JEE MAIN Quadratic Equations Previous Year Question Paper and Solution with PDF

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JEE MAIN

JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Quadratic Equations PYQ

Qus : 1 JEE MAIN PYQ

The number of real solutions of the equation, x² $-$ |x| $-$ 12 = 0 is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Evening Shift) PYQ

Solution

Qus : 2 JEE MAIN PYQ

Let a , b, c , d and p be any non zero distinct real numbers such that(a² + b² + c²)p² – 2(ab + bc + cd)p + (b² + c² + d²) = 0. Then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Morning) PYQ

Solution

Qus : 3 JEE MAIN PYQ

The minimum value of the sum of the squares of the roots of $x^{2}+(3-a)x+1=2a$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 July Evening Shift) PYQ

Solution

Qus : 4 JEE MAIN PYQ

If $\alpha $ and $\beta $ be two roots of the equation x² – 64x + 256 = 0. Then the value of${\left( {{{{\alpha ^3}} \over {{\beta ^5}}}} \right)^{1/8}} + {\left( {{{{\beta ^3}} \over {{\alpha ^5}}}} \right)^{1/8}}$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Morning) PYQ

Solution

Qus : 5 JEE MAIN PYQ

The value of $\lambda$ such that the sum of the squares of the roots of the quadratic equation $x^2 + (3 - \lambda)x + 2 = \lambda$ has the least value, is –

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Evening Shift) PYQ

Solution

Qus : 6 JEE MAIN PYQ

The number of integral values of $k$ for which one root of the equation $2x^{2}-8x+k=0$ lies in the interval $(1,2)$ and its other root lies in the interval $(2,3)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (1 February Evening Shift) PYQ

Solution

Qus : 7 JEE MAIN PYQ

Let $\alpha$ and $\beta$ be the roots of equation $px^{2}+qx+r=0$, $p\ne 0$. If $p,q,r$ are in A.P. and $\dfrac{1}{\alpha}+\dfrac{1}{\beta}=4$, then the value of $|\alpha-\beta|$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2014 (Offline) PYQ

Solution

Qus : 8 JEE MAIN PYQ

If $\alpha $ and $\beta $ are the roots of the equation2x(2x + 1) = 1, then $\beta $ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Evening) PYQ

Solution

Qus : 9 JEE MAIN PYQ

If $\lambda \in \mathbb{R}$ is such that the sum of the cubes of the roots of the equation $x^{2} + (2-\lambda)x + (10-\lambda)=0$ is minimum, then the magnitude of the difference of the roots of this equation is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Morning Shift) PYQ

Solution

Qus : 10 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Morning Shift) PYQ

Solution

Qus : 11 JEE MAIN PYQ

The sum of all the roots of the equation $\lvert x^{2}-8x+15\rvert-2x+7=0$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Morning Shift) PYQ

Solution

Qus : 12 JEE MAIN PYQ

The product of all the rational roots of the equation $ (x^2 - 9x + 11)^2 - (x - 4)(x - 5) = 3 $ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Morning Shift) PYQ

Solution

Qus : 13 JEE MAIN PYQ

If the equations $x^{2}+2x+3=0$ and $ax^{2}+bx+c=0$, $a,b,c\in\mathbb{R}$, have a common root, then $a:b:c$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2013 (Offline) PYQ

Solution

Qus : 14 JEE MAIN PYQ

If $\alpha, \beta$ are the roots of the equation $x^{2} - \left(5 + 3\sqrt{\log_{3}5} - 5\sqrt{\log_{5}3}\right)x + 3\left(3^{\tfrac{1}{3}\log_{3}5} - 5^{\tfrac{2}{3}\log_{5}3} - 1\right) = 0$, then the equation, whose roots are $\alpha + \tfrac{1}{\beta}$ and $\beta + \tfrac{1}{\alpha}$, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Evening Shift) PYQ

