Quadratic Equations Questions - JEE

If positive numbers $a, b, c$ are in H.P., then equation $x^2 - kx + 2b^{101} - a^{101} - c^{101} = 0$ ($k \in R$) has

For the quadratic equation 6x2 − 11x + a = 0, find the number of integer values of a for which the roots are rational.

The minimum possible value of a is

Let a, b, c ∈ ℝ such that two of them are equal and satisfy 2abcbc2ac2ab = 0, then equation 24ax2 + 4bx + c = 0 has

For Problems 33 and 34The real numbers $x_1, x_2, x_3$ satisfying the equation $x^3 - x^2 + \beta x + \gamma = 0$ are in A.P.All possible values of $\beta$ are

The quadratic equation $x(x+1) + (x+1)(x+2) + \cdots + [x+(n-1)](x+n) = 10n$ has two consecutive integral solutions. Find the value of $n$.

If the roots of the equation $x^2 + 2ax + b = 0$ are real and distinct and they differ by at most $2m$, then $b$ lies in the interval

The total number of integral values of $a$ so that $x^2 - (a+1)x + a - 1 = 0$ has integral roots is equal to

If α, β are the roots of the equation ax2 + bx + c = 0 and An = αn + βn, then aAn+2 + bAn+1 + cAn is equal to

If x is real, the maximum value of $\dfrac{3x^2+9x+17}{3x^2+9x+7}$ is

Let $ p(x) = 51x^2 + mx + c $ and $ q(x) = 3x^2 + bx + a $ are two quadratic polynomials with integer coefficients such that $ p(r) = q(r) = 0 $. If $ r $ is an irrational number, then the value of $ \dfrac{c}{a} $ is:

For a constant k, the two roots of the quadratic equation $3x^2 - x + k = 0$ are $\sin\theta$ and $\cos\theta$. The value of $54(\sin^3\theta + \cos 3\theta)$ is:

For $x^2 - (a+3)|x| + 4 = 0$ to have real solutions, the range of $a$ is

If $\alpha, \beta$ and $\gamma$ are roots of $x^3 - 2x^2 + 6x - 1 = 0$, find the value of the following expression:\[\alpha\left(\frac{\alpha^2+\alpha+1}{\alpha^2-\alpha+1}\right)+\beta\left(\frac{\beta^2+\beta+1}{\beta^2-\beta+1}\right)+\gamma\left(\frac{\gamma^2+\gamma+1}{\gamma^2-\gamma+1}\right)\]

If $\alpha \neq \beta$ but $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$ then the equation having $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ as its roots is

If x2 + 3x + 5 = 0 and ax2 + bx + c = 0 have a common root and a, b, c ∈ ℕ, the minimum value of a + b + c is

Let the original equation have two roots $\alpha$ and $\beta$. Then $\alpha\beta = \alpha^2\beta^2$ ... (i) and $\alpha^2 + \beta^2 = \alpha + \beta$ ... (ii). Find the number of quadratic equations satisfying these conditions.

The equation formed by decreasing each root of $ax^2 + bx + c = 0$ by 1 is $2x^2 + 8x + 2 = 0$, then

Number of integral value(s) of k for which the equation 4x2 − 16x + k = 0 has one root lie between 1 and 2 and other root lies between 2 and 3, is

Solve for x, $(x^2 + 3x + 1)(x^2 + 3x - 3) \geq 5$

If f(x) = ax2 − bx + c has two distinct roots α and β, and f(0) and f(1) are of the same sign, with α(1 − α) ≤ 1/4 for all α ∈ (0,1), find the least value of a.

If x_1 and x_2 are the roots of x^2 + (1 - \sin \theta)x - \frac{\cos^2 \theta}{2} = 0, then the maximum value of x_1^2 + x_2^2 is

If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, then the value of q is

If both the roots of the quadratic equation $x^2 - mx + 4 = 0$ are real and distinct and they lie in the interval $[1, 5]$ then $m$ lies in the interval

Let α, β be the roots of the equation x2 − x + p = 0 and γ, δ be the roots of the equation x2 − 4x + q = 0. If α, β, γ and δ are in GP, the integral values of p and q respectively, are

Let the roots of the equation $x^2 + (2-\lambda)x + (10-\lambda) = 0$ be $\alpha$ and $\beta$. The minimum value of $\alpha^3 + \beta^3$ occurs at $\lambda = 4$. What is $|\alpha - \beta|$?

