Quadratic Equations — JEE Main PYQs

Let $\alpha, \beta$ be the roots of the equation $x^2 - 3x + r = 0$, and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 + 3x + r = 0$. If the roots of the equation $x^2 + 6x = m$ are $2\alpha + \beta + 2r$ and $\alpha - 2\beta - \frac{r}{2}$, then $m$ is equal to :

If the set of all solutions of $|x^2+x-9| = |x| + |x^2-9|$ is $[\alpha, \beta] \cup [\gamma, \infty)$, then $(\alpha^2+\beta^2+\gamma^2)$ is equal to:

If $\alpha = 1$ and $\beta = 1 + i\sqrt{2}$, where $i = \sqrt{-1}$, are two roots of the equation $x^3 + ax^2 + bx + c = 0$, $a, b, c \in \mathbb{R}$, then $\displaystyle\int_{-1}^{1}\left(x^3 + ax^2 + bx + c\right)dx$ is equal to:

If the quadratic equation $(\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0$, $\lambda \neq -2$, has two positive roots, then the number of possible integral values of $\lambda$ is:

Let $\alpha, \beta$ be the roots of the equation $x^2 - x + p = 0$ and $\gamma, \delta$ be the roots of the equation $x^2 - 4x + q = 0$; $p, q \in \mathbf{Z}$. If $\alpha, \beta, \gamma, \delta$ are in G.P., then $|p + q|$ equals :

Let one root of the quadratic equation in $x$:

$$(k^2-15k+27)x^2+9(k-1)x+18=0$$

be twice the other. Then the length of the latus rectum of the parabola $y^2=6kx$ is equal to:

Consider the quadratic equation $(n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0,\ n \in \mathbb{R}$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $\alpha$ and the common ratio is $\frac{\alpha}{\beta}$, is :

The sum of squares of all the real solutions of the equation $\log_{(x+1)}\left(2x^2 + 5x + 3\right) = 4 - \log_{(2x+3)}\left(x^2 + 2x + 1\right)$ is equal to __________.

Let $\mathrm{P_n} = \alpha^\mathrm{n} + \beta^\mathrm{n}$, $\mathrm{n} \in \mathbf{N}$. If $\mathrm{P_{10}} = 123$, $\mathrm{P_9} = 76$, $\mathrm{P_8} = 47$ and $\mathrm{P_1} = 1$, then the quadratic equation having roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is :

If the set of all $a \in \mathbf{R}-\{1\}$, for which the roots of the equation $(1-a)x^2+2(a-3)x+9=0$ are positive is $(-\infty, -\alpha] \cup [\beta, \gamma)$, then $2\alpha+\beta+\gamma$ is equal to _________.

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 $\frac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} + \frac{Q_{25} - Q_{23}}{Q_{24}}$ is equal to

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

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

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 equal to

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

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

The product of all solutions of the equation $e^{5(\log_e x)^2+3}=x^8, x>0$, is :

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 :

If the equation $a(b - c)x^2 + b(c - a)x + c(a - b) = 0$ has equal roots, where $a + c = 15$ and $b = \frac{36}{5}$, then $a^2 + c^2$ is equal to____

Let $\alpha, \beta$ be the roots of the equation $x^2 - \mathrm{a}x - \mathrm{b} = 0$ with $\text{Im}(\alpha) < \text{Im}(\beta)$. Let $\mathrm{P_n} = \alpha^{\mathrm{n}} - \beta^{\mathrm{n}}$. If $\mathrm{P_3} = -5\sqrt{7}\, i$, $\mathrm{P_4} = -3\sqrt{7}\, i$, $\mathrm{P_5} = 11\sqrt{7}\, i$ and $\mathrm{P_6} = 45\sqrt{7}\, i$, then $|\alpha^4 + \beta^4|$ is equal to _________.

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

The number of real solution(s) of the equation $x^2 + 3x + 2 = \min\{|x - 3|, |x + 2|\}$ is :

The sum of the squares of all the roots of the equation $x^2 + |2x - 3| - 4 = 0$ is

Let $f : \mathbf{R} - \{0\} \to (-\infty,\ 1)$ 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 sum of squares of all possible values of $\mathrm{K}$ is :

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

Let $S = \{\sin^2 2\theta : (\sin^4\theta + \cos^4\theta)x^2 + (\sin 2\theta)x + (\sin^6\theta + \cos^6\theta) = 0$ has real roots$\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set S, respectively, then $3((\alpha - 2)^2 + (\beta - 1)^2)$ equals _________

The number of distinct real roots of the equation $|x|\,|x+2| - 5|x+1| - 1 = 0$ is ________.

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

Let $\alpha$, $\beta$ be the distinct roots of the equation $x^2 - (t^2 - 5t + 6)x + 1 = 0$, $t \in \mathbb{R}$ and $a_n = \alpha^n + \beta^n$. Then the minimum value of $\dfrac{a_{2023} + a_{2025}}{a_{2024}}$ is

Let $x_1, x_2, x_3, x_4$ be the solution of the equation $4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0$ and $\left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \dfrac{125}{16}m$. Then the value of $m$ is ________

Let $\alpha, \beta$ be roots of $x^2+\sqrt{2}x-8=0$. If $\mathrm{U_n}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}$, then $\dfrac{\mathrm{U}_{10}+\sqrt{2}\,\mathrm{U}_9}{2\mathrm{U}_8}$ is equal to ________.

The sum of all the solutions of the equation $(8)^{2x}-16\cdot(8)^{x}+48=0$ is :

The number of distinct real roots of the equation $|x+1|\,|x+3| - 4|x+2| + 5 = 0$, is ________

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 $\frac{1}{10}\left(\alpha^6+\beta^6\right)$, is :

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

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

Let $\alpha$ and $\beta$ be the roots of the equation $px^2+qx-r=0$, where $p \neq 0$. If $p$, $q$ and $r$ be the consecutive terms of a non constant G.P. and $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}$, then the value of $(\alpha-\beta)^2$ is :

Let $\alpha, \beta \in\mathbf{N}$ be roots of the equation $x^2-70x+\lambda=0$, where $\frac{\lambda}{2}, \frac{\lambda}{3}\notin\mathbf{N}$. If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$ is equal to :

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

Let S be the set of postive integral values of $a$ for which $\frac{ax^2+2(a+1)x+9a+4}{x^2-8x+32}<0, \forall x \in \mathbb{R}$. Then, the number of elements in S is :

Let $a$ be the sum of all coefficients in the expansion of $(1-2x+2x^2)^{2023}(3-4x^2+2x^3)^{2024}$ and $b=\lim_{x \to 0}\left(\frac{\int_0^x \frac{\log(1+t)}{t^{2024}+1}dt}{x^2}\right)$. If the equations $cx^2+dx+e=0$ and $2bx^2+ax+4=0$ have a common root, where $c, d, e \in \mathbb{R}$, then $\mathrm{d}:\mathrm{c}:\mathrm{e}$ equals

For $0<c<b<a$, let $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ and $\alpha \neq 1$ be one of its root. 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 $c$

Let $a, b, c$ be the lengths of three sides of a triangle satisfying the condition $\left(a^{2}+b^{2}\right) x^{2}-2 b(a+c) x+\left(b^{2}+c^{2}\right)=0$. If the set of all possible values of $x$ is the interval $(\alpha, \beta)$, then $12\left(\alpha^{2}+\beta^{2}\right)$ is equal to ______.