Quadratic Equations(PYQ’s)

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
(A) 24
(B) 26
(C) 36
(D) 30
The number of real roots of the equation $x|x-2| + 3|x-3| + 1 = 0$ is:
(A) 4
(B) 3
(C) 2
(D) 1
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
(A) 5
(B) 6
(C) 0
(D) 9
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
(A) 6
(B) 5
(C) 8
(D) 7
Let the equation $x(x+2)(12-k) = 2$ have equal roots. Then the distance of the point $(k, \frac{k}{2})$ from the line $3x + 4y + 5 = 0$ is
(A) 15
(B) 12
(C) $5\sqrt{3}$
(D) $15\sqrt{5}$
Let $\alpha_\theta$ and $\beta_\theta$ be the roots of $2x^2 + (\cos\theta)x - 1 = 0$ , $\theta \in (0, 2\pi)$ , 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 $\frac{P_n + \sqrt{3} P_n}{2 P_n} + \frac{Q_n - Q_n}{4 P_n}$ is equal to
(A) 4
(B) 3
(C) 5
(D) 7
Let $P_n = \alpha^n + \beta^n$ , $n \in \mathbb{N}$ . If $P_{10} = 123$ , $P_9 = 76$ , $P_8 = 47$ and $P_1 = 1$ , then the quadratic equation having roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is:
(A) $x^2 + x - 1 = 0$
(B) $x^2 - x + 1 = 0$
(C) $x^2 + x + 1 = 0$
(D) $x^2 - x - 1 = 0$
If the set of all $a \in \mathbb{R}$ , for which the equation $2x^2 + (a-5)x + 15 = 3a$ has no real root, is the interval $(\alpha, \beta)$ , and $X = \{x \in \mathbb{Z} : \alpha < x < \beta\}$ , then $\sum_{x \in X} x^2$ is equal to:
(A) 2139
(B) 2119
(C) 2109
(D) 2129
The number of solutions of the equation
$\left( \frac{9}{x} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{x} - \frac{7}{\sqrt{x}} + 3 \right) = 0$
is:

(A) 3
(B) 2
(C) 1
(D) 4
Let $f: \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right)$ . If $f(K) = -2K$ , then the sum of squares of all possible values of $K$ is:
(A) 9
(B) 1
(C) 6
(D) 7
The sum of the squares of all the roots of the equation $x^2 + |2x - 3| - 4 = 0$ , is
(A) $6(2 - \sqrt{2})$
(B) $3(3 - \sqrt{2})$
(C) $3(2 - \sqrt{2})$
(D) $6(3 - \sqrt{2})$
The number of real solution(s) of the equation $x^2 + 3x + 2 = \min \{|x-3|, |x+2|\}$ is:
(A) 2
(B) 3
(C) 1
(D) 0
The product of all the rational roots of the equation $(x^2 - 9x + 11)^2 - (x-4)(x-5) = 3$ , is equal to
(A) 7
(B) 21
(C) 28
(D) 14
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:
(A) 27
(B) 17
(C) 25
(D) 24
If the set of all $a \in \mathbb{R} \setminus \{1\}$ , 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
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
The number of distinct real roots of the equation $|x+1||x+3| - 4|x+2| + 5 = 0$ , is
Let $x_1, x_2, x_3, x_4$ be the solutions of the equation $4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0$ and $(4 + x_1^2)(4 + x_2^2)(4 + x_3^2)(4 + x_4^2) = \frac{125}{16} m$ . Then the value of $m$ is
The number of real solutions of the equation $x|x+5| + 2|x+7| - 2 = 0$ is
The number of distinct real roots of the equation $|x||x+2| - 5|x+1| - 1 = 0$ is
Let $a, b, c$ be the lengths of three sides of a triangle satisfying the condition $(a^2 + b^2)x^2 - 2b(a+c)x + (b^2 + c^2) = 0$ . If the set of all possible values of $x$ is the interval $(\alpha, \beta)$ , then $12(\alpha^2 + \beta^2)$ is equal to
The number of real solutions of the equation $x(x^2 + 3|x| + 5|x-1| + 6|x-2|) = 0$ is