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

If for a positive integer n, the quadratic equation $x(x+1) + (x+1)(x+2) + \cdots + (x+n-1)(x+n) = 10n$ has two consecutive integral solutions, then n is equal to

Ex. 80: Given $h(x) = (1 - \sin\theta)x^2 + 2(1 - \sin\theta)x - 3\sin\theta$ where $\theta \in \mathbb{R} - \left\{(4n+1)\frac{\pi}{2}, n \in \mathbb{Z}\right\}$. If the quadratic equation $h(x) = 0$ has both roots complex, then $\theta$ belongs to:

Find the complete set of values of a for which f(θ) > 0 for all θ ∈ [0, π/2), where f(θ) = tan²θ + (a + 1)tanθ – (a – 3)

The product of all values of x satisfying the equation\[\frac{1}{x^2+2x} + \frac{1}{x^2+6x+8} + \frac{1}{x^2+10x+24} = \frac{1}{5} - \frac{1}{x^2+14x+48}\]is

The sum of all real values of x satisfying the equation $\left(x^2 + 4x - 60\right)^{\left(x^2 - 5x + 5\right)} = 1$ is

The equation $|x+1|\,|x-1| = a^2 - 2a - 3$ can have real solution in x if a belongs to

The number of non-negative integral ordered pair(s) $(x, y)$ for which $(xy - 7)^2 = x^2 + y^2$ holds is greater than or equal to:

Find the number of positive integers x for which f(x) = x3 − 8x2 + 20x − 13 is a prime number.

Let r1, r2, r3, ..., rn be n positive integers, not necessarily distinct, such that (x − r1)(x − r2)···(x − rn) = xn − 56xn−1 + ... − 2009.The possible value of n is

Given that $m$ is a real number not less than $-1$, such that equation $x^2 + 2(m-2)x + m^2 - 3m + 3 = 0$ has two distinct real roots $x_1$ and $x_2$. Find the maximum value of $\frac{1}{2}\left(\frac{mx_1^2}{1-x_1} + \frac{mx_2^2}{1-x_2}\right)$

Let $l \neq 0$ be in $\mathbb{R}$. If a and β are the roots of the equation $x^2 - x + 2l = 0$ and a and γ are the roots of the equation $3x^2 - 10x + 27l = 0$, then l is equal to

Two non-integer roots of $\left(\frac{3x-1}{2x+3}\right)^4 - 5\left(\frac{3x-1}{2x+3}\right)^2 + 4 = 0$ are

If the other roots of equations are reciprocal to each other, then (q - b)^2 is equal to

If a, b, c are complex numbers and $a + b + c = ab + bc + ca = abc = 1$, then find $a + b^4 + c^4$

Find the complete set of values of a for which f(θ) f(θ) = tan²θ + (a + 1)tanθ – (a – 3)

Let f(x) be a polynomial function of second degree. If f(1) = f(-1) and a, b, c are in AP, then f'(a), f'(b) and f'(c) are in

Given f(x) = ax^4 - 2ax^2 + e, if f(x) = f(0), find the sum of squares of all elements in the set T = \{x \in \mathbb{R} | f(x) = f(0)\}.

Let $Q(x) = (x^3 - 2x^2 + x - 1)(x+1) = 0 \Rightarrow \delta = -1$.$\alpha + \beta + \gamma - 2,\ \alpha\beta + \beta\gamma + \gamma\alpha - 1$.$P(x) = (x^2+1)Q(x) + x^2 - x + 1$.Find $P(\alpha) + P(\beta) + P(\gamma) + P(-1)$.

If $a \in \mathbb{R}$ and the equation $-3(x - [x])^2 + 2(x - [x]) + a^2 = 0$ (where $[x]$ denotes the greatest integer $\leq x$) has no integral solution, then all possible values of $a$ lie in the interval

The real positive number x when added to its inverse gives the minimum value of the sum at x equal to

The least non-negative integral value of $\lambda$ for which the equation $2x^2 - 2(2\lambda+1)x + \lambda(\lambda+1) = 0$ has one root less than $\lambda$ and other root greater than $\lambda$, is equal to:

Consider the equation $x^4 - (k-1)x^2 + (2-k) = 0$. The complete set of possible values of real $k$ for which the equation has four distinct real roots is

If $ax^2 + bx + 8 = 0$, where $a, b \in \mathbb{R}$, $a \neq 0$ has no distinct real roots, then the least value of $4a + b$ is

Quadratic Equations Questions (527)