Solution

Qus : 15 JEE MAIN PYQ

Let the set of all values of $p\in\mathbb{R}$, for which both the roots of the equation $x^{2}-(p+2)x+(2p+9)=0$ are negative real numbers, be the interval $(\alpha,\beta)$. Then $\beta-2\alpha$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (7 April Morning Shift) PYQ

Solution

Qus : 16 JEE MAIN PYQ

If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation 3x² + $\lambda$x $-$ 1 = 0 is 15, then 6($\alpha$³ + $\beta$³)² is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (24 June Morning Shift) PYQ

Solution

Qus : 17 JEE MAIN PYQ

If $f(x)$ is a quadratic expression such that $f(1)+f(2)=0$, and $-1$ is a root of $f(x)=0$, then the other root of $f(x)=0$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Evening Shift) PYQ

Solution

Qus : 18 JEE MAIN PYQ

The number of distinct real roots of the equation

x⁷ $-$ 7x $-$ 2 = 0 is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (24 June Evening Shift) PYQ

Solution

Qus : 19 JEE MAIN PYQ

$ \text{Let } \alpha, \beta \text{ be the roots of the equation } x^{2} - \sqrt{2}x + \sqrt{6} = 0 \text{ and } \dfrac{1}{\alpha^{2}} + 1, ; \dfrac{1}{\beta^{2}} + 1 \text{ be the roots of the equation } x^{2} + ax + b = 0. $ $\text{Then the roots of the equation } x^{2} - (a+b-2)x + (a+b+2) = 0 \text{ are :}$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 July Evening Shift) PYQ

Solution

Qus : 20 JEE MAIN PYQ

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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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Evening Shift) PYQ

Solution

Qus : 21 JEE MAIN PYQ

$ \text{Let } f(x)=ax^{2}+bx+c \text{ be such that } f(1)=3,\ f(-2)=\lambda \text{ and } f(3)=4. $ $ \text{If } f(0)+f(1)+f(-2)+f(3)=14,\ \text{then } \lambda \text{ is equal to:} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 July Evening Shift) PYQ

Solution

Qus : 22 JEE MAIN PYQ

If $\lambda$ be the ratio of the roots of the quadratic equation in $x$, \[ 3m^{2}x^{2}+m(m-4)x+2=0, \] then the least value of $m$ for which $\displaystyle \lambda+\frac{1}{\lambda}=1$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Morning Shift) PYQ

Solution

Qus : 23 JEE MAIN PYQ

Let a, b $\in$ R be such that the equation $a{x^2} - 2bx + 15 = 0$ has a repeated root $\alpha$. If $\alpha$ and $\beta$ are the roots of the equation ${x^2} - 2bx + 21 = 0$, then ${\alpha ^2} + {\beta ^2}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 June Evening Shift) PYQ

Solution

Qus : 24 JEE MAIN PYQ

The set of all values of K > $-$1, for which the equation ${(3{x^2} + 4x + 3)^2} - (k + 1)(3{x^2} + 4x + 3)(3{x^2} + 4x + 2) + k{(3{x^2} + 4x + 2)^2} = 0$ has real roots, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Evening Shift) PYQ

Solution

Qus : 25 JEE MAIN PYQ

Let $p, q$ and $r$ be real numbers $(p \ne q,, r \ne 0)$, such that the roots of the equation $\dfrac{1}{x+p} + \dfrac{1}{x+q} = \dfrac{1}{r}$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (16 April Morning Shift) PYQ

Solution

Qus : 26 JEE MAIN PYQ

The number of integral values of $m$ for which the quadratic expression $(1+2m)x^2-2(1+3m)x+4(1+m)$, $x\in\mathbb{R}$, is always positive, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Evening Shift) PYQ

Solution

Qus : 27 JEE MAIN PYQ

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Morning) PYQ

Solution

Let the cubic polynomial be $$p(x) = a x^3 + b x^2 + c x + d$$ where $a, b, c, d$ are real constants.

