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

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

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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 $\alpha$ 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 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

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

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Q JEE MAIN_2026

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

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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 $\alpha$ 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_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 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 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

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

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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 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 $\alpha$ 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 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

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

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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 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

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 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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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 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

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 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_2021

The set of all values of $k>-1$, for which the equation $...

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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 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

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

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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Q JEE MAIN 2019

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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Q JEE MAIN 2019

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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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 $\lambda \neq 0$ 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 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

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_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 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 $a+b+c=1, a b+b c+c a=2$ and $a b c=3$, then the value of $a 4+b 4+c 4$ is equal...

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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

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 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

The number of real solutions of the equation, $\mathrm{x}^2-|\mathrm{x}|-12=0$ is:

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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

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

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

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_

VALUE OF

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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

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_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

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 $p$ 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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Q JEE MAIN 2022

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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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

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

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

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

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

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

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

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

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

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

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

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

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

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 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

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

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

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

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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Q JEE-Main 2024

Let S={x∈R:(√3+√2 )^x+(√3-√2 )^x=10}. Then the number of elements in S is :

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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 $\alpha+i \beta$ 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

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

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 $\alpha$ and $\beta $ be the roots of ${x^2} + \sqrt 3 x - 16 = 0$, and $...

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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 $\alpha, \beta$ 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 number of solutions of the equation $...

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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

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

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...

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