Let $\alpha, \beta \in \mathbb{N}$ be roots of the equation $x^2 - 70x + \lambda = 0$ , where $\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{N}$ . If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha-1} + \sqrt{\beta-1})(\lambda + 35)}{|\alpha - \beta|}$ is equal to
Let the set $C = \{(x, y) \mid x^2 - 2^y = 2023, x, y \in \mathbb{N}\}$ . Then $\sum_{(x, y) \in C}(x + y)$ is equal to
Let $[\alpha]$ denote the greatest integer $\leq \alpha$ . Then $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \ldots + [\sqrt{120}]$ is equal to
The number of points where the curve $f(x) = \mathrm{e}^{8x} - \mathrm{e}^{6x} - 3\mathrm{e}^{4x} - \mathrm{e}^{2x} + 1, x \in \mathbb{R}$ cuts the $x$ -axis, is equal to
If $a$ and $b$ are the roots of the equation $x^2 - 7x - 1 = 0$ , then the value of $\frac{a^{21} + b^{21} + a^{17} + b^{17}}{a^{19} + b^{19}}$ is equal to
Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2 - 12x + [x] + 31 = 0$ and $x^2 - 5|x+2| - 4 = 0$ respectively, where $[x]$ denotes the greatest integer $\leq x$ . Then $m^2 + mn + n^2$ is equal to
If the value of real number $a > 0$ for which $x^2 - 5ax + 1 = 0$ and $x^2 - ax - 5 = 0$ have a common real root is $\frac{3}{\sqrt{2\beta}}$ , then $\beta$ is equal to
Let $\alpha_1, \alpha_2, \ldots, \alpha_7$ be the roots of the equation $x^7 + 3x^5 - 13x^3 - 15x = 0$ and $|\alpha_1| \geq |\alpha_2| \geq \ldots \geq |\alpha_7|$ . Then $\alpha_1 \alpha_2 - \alpha_3 \alpha_4 + \alpha_5 \alpha_6$ is equal to
Let $\alpha \in \mathbb{R}$ and let $\alpha, \beta$ be the roots of the equation $x^2 + 60^{1/4}x + a = 0$ . If $\alpha^4 + \beta^4 = -30$ , then the product of all possible values of $a$ is
Let $\lambda \in \mathbb{R}$ and let the equation $E$ be $|x|^2 - 2|x| + |\lambda - 3| = 0$ . Then the largest element in the set $S = \{x + \lambda : x \text{ is an integer solution of } E\}$ is
Let $\alpha, \beta (\alpha > \beta)$ be the roots of the quadratic equation $x^2 - x - 4 = 0$ . If $P_n = \alpha^n - \beta^n, n \in \mathbb{N}$ , then $\frac{P_{15} P_{16} - P_{14} P_{16} - P_{15}^2 + P_{14} P_{15}}{P_{13} P_{14}}$ is equal to
The sum of all real values of $x$ for which $\frac{3x^2 - 9x + 17}{x^2 + 3x + 10} = \frac{5x^2 - 7x + 19}{3x^2 + 5x + 12}$ is equal to
If for some $p, q, r \in \mathbb{R}$ , not all have the same sign, one of the roots of the equation $(p^2 + q^2)x^2 - 2q(p + r)x + q^2 + r^2 = 0$ is also a root of the equation $x^2 + 2x - 8 = 0$ , then $\frac{q^2 + r^2}{p^2}$ is equal to
The number of distinct real roots of the equation $x^5(x^3 - x^2 - x + 1) + x(3x^3 - 4x^2 - 2x + 4) - 1 = 0$ is
The number of real solutions of the equation $e^{4x} + 4e^{3x} - 58e^{2x} + 4e^x + 1 = 0$ is
Let $\alpha, \beta$ be the roots of the equation $x^2 - 4\lambda x + 5 = 0$ and $\alpha, \gamma$ be the roots of the equation $x^2 - (3\sqrt{2} + 2\sqrt{3})x + 7 + 3\lambda\sqrt{3} = 0, \lambda > 0$ . If $\beta + \gamma = 3\sqrt{2}$ , then $(\alpha + 2\beta + \gamma)^2$ is equal to
If the sum of all the roots of the equation $e^{2x} - 11e^x - 45e^{-x} + \frac{81}{2} = 0$ is $\log_e p$ , then $p$ is equal to
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
The sum of the cubes of all the roots of the equation $x^4 - 3x^3 - 2x^2 + 3x + 1 = 0$ is