Given:
Local maximum at $x = 1 \Rightarrow p'(1) = 0,\ p(1) = 8$
Local minimum at $x = 2 \Rightarrow p'(2) = 0,\ p(2) = 4$

Derivative:
$$p'(x) = 3a x^2 + 2b x + c$$ Apply the stationary point conditions:
\[ \begin{cases} 3a(1)^2 + 2b(1) + c = 0 \\[4pt] 3a(2)^2 + 2b(2) + c = 0 \end{cases} \] Simplify: \[ \begin{cases} 3a + 2b + c = 0 \\[4pt] 12a + 4b + c = 0 \end{cases} \] Subtracting gives: \[ 9a + 2b = 0 \Rightarrow b = -\frac{9a}{2}. \] Substitute into $3a + 2b + c = 0$: \[ 3a + 2\left(-\frac{9a}{2}\right) + c = 0 \Rightarrow 3a - 9a + c = 0 \Rightarrow c = 6a. \]
Using the value conditions:
\[ \begin{cases} p(1) = a + b + c + d = 8 \\[4pt] p(2) = 8a + 4b + 2c + d = 4 \end{cases} \] Substitute $b = -\frac{9a}{2},\ c = 6a$:
\[ a - \frac{9a}{2} + 6a + d = 8 \Rightarrow \frac{5a}{2} + d = 8 \Rightarrow d = 8 - \frac{5a}{2}. \] and \[ 8a + 4\left(-\frac{9a}{2}\right) + 2(6a) + d = 4 \Rightarrow 2a + d = 4. \] Substitute $d = 8 - \frac{5a}{2}$: \[ 2a + 8 - \frac{5a}{2} = 4 \Rightarrow -\frac{a}{2} = -4 \Rightarrow a = 8. \]
Now, \[ b = -\frac{9a}{2} = -36, \quad c = 48, \quad d = 8 - \frac{5(8)}{2} = -12. \]
Therefore, $$p(0) = d = -12.$$
Final Answer: $$\boxed{p(0) = -12}$$

Qus : 28 JEE MAIN PYQ

Let $a,b,c\in \mathbb{R}$. If $f(x)=ax^{2}+bx+c$ is such that $a+b+c=3$ and $f(x+y)=f(x)+f(y)+xy,\ \forall x,y\in \mathbb{R}$, then $\displaystyle \sum_{n=1}^{10} f(n)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (Offline) PYQ

Solution

Qus : 29 JEE MAIN PYQ

If for a positive integer $n$, the quadratic equation $x(x+1) + (x+1)(x+2) + \ldots + (x+n-1)(x+n) = 10n$ has two consecutive integral solutions, then $n$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (Offline) PYQ

Solution

Qus : 30 JEE MAIN PYQ

Let $x=\dfrac{m}{n}$ ($m,n$ are co-prime natural numbers) be a solution of the equation $\cos\!\left(2\sin^{-1}x\right)=\dfrac{1}{9}$ and let $\alpha,\beta\ (\alpha>\beta)$ be the roots of the equation $m x^{2}-n x-m+n=0$. Then the point $(\alpha,\beta)$ lies on the line

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Evening Shift) PYQ

Solution

Qus : 31 JEE MAIN PYQ

Let f(x) be a quadratic polynomial such thatf(–1) + f(2) = 0. If one of the roots of f(x) = 0is 3, then its other root lies in :

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JEE MAIN JEE Mains PYQ JEE MAIN PYQ

Solution

Let the quadratic polynomial be $ f(x) = a x^2 + b x + c $.

Given that $$ f(-1) + f(2) = 0 $$ and one of the roots of $ f(x) = 0 $ is $3$.

Let the other root be $\alpha$. Hence, $ f(x) = k(x - 3)(x - \alpha) $, where $k \ne 0$.

