Quadratic Equations Questions - JEE

Given equation is $[1 - a^2]x^2 + 2ax - 1 = 0$. Its discriminant $D = 4$ and roots are $-\frac{1}{a+1}, -\frac{1}{a-1}$. Find the range of $a$ for which this holds.

Solve the equation $(1 + i)x^2 + (1 - i)x - 2i = 0$ and find $|\alpha - \beta|^2$ where $\alpha, \beta$ are the roots.

If

Find the value of m for which the expression 12x² − 10xy + 2y² − 11x − 5y + m can be resolved into two rational linear factors.

If the roots of the equation \frac{1}{x+p} + \frac{1}{x+q} = \frac{1}{r}\ are equal in magnitude but opposite in sign, then the product of the roots is:

When $x > \frac{3}{2}$, the equation $x^2 + 2x - 7 = 0$ gives roots $x = -1 \pm 2\sqrt{2}$. For \(x

68. If \((b^2 - 4ac)^2(1 + 4a^2)

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

Let $f(x) = x^2 + ax + b; a, b \in \mathbb{R}$. If $f(1) + f(2) + f(3) = 0$, then the roots of the equation $f(x) = 0$

If $\alpha$ and $\beta$ are the roots of equation $2x^2 - 5x + 7 = 0$, then the equation whose roots are $2\alpha + 3\beta$ and $3\alpha + 2\beta$ is

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

Find the number of integral values of $\lambda$ such that \((\lambda^2 + \lambda - 2)x^2 + (\lambda + 2)x

Find the number of positive integral values of k for which \(kx^2 + (k - 3)x + 1 x.

If f(x) = ax^2 + bx + c, where a, b, c \in \mathbb{R} and the equation f(f(x)) - x = 0 has imaginary roots \alpha, \beta, and \gamma and \delta be the roots of f(f(x)) - x = 0, then \frac{\beta^2\alpha + \delta}{\gamma\beta + 1} is

The number of integral values of $a$ for which the quadratic equation $(x - a)(x - 1991) + 1 = 0$ has integral roots are

Suppose that $f(x)$ is a quadratic expression positive for all real $x$. If $g(x) = f(x) + f'(x) + f''(x)$, then for any real $x$ (where $f'(x)$ and $f''(x)$ represent 1st and 2nd derivative, respectively)

The inequality \frac{x + 3}{x^2 - x - 2} \geq \frac{1}{x - 4}\ holds for all x satisfying:

81. The set of all possible real values of $a$ such that the inequality \((x - (a-1))(x - (a^2 + 2))

Solution set of the equation 3^{2x^2} - 3^{x^2 + x + 6} + 3^{2(x + 6)} = 0 is

74. $x_1$ and $x_2$ are the roots of $ax^2 + bx + c = 0$ and \(x_1 x_2

Given that α, β, γ are all real roots of the equation x3 − 2007x + 2002 = 0, then the value of $\frac{α − 1}{α + 1} + \frac{β − 1}{β + 1} + \frac{γ − 1}{γ + 1}$ is equal to

The solution set of the equation $(\cos p - 1)x^2 + (\cos p)x + \sin p = 0$ are real, then

If $x, y$ and $z$ are real such that $x + y + z = 4$, $x^2 + y^2 + z^2 = 6$, find the range of $x$.

Consider two quadratic expressions f(x) = ax^2 + bx + c and g(x) = ax^2 + px + q (a, b, c, p, q \in \mathbb{R}, b \neq p) such that their discriminants are equal. If f(x) = g(x) has a root x = \alpha, then

Let $a, b, c \in \mathbb{R}$ with $a > 0$ such that the equation $ax^2 + bcx + b^3 + c^3 - 4abc = 0$ has non-real roots. If $P(x) = ax^2 + bx + c$ and $Q(x) = ax^2 + cx + b$, then

If $\cos A$, $\cos B$ and $\cos C$ are the roots of cubic $x^3 + ax^2 + bx + c = 0$, where A, B, C are the angles of a triangle then find the value of $a^2 - 2b - 2c$.

