JEE Quadratic Equations PYQs with Solutions PDF, Download

Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^{2}+2ax+\left(3a+10\right)=0$$ such that $$\alpha < 1 < \beta$$. Then the set of all possible values of $$a$$ is :

The positive integer n, for which the solutions of the equation x(x + 2) + (x + 2)(x + 4) + .... + (x + 2n - 2)(x + 2n) = $$\dfrac{8n}{3}$$ are two consecutive even integers, is:

The sum of all the roots of the equation $$(x-1)^2-5\mid x-1\mid+\ 6=0$$ is:

let $$\alpha, \beta$$ be the roots of the quadratic equation $$12x^{2}-20x+3\lambda=0, \lambda\in \mathbb{Z}$$. If $$\frac{1}{2}\leq |\beta-\alpha|\leq\frac{3}{2}$$, then the sum of all possible values of $$\lambda$$ is :

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

If $$\alpha$$ and $$\beta$$ ($$\alpha < \beta$$) are the roots of the equation $$(-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6\sqrt{x})+(9-2\sqrt{3})=0,x\geq0\text{ then }\sqrt{\frac{\beta}{\alpha}}+\sqrt{\alpha\beta}$$ is equal to:

The number of distinct real solutions of the equation $$x\lvert x+4 \rvert + 3\lvert x+2 \rvert + 10 = 0$$ is

The smallest positive integral value of a, for which all the roots of $$x^{4} - ax^{2} + 9 = 0$$ are real and distinct, is equal to

If $$\alpha, \beta$$, where $$\alpha < \beta$$, are the roots of the quadratic equation  $$\lambda x^{2}-(\lambda + 3)x+3=0$$ and  $$\dfrac{1}{\alpha}-\dfrac{1}{\beta}=\dfrac{1}{3}$$, then the sum of all possible values of $$\lambda$$ is

Let $$\alpha_{\theta}$$ anf $$\beta_{\theta}$$ 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 :

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

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 $$x^2 + 3x + 2 = \min\{|x-3|,|x+2|\}\text{ is:}$$

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

The number of solutions of the equation $$\left(\dfrac{9}{x}-\dfrac{9}{\sqrt{x}}+2\right)\left(\dfrac{2}{x}-\dfrac{7}{\sqrt{x}}+3\right)=0$$ is:

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

If the set of all values of a, for which the equation $$5x^{3}-15x-a=0$$ has three distinct real roots, is the interval $$(\alpha, \beta)$$, then $$\beta-2\alpha $$ is equal to ______

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 :

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 $$\alpha$$ 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:

If $$\alpha, \beta$$ are the roots of the equation, $$x^2 - x - 1 = 0$$ and $$S_n = 2023\alpha^n + 2024\beta^n$$, then

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

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 $$\alpha, \beta \in \mathbb{R}$$ be roots of equation $$x^2 - 70x + \lambda = 0$$, where $$\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{Z}$$. 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 $$x(x^2 + 3|x| + 5|x-1| + 6|x-2|) = 0$$ is ______.

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:

For $$0 \lt c \lt b \lt a$$, 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 $$a$$ be the sum of all coefficients in the expansion of $$(1 - 2x + 2x^2)^{2023}(3 - 4x^2 + 2x^3)^{2024}$$ and $$b = \lim_{x \to 0} \frac{\int_0^x \frac{\log(1+t)}{t^{2024}+1}dt}{x^2}$$. 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 $$d : c : e$$ equals

The number of solutions, of the equation $$e^{\sin x} - 2e^{-\sin x} = 2$$ 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 $$\alpha, \beta$$ 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

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 $$(4 + x_1^2)(4 + x_2^2)(4 + x_3^2)(4 + x_4^2) = \frac{125}{16}m$$. Then the value of $$m$$ is

If $$z_1, z_2$$ are two distinct complex numbers such that $$\left|\frac{z_1 - 2z_2}{\frac{1}{2} - z_1\bar{z}_2}\right| = 2$$, then

Let $$\alpha, \beta$$ be roots of $$x^2 + \sqrt{2}x - 8 = 0$$. If $$U_n = \alpha^n + \beta^n$$, then $$\frac{U_{10} + \sqrt{2}U_9}{2U_8}$$ is equal to ___________

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

If the function $$f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$$, $$a > 0$$ has a local maximum at $$x = \alpha$$ and a local minimum at $$x = \alpha^2$$, then $$\alpha$$ and $$\alpha^2$$ are the roots of the equation :

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 $$\frac{1}{10}(\alpha^6 + \beta^6)$$, 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 $$(11\sqrt{3} - 10\sqrt{2})P_{10} + (11\sqrt{2} + 10)P_{11} - 11P_{12}$$ is equal to