Substitute the condition $ f(-1) + f(2) = 0 $: $$ k(-1 - 3)(-1 - \alpha) + k(2 - 3)(2 - \alpha) = 0 $$ Simplify: $$ (-4)(-1 - \alpha) + (-1)(2 - \alpha) = 0 $$ $$ 4(1 + \alpha) - 2 + \alpha = 0 $$ $$ 2 + 5\alpha = 0 \Rightarrow \alpha = -\frac{2}{5}. $$
Therefore, the other root is $-\dfrac{2}{5}$, which lies in $$\boxed{(-1,\ 0)}.$$

Qus : 32 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (24 January Evening Shift) PYQ

Solution

Qus : 33 JEE MAIN PYQ

Let P(x) = x² + bx + c be a quadratic polynomial with real coefficients such that $\int_0^1 {P(x)dx} $ = 1 and P(x) leaves remainder 5 when it is divided by (x $-$ 2). Then the value of 9(b + c) is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (16 March Evening Shift) PYQ

Solution

Qus : 34 JEE MAIN PYQ

Let p(x) be a quadratic polynomial such that p(0)=1. If p(x) leaves remainder 4 when divided by x-1 and it leaves remainder 6 when divided by x+1, then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

Qus : 35 JEE MAIN PYQ

If $\alpha $ and $\beta $ are the roots of the equation x² + px + 2 = 0 and ${1 \over \alpha }$ and ${1 \over \beta }$ are the roots ofthe equation 2x² + 2qx + 1 = 0, then $\left( {\alpha - {1 \over \alpha }} \right)\left( {\beta - {1 \over \beta }} \right)\left( {\alpha + {1 \over \beta }} \right)\left( {\beta + {1 \over \alpha }} \right)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Morning) PYQ

Solution

Qus : 36 JEE MAIN PYQ

The number of integral values of $m$ for which the equation $(1+m^{2})x^{2}-2(1+3m)x+(1+8m)=0$ has no real root is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Evening Shift) PYQ

Solution

Qus : 37 JEE MAIN PYQ

Consider the two sets :
A = {m $ \in $ R : both the roots of x² – (m + 1)x + m + 4 = 0 are real} and B = [–3, 5).
Which of the following is not true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Morning) PYQ

Solution

Qus : 38 JEE MAIN PYQ

Let $\alpha, \beta$ be the roots of the quadratic equation $x^{2}+\sqrt{6}x+3=0$. Then $\dfrac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (12 April Morning Shift) PYQ

Solution

Qus : 39 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Morning Shift) PYQ

Solution

Qus : 40 JEE MAIN PYQ

Let p and q be two positive numbers such that p + q = 2 and p⁴+q⁴ = 272. Then p and q areroots of the equation :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Morning Shift) PYQ

Solution

Qus : 41 JEE MAIN PYQ

Let f(x) be a quadratic polynomial such that f($-$2) + f(3) = 0. If one of the roots of f(x) = 0 is $-$1, then the sum of the roots of f(x) = 0 is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 June Evening Shift) PYQ

Solution

Qus : 42 JEE MAIN PYQ

For $0 < c < b < a$, let $(a+b-2c)x^{2} + (b+c-2a)x + (c+a-2b) = 0$ and let $\alpha \ne 1$ be one of its roots. Then, among the two statements: (I) If $\alpha \in (-1,0)$, then $b$ cannot be the geometric mean of $a$ and $c$. (II) If $\alpha \in (0,1)$, then $b$ may be the geometric mean of $a$ and $c$.

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Morning Shift) PYQ

Solution

Qus : 43 JEE MAIN PYQ

Let $\lambda\ne 0$ be a real number. Let $\alpha,\beta$ be the roots of the equation $14x^{2}-31x+3\lambda=0$ and $\alpha,\gamma$ be the roots of the equation $35x^{2}-53x+4\lambda=0$. Then $\dfrac{3\alpha}{\beta}$ and $\dfrac{4\alpha}{\gamma}$ are the roots of the equation

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Morning Shift) PYQ

Solution

Qus : 44 JEE MAIN PYQ

Let $\alpha, \beta$ 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 $\dfrac{1}{10} (\alpha^{6} + \beta^{6})$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Morning Shift) PYQ

Solution

Qus : 45 JEE MAIN PYQ

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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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Evening Shift) PYQ