Let $f(x) = ax^2 - bx + c^2$, $b \neq 0$ and $f(x) \neq 0$ for all $x \in \mathbb{R}$. Then

The graph of a quadratic polynomial y = ax^2 + bx + c; a, b, c \in \mathbb{R} is as shown. Which one of the following is not correct?

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 the set $C=\{(x,y)\mid x^2-2^y=2023,\,x,y\in\mathbb{N}\}$. Then $\displaystyle\sum_{(x,y)\in C}(x+y)$ is equal to

The number of real roots of the equation $5 + |2^x - 1| = 2^x(2^x - 2)$ is __________.

If x_1 and x_2 are the arithmetic and harmonic means of the roots of the equation ax^2 + bx + c = 0, the quadratic equation whose roots are x_1 and x_2, is

If one root of equation $x^2 + ax + 12 = 0$ is 4 while the equation $x^2 + ax + b = 0$ has equal roots, then the value of b is

The number of ordered pairs (a, b), where a, b are integers satisfying the inequality $\min\left(x^2 + (a-b)x + (1-a-b)\right) > \max\left(-x^2 + (a+b)x - (1+a+b)\right)$ for all $x \in \mathbb{R}$, is:

If a, b, c are real and a \neq b, the roots of the equation 2(a - b)x^2 - 11(a + b + c)x - 3(a - b) = 0 are

If the equations $ax^2 + bx + c = 0$ and $cx^2 + bx + a = 0$, where $a, b, c \in \mathbb{R}$ and $ac \neq 0$, have a common non-real root, then

If the roots of the equation $x^2 - 8x + a^2 - 6a = 0$ are real and distinct, then find all possible values of $a$.

If $k \in ?$

Let f(x) = 0 be an equation of degree six, having integer coefficients and whose one root is $2\cos\dfrac{\pi}{18}$. Then, the sum of all the roots of f'(x) = 0, is

167. The number of values of $k$ for which the equation $(x^2 + (2k-6)x + 7 - 3k)(x^2 + (2k-2)x + 3k - 5) = 0$ has two different pairs of equal roots, is equal to:

If $-3 \leq \frac{x^2 + bx + 1}{x^2 - bx + 1} \leq 2$ for all $x \in \mathbb{R}$, then the value of $b$ belongs to

For the equation $|x^2 - 2x - 3| = b$, which of the following statements is true?

$f(t) = 4t^2 - 16t + t$. Opening upward parabola. Find the number of integer values of $t$ for which $f(1) > 0$, $f(2) < 0$, $f(3) > 0$.

If $(x+1)$ is a factor of $x^3 + kx^2 - 3x + k + 2$, then $k$ is equal to:

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

If the absolute value of the expression \[\frac{\alpha - 1}{\alpha + 2} + \frac{\beta - 1}{\beta + 2} + \frac{\gamma - 1}{\gamma + 2}\] can be expressed as m/n, where m and n are co-prime, the value of m2/[(m − n)(m + n)] is

$7z^2 + 1 = 0$

If $1, \alpha + \beta, \alpha\beta$ are in A.P. and $1, \frac{1}{\alpha}, \frac{1}{\beta}$ are in A.P., find the value.

Find the number of integral values of parameter 'a' so that the inequality $(2a - a^2) \leq x^2 - 3x + 2 \leq 3 - a^2$ holds for any real x in the interval $[0, 2]$.

If $\alpha_1, \alpha_2, \alpha_3$ and $\alpha_4$ are the roots of the equation $x^4 + (2 - \sqrt{3})x^2 + (2 + \sqrt{3}) = 0$, then the value of $(1 - \alpha_1)(1 - \alpha_2)(1 - \alpha_3)(1 - \alpha_4)$ is equal to:

Quadratic Equations Questions (527)