEE Main Mathematics Medium

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

JEE Main Mathematics Hard

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Q JEE MAIN 2019

JEE Main Mathematics Hard

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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 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 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

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

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 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

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

JEE Main Mathematics Hard

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Q JEE MAIN 2020

JEE Main Mathematics Hard

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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 2019

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

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 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 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_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

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

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

Let f(x) be a polynomial of degree 3 such that f(–1) = 10, f(1) = –6, f(x) has a critical...

JEE Main Mathematics Medium

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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;...

JEE Main Mathematics Easy

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Q JEE-MAIN 2020

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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Q JEE MAIN 2020

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 $...

JEE Main Mathematics Medium

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Q JEE MAIN 2020

Let S be the set of all functions f : [0, 1]  R, which are continuous on [0, 1]...

JEE Main Mathematics Hard

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Q JEE MAIN 2019

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...

JEE Main Mathematics Hard

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Q JEE Main 2021

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...

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

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_

Let $M$ and $m$ respectively be the maximum and minimum values of the function $f(x)=\tan ^{-1}(\sin x+\cos x)$ in $...

JEE Main Mathematics Easy

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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 :

JEE Main Mathematics Medium

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Q JEE Main 2019

Let $0

JEE Main Mathematics Hard

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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, $...

JEE Main Mathematics Easy

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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

JEE Main Mathematics Medium

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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...

JEE Main Mathematics Medium

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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

JEE Main Mathematics Medium

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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...

JEE Main Mathematics Medium

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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 $A=\left[a_{i j}\right]$ be a $3 \times 3$ matrix, where $...

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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 $\left(0, \frac{\pi}{2}\right)$...

JEE Main Mathematics Medium

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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 $...

JEE Main Mathematics Hard

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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...

JEE Main Mathematics Easy

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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

The triangle of maximum area that can be inscribed in a given circle of radius ' r ' is :

JEE Main Mathematics Medium

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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

JEE Main Mathematics Medium

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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...

JEE Main Mathematics Medium

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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...

JEE Main Mathematics Medium

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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...

JEE Main Mathematics Medium

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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 $f(\alpha)$,...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

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...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

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...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $...

JEE Main Mathematics Medium

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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...

JEE Main Mathematics Hard

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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...

JEE Main Mathematics Medium

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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)...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

The number of distinct real roots of $x^4-4 x+1=0$ is :

JEE Main Mathematics Medium

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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 $[-1,2]$...

JEE Main Mathematics Easy

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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...

JEE Main Mathematics Hard

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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...

JEE Main Mathematics Medium

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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...

JEE Main Physics Easy

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Q JEE MAIN 2025

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...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

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...

JEE Main Mathematics Medium

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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...

JEE Main Mathematics Medium

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Q JEE MAIN

Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $\left(1+x\left(\lambda^2-x^2\right)\right)$...

JEE Main Mathematics Medium

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Q JEE MAIN

A variable line L passes through the point (3,5) and intersects the positive coordinate axes at the points A and...

JEE Main Mathematics Medium

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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...

JEE Main Mathematics Medium

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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:

JEE Main Mathematics Easy

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Q JEE MAIN 2024

The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the...

JEE Main Mathematics Medium

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Q JEE MAIN

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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Q JEE MAIN 2025

Let $x = - 1$ and $x = 2$ be the critical points of the function $...

JEE Main Mathematics Medium

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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...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

The sum of all local minimum values of the function $...

JEE Main Mathematics Medium

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