Solution

Qus : 46 JEE MAIN PYQ

Let $a$ be the sum of all coefficients in the expansion of $\big(1-2x+2x^{2}\big)^{2023}\big(3-4x^{2}+2x^{3}\big)^{2024}$ and $b=\lim_{x\to 0}\left(\frac{\displaystyle \int_{0}^{x}\frac{\log(1+t)}{2t^{2}+t}\,dt}{x^{2}}\right).$ If the equations $c x^{2}+d x+e=0$ and $2b\,x^{2}+a x+4=0$ have a common root, where $c,d,e\in\mathbb{R}$, then $d:c:e$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Morning Shift) PYQ

Solution

Qus : 47 JEE MAIN PYQ

Let $\alpha,\beta;\ \alpha>\beta,$ be the roots of the equation $x^{2}-\sqrt{2},x-\sqrt{3}=0$. Let $P_{n}=\alpha^{n}-\beta^{n},\ n\in\mathbb{N}$. Then $(11\sqrt{3}-10\sqrt{2}),P_{10}+(11\sqrt{2}+10),P_{11}-11,P_{12}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Evening Shift) PYQ

Solution

Qus : 48 JEE MAIN PYQ

If $m$ is chosen in the quadratic equation $(m^{2}+1)x^{2}-3x+(m^{2}+1)^{2}=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Evening Shift) PYQ

Solution

Qus : 49 JEE MAIN PYQ

The set of all real values of $\lambda $ for which thequadratic equations,
($\lambda $² + 1)x² – 4$\lambda $x + 2 = 0 always have exactly one root in the interval (0, 1) is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Evening) PYQ

Solution

Qus : 50 JEE MAIN PYQ

If the equations $x^{2} + bx - 1 = 0$ and $x^{2} + x + b = 0$ have a common root different from $-1$, then $|b|$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (9 April Morning Shift) PYQ

Solution

Qus : 51 JEE MAIN PYQ

Let $\alpha $ and $\beta $ be the roots of x² - 3x + p=0 and $\gamma $ and $\delta $ be the roots of x² - 6x + q = 0. If $\alpha, \beta, \gamma, \delta $form a geometric progression.Then ratio (2q + p) : (2q - p) is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Morning) PYQ

Solution

Qus : 52 JEE MAIN PYQ

The probability of selecting integers a$\in$[$-$ 5, 30] such that x² + 2(a + 4)x $-$ 5a + 64 > 0, for all x$\in$R, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Morning Shift) PYQ

Solution

Qus : 53 JEE MAIN PYQ

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^{2}+x\sin\theta-2\sin\theta=0,\ \theta\in\left(0,\dfrac{\pi}{2}\right)$, then $\displaystyle \frac{\alpha^{12}+\beta^{12}}{\left(\alpha^{-12}+\beta^{-12}\right)}\cdot(\alpha-\beta)^{24}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Morning Shift) PYQ

Solution

Qus : 54 JEE MAIN PYQ

Let [t] denote the greatest integer $ \le $ t. Then the equation in x, [x]² + 2[x+2] - 7 = 0 has :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Morning) PYQ

Solution

Qus : 55 JEE MAIN PYQ

Let $\alpha$ and $\beta$ be two roots of the equation $x^{2}+2x+2=0$. Then $\alpha^{15}+\beta^{15}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Morning Shift) PYQ

Solution

Qus : 56 JEE MAIN PYQ

The numbers of pairs (a, b) of real numbers, such that whenever $\alpha$ is a root of the equation x² + ax + b = 0, $\alpha$² $-$ 2 is also a root of this equation, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (1 September Evening Shift) PYQ

Solution

Qus : 57 JEE MAIN PYQ

Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}x - 16 = 0$, and $\gamma$ and $\delta$ be the roots of $x^2 + 3x - 1 = 0$. If $P_n = \alpha^n + \beta^n$ and $Q_n = \gamma^n + \delta^n$, then $\dfrac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} + \dfrac{Q_{25} - Q_{23}}{Q_{24}}$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