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 $52 - 10f(10)$ is equal to
Let $\lambda \neq 0$ be in $\mathbb{R}$ . If $\alpha$ and $\beta$ are the roots of the equation $x^2 - x + 2\lambda = 0$ , and $\alpha$ and $\gamma$ are the roots of equation $3x^2 - 10x + 27\lambda = 0$ , then $\frac{\beta \gamma}{\lambda}$ is equal to:
(A) 36
(B) 9
(C) 27
(D) 18
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 roots, is
The number of real roots of the equation $e^{4x} - e^{3x} - 4e^{2x} - e^x + 1 = 0$ is equal to
If $a + b + c = 1$ , $ab + bc + ca = 2$ and $abc = 3$ , then the value of $a^4 + b^4 + c^4$ is equal to
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 value of $\left(\frac{P_{17} P_{20} + 5\sqrt{2} P_{17} P_{19}}{P_{18} P_{19} + 5\sqrt{2} P_{18}^2}\right)$ is equal to
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 some integer $n \geq 1$ . Then, the value of $p_n^2$ is
The sum of the $162^{\text{th}}$ power of the roots of the equation $x^3 - 2x^2 + 2x - 1 = 0$ is
The number of real roots of the equation $(x+1)^2 + |x-5| = \frac{27}{4}$ is
The least positive value of $a$ for which the equation $2x^2 + (a-10)x + \frac{33}{2} = 2a$ has real roots is
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
(A) $10\sqrt{3} P_9$
(B) $11\sqrt{3} P_9$
(C) $11\sqrt{2} P_9$
(D) $10\sqrt{2} P_9$
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 $\frac{1}{10}(\alpha^6 + \beta^6)$ , is:
(A) $x^2 - 180x + 9506 = 0$
(B) $x^2 - 195x + 9506 = 0$
(C) $x^2 - 190x + 9466 = 0$
(D) $x^2 - 195x + 9466 = 0$
The sum of all the solutions of the equation $(8)^{2x} - 16 \cdot (8)^x + 48 = 0$ is:
(A) $1 + \log_8(6)$
(B) $1 + \log_6(8)$
(C) $\log_8(6)$
(D) $\log_8(4)$
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 $\frac{a_{2023} + a_{2025}}{a_{2024}}$ is
(A) $-\frac{1}{2}$
(B) $-\frac{1}{4}$
(C) $\frac{1}{4}$
(D) $\frac{1}{2}$
The coefficients $a, b, c$ in the quadratic equation $ax^2 + bx + c = 0$ are from the set $\{1, 2, 3, 4, 5, 6\}$ . If the probability of this equation having one real root bigger than the other is $p$ , then $216p$ equals:
(A) 38
(B) 7
(C) 57
(D) 19
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:
(A) $x^2 + 8x + 12 = 0$
(B) $2x^2 + 11x + 12 = 0$
(C) $4x^2 + 14x + 12 = 0$
(D) $x^2 + 10x + 16 = 0$
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:
(A) 8
(B) 9
(C) $\frac{20}{3}$
(D) $\frac{80}{9}$
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:
(A) 4
(B) 0
(C) 2
(D) 1
Let $S$ be the set of positive 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:
(A) 0
(B) $\infty$
(C) 3
(D) 1
If $\alpha, \beta$ are the roots of the equation $x^2 - x - 1 = 0$ and $S_n = 2023\alpha^n + 2024\beta^n$ , then:
(A) $2S_{12} = S_{11} + S_{10}$
(B) $S_{12} = S_{11} + S_{10}$
(C) $S_{11} = S_{10} + S_{12}$
(D) $2S_{11} = S_{12} + S_{10}$
The number of real roots of the equation $x|x| - 5|x+2| + 6 = 0$ , is:
(A) 4
(B) 3
(C) 5
(D) 6
Let $\alpha, \beta$ be the roots of the equation $x^2 - \sqrt{2}x + 2 = 0$ . Then $\alpha^{14} + \beta^{14}$ is equal to
(A) -64
(B) $-64\sqrt{2}$
(C) $-128\sqrt{2}$
(D) -128
The set of all for which the equation has exactly one real root, is:
(A) $(-\infty, \infty)$
(B) $(-6, \infty)$
(C) $(-\infty, -3)$
(D) $(-6, -3)$