Quadratic Equations Questions - JEE

If no value of $\theta$

Find the number of positive integers satisfying the inequality \(x^2 - 10x + 16

If the equation $(1+m)x^2 - 2(1+3m)x + (1+8m) = 0$, where $m \in \mathbb{R} \setminus \{-1\}$, has at least one root is negative, then

23. Suppose that $x_1$ and $x_2$ are the positive real solution of $x^2 - bx + c = 0$ provided that $x_1^2 + \sqrt{x_2^2 - 2x_2} = 2x_1 - 1$. The minimum value of $(b + c)$, 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 $\dfrac{1}{10}(\alpha^6+\beta^6)$, is:

Given that the quadratic equation $x^2 - 64x + 256 = 0$ has roots $a$ and $b$, find the value of $\left(\frac{a^3}{b^5}\right)^{1/8} + \left(\frac{b^3}{a^5}\right)^{1/8}$.

If $\sin\theta = \frac{7\sin\theta}{11\cos\theta}$

Let $\sum_{r=1}^{10}(r+r \times \binom{10}{r}) = 2^{10}(a \times 4^5 + b)$ where $a, b \in \mathbb{N}$ and $f(x) = x^2 - 2x - k^2 + 1$.If a and b lie between the roots of $f(x) = 0$, then find the smallest positive integral value of k.

For $x\in\mathbb{R}$, the expression $\dfrac{x^2+2x+c}{x^2+4x+3c}$ can take all real values if $c\in$

The sum of the squares of the roots of $|x + 2| + |x - 2| - 2 = 0$ and the squares of the roots of $2x^2 - 2|x - 3| - 5 = 0$ is

77. All the values of $m$ for which both the roots of the equation $x^2 - 2mx + m^2 - 1 = 0$ are greater than $-2$ but less than 4 lie in the interval

76. If roots of $x^2 - (a-3)x + a = 0$ are such that at least one of them is greater than 2, then

The number of real solutions of $|x - 2\sqrt{5 - 4x - x^2}| = 16$ is/are

The sum of values of x satisfying the equation $(31 + 8\sqrt{15})^{x^2 - 3} + 1 = (32 + 8\sqrt{15})^{x^2 - 3}$ is

tan α and tan β are the roots of the equation $x^2 + ax + b = 0$, then the value of $\sin^2(α + β) + a\sin(α + β)\cos(α + β) + b\cos^2(α + β)$ is equal to

Two students while solving a quadratic equation in x, one copied the constant term incorrectly and got the roots 3 and 2. The other copied the coefficient of x² correctly as -6 and 1 respectively the correct roots are

Solve $\sqrt{x-2}\,(x^2-4x-5)=0$.

Ex. 62: If the sum of the base 2 logarithms of the roots of the cubic $f(x) = 0$ is 5, then the value of $a$ is

If $x^4 + 3x^3 + 2(1-a)x^2 - 3ax + a^2 = 0$ has only real roots then which of the following may be the value of $a$?

The given equation is $-3(x - [x])^2 + 2(x - [x]) + a^2 = 0$. The values of $a$ for which the equation has non-integral solutions satisfy:

If roots of the equation $\dfrac{1}{x-a} + \dfrac{1}{x-b} + \dfrac{1}{x-c} + \dfrac{1}{x-d} + \dfrac{(x-2)(x^2+2x+4)}{(x-a)(x-b)(x-c)(x-d)} = 0$ are $\alpha$, $\beta$ and $\gamma$, then sum of the roots of the equation $5(x-\alpha)(x-\beta)(x-\gamma) + 8 - x^3 = 0$ is:

Given equation is $bx^3 + cx + a = 0$. The roots are imaginary if \(c^2 - 4ab

If $x^4 + 3x^3 + 2(1-a)x^2 - 3ax + a^2 = 0$ has only real roots then which of the following may be the value of $a$?

The range of value of $\lambda$ for which the expression $\dfrac{2x^2 - 5x + 3}{4x - \lambda}$ can take all real values for $x \in R - \left\{\dfrac{\lambda}{4}\right\}$, is:

If the expression $[mx - 1 + (1/x)]$ is non-negative for all positive real $x$, then the minimum value of $m$ must be

Let a > 2 be a constant. If there are just 18 positive integers satisfying the inequality $(x - a)(x - 2a)(x - a^2) < 0$, then find the value of a.