Qus : 58 JEE MAIN PYQ

Let $\lambda \ne 0$ be in R. If $\alpha $ and $\beta $ are the roots of the equation, x² - x + 2$\lambda $ = 0 and $\alpha $ and $\gamma $ are the roots of the equation, $3{x^2} - 10x + 27\lambda = 0$, then ${{\beta \gamma } \over \lambda }$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Evening) PYQ

Solution

Qus : 59 JEE MAIN PYQ

If $x$ is a solution of the equation $\sqrt{2x+1} - \sqrt{2x-1} = 1,\ (x \ge \tfrac12)$, then $\sqrt{4x^{2}-1}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (10 April Morning Shift) PYQ

Solution

Qus : 60 JEE MAIN PYQ

If $\alpha $ is positive root of the equation, p(x) = x² - x - 2 = 0, then$\mathop {\lim }\limits_{x \to {\alpha ^ + }} {{\sqrt {1 - \cos \left( {p\left( x \right)} \right)} } \over {x + \alpha - 4}}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 5 September 2020 (Morning) PYQ

Solution

Qus : 61 JEE MAIN PYQ

The number of real roots of the equation $\sqrt{x^{2}-4x+3}+\sqrt{x^{2}-9}=\sqrt{4x^{2}-14x+6}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (31 January Morning Shift) PYQ

Solution

Qus : 62 JEE MAIN PYQ

Let \alpha_\theta and \beta_\theta be the distinct roots of $2x^2+(\cos\theta)x-1=0$, $\theta\in(0,2\pi)$. If $m$ and $M$ are the minimum and the maximum values of $\alpha_\theta^{4}+\beta_\theta^{4}$, then $16(M+m)$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (22 January Evening Shift) PYQ

Solution

Qus : 63 JEE MAIN PYQ

If both the roots of the quadratic equation $x^{2}-mx+4=0$ are real and distinct and they lie in the interval $[1,5]$, then $m$ lies in the interval:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Evening Shift) PYQ

Solution

Qus : 64 JEE MAIN PYQ

Let $\alpha$ and $\beta$ be the roots of the equation $p x^{2}+q x-r=0$, where $p\ne 0$. If $p,q,r$ are consecutive terms of a non-constant G.P. and $\dfrac1\alpha+\dfrac1\beta=\dfrac34$, then the value of $(\alpha-\beta)^{2}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Evening Shift) PYQ

Solution

Qus : 65 JEE MAIN PYQ

Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $\left(k,\dfrac{k}{2}\right)$ from the line $3x+4y+5=0$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Evening Shift) PYQ

Solution

Qus : 66 JEE MAIN PYQ

The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation $6x^{2}-11x+\alpha=0$ are rational numbers is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Evening Shift) PYQ

Solution

Qus : 67 JEE MAIN PYQ

The product of the roots of the equation 9x² - 18|x| + 5 = 0 is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 5 September 2020 (Morning) PYQ

Solution

Qus : 68 JEE MAIN PYQ

Consider the equation $x^{2}+4x-n=0$, where $n\in[20,100]$ is a natural number. Then the number of all distinct values of $n$, for which the given equation has integral roots, is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Morning Shift) PYQ

Solution

Qus : 69 JEE MAIN PYQ

Consider the quadratic equation $(c - 5)x^2 - 2cx + (c - 4) = 0,\ c \ne 5.$ Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0, 2)$ and its other root lies in the interval $(2, 3).$ Then the number of elements in $S$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Morning Shift) PYQ

Solution

Qus : 70 JEE MAIN PYQ

If $\alpha $ and $\beta $ are the roots of the equation,7x² – 3x – 2 = 0, then the value of${\alpha \over {1 - {\alpha ^2}}} + {\beta \over {1 - {\beta ^2}}}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 5 September 2020 (Evening) PYQ

Solution

Qus : 71 JEE MAIN PYQ

If $2$ and $6$ are roots of the equation $ax^{2}+bx+1=0$, then the quadratic equation whose roots are $\dfrac{1}{2a+b}$ and $\dfrac{1}{6a+b}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Morning Shift) PYQ

Solution

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