If the equation $ax^2 + bx + c = 0$, where $a, b, c \in \mathbb{R}$ and $a > 0$, has two real roots $\alpha$ and $\beta$ such that $\alpha and \(\beta > 2$, then

If $a \in (-1, 1)$, then roots of the quadratic equation $(a-1)x^2 + ax + \sqrt{1 - a^2} = 0$ are

69. If the equation $|x^2 + bx + c| = k$ has four real roots, then

Find sum of all the possible values of m for which the equation $16x^4 - mx^3 + (2m+17)x^2 - mx + 16 = 0$ has four distinct roots forming a geometric progression.

163. If $a, b, c \in \mathbb{R}$ and $a^2 + b^2 + c^2 + 4 = ab + bc + 2c + 2a$, then roots of $ax^2 + bx + c = 0$ are:

For the equation $x^2 - 4\sqrt{2}kx + 2k^4 - 1 = 0$, if $\alpha + \beta = 4\sqrt{2}k$ and $\alpha^3 + \beta^3 = 280\sqrt{2}$, then the value of $k$ is:

Let $f(x) = (k-3)x^2 - 2kx + 3k - 6$ where $x \in R$. If the range of $f(x)$ is $[0, \infty)$, then the value of $k$ can be:

67. If both roots of the equation $ax^2 + x + c - a = 0$ are imaginary and $c > -1$, then

The number of integers n such that the equation $nx^2 + (n+1)x + (n+1) = 0$ has only rational roots, is equal to:

Let f(x) be a polynomial of degree 8 such that $f(r) = \frac{1}{r}$, r = 1, 2, 3, ..., 8, 9, then find $\frac{1}{f(10)}$.

If $ x+1 $ is a factor of $ f(x) = x^3 + kx^2 - 3x + k + 2 $, then the value of $ k $ is:

Let $\alpha$ be the only real root of $x^5 - x^3 + x - 2 = 0$. Then the value of $(\alpha^2 + 1)(\alpha^4 - \alpha^2 + 1)$ is:

If the roots of the equation $ax^2 - bx + c = 0$ are α, β, then the roots of the equation $b^2cx^2 - ab^2x + a^3 = 0$ are

Let α and β be the roots of $x^2 - 6x - 2 = 0$, with α > β. If $a_n = \alpha^n - \beta^n$ for $n \geq 1$, then the value of $\dfrac{a_{10} - 2a_8}{2a_9}$ is

If the equations $ax^2 + bx + c = 0$ and $x^3 + 3x^2 + 3x + 2 = 0$ have two common roots, then

Solve $\sqrt{x^2+4x-21}+\sqrt{x^2-x-6}=\sqrt{6x^2-5x-39}$.

Question nos. 690 to 692Column-1 represents a quadratic equation with some given conditions. Column-2 represents number of non-positive integral values of 'k' and column-3 represents number of prime values of 'k'. Then match the following.Column-1Column-2Column-3(I) Let α and β are real roots of $x^2 - 8x + k^2 - 6k = 0$ such that $\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha} = 2$.(i) 0(P) 0(II) If one root of the equation $(k-2)x^2 - (8-2k)x + (3k+8) = 0$ is negative and other is positive.(ii) 1(Q) 1(III) If difference between the real roots of equation $4x^2 - 2kx + 1 = 0$ is less than $\sqrt{3}$.(iii) 2(R) 2(IV) If quadratic expression $2kx^2 - (4k-5)x - 10$ is negative for exactly three distinct integral values of $x$.(iv) 3(S) 3Which of the following options is the only correct combination?

If $0 < n < 1$

Let $f(x) = (x-1)(x-2)(x-3)(c-x) + x^4 - x$. If the coefficient of $x^3$ in $f(x)$ is 1, find $c$.

708. Let $ f(x) = x^2 + \alpha x + \beta $ where $ \alpha, \beta \in R $ and $ f(f(x)) = 0 $ has 2 roots 1 and 2. Find the value of $ 2|f(0)| $.

72. If $ax^2 + bx + c = 0$ has imaginary roots and $a - b + c > 0$, then the set of points $(x, y)$ satisfying the equation $\left|a\left(x^2 + \dfrac{y}{a}\right) + (b+1)x + c\right| = |ax^2 + bx + c| + |x + y|$ consists of the region in the $xy$-plane which is

Ex. 78: Given $f(x) = (x + 2a)(x + a - 4)$ where $a \in \mathbb{R}$. If \(f(x)

Two equations have irrational roots occurring in pairs such that $\dfrac{51}{3} = 17 = \dfrac{m}{b} = \dfrac{c}{a}$. Find $\dfrac{c}{a}$.

Given one root of $f(x)$ is $-1$. Then $f(x) = a(x+1)(x-\alpha)$, where $\alpha$ is the other root of the quadratic equation. If $f(1) + f(2) = 0$, find $\alpha$.

Quadratic Equations Questions (527)