Quadratic Equations Questions - JEE

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

For Problems 35–37Consider the equation $x^4 - \lambda x^2 + 9 = 0$.If the equation has no real root, then $\lambda$ lies in the interval

The solution set of the equation $pqx^2 - (p + q)^2 x + (p + q)^2 = 0$ is

\((\lambda^2 + \lambda - 2)x^2 + (\lambda + 2)x - 1

Given $p, q, r$ are real numbers ($p \neq q,\ r \neq 0$) and \[\frac{1}{x+p} + \frac{1}{x+q} = \frac{1}{r}\] The roots $\alpha$ and $\beta$ of the resulting quadratic satisfy which general form?

If equations $ax^2 + bx + c = 0$, $(a, b, c \in R,\ a \neq 0)$ and $2x^2 + 3x + 4 = 0$ have a common root then $a : b : c$ equals

One of the roots of $ax^2 + bx + c = 0$ is greater than 2 and the other is less than −1. If the roots of $cx^2 + bx + a = 0$ are α and β, then

For how many values of $n$ in the range $n \in [5, 100]$ is $D$ a perfect square, where $D$ is the discriminant of $x^2 + 2x - n = 0$?

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 3 distinct real roots is

For the quadratic $ x^2 + (n+1)x + (n+2) = 0 $ (where n is an integer), the discriminant must be a perfect square and positive. Given $ D = (n+1)^2 - 4n(n+2) = -3n^2 - 6n + 1 = 4 - 3(n+1)^2 $, find the possible values of n.

If $p^3 + q^3 + q(1 - 3p) = 0$, find $(\alpha + \alpha^3)^2$ where $\alpha$ and $\alpha^3$ are roots of $x^2 + px - q = 0$.

Given the quadratic equation $81x^2 + kx + 256 = 0$ and one real root is the cube of the other root. Find the value of $k$.

Find the value of a for which one root of the quadratic equation $(a^2 - 5a + 3)x^2 + (3a-1)x + 2 = 0$ is twice as large as the other.

Given the equation $x^2 - 1154x + 1 = 0$ with roots $a$ and $b$, find $a + b$ and $ab$.

If $\alpha$ and $\beta$ are roots of $x^2 - 2px + p^2 - 1 = 0$ and $\left|\frac{\alpha^2+\beta^2}{\alpha\beta}+3\right| \leq 5$, then $p \in [a,\,b]$. Find the value of $[2(a^2+b^2)]$.

If $a$, $b$ and $c$ are side lengths of a triangle $ABC$ such that $x^2 - 2(a+b+c)x + 3k(ab+bc+ca) = 0$, where \(k

If the roots of equation $x^2 - bx + m - 1 = 0$ are equal but opposite in sign, then the value of m will be

83. If $a, b, c$ are distinct positive numbers, then the nature of roots of the equation $\dfrac{1}{x-a} + \dfrac{1}{x-b} + \dfrac{1}{x-c} = \dfrac{1}{x}$ is

Let f(x) be a quadratic expression positive for all real x. If g(x) = f(x) − f'(x) + f''(x), then for any real x

Difference between the corresponding roots of $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ is same and $a \neq b$, then

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

The coefficient of x in the equation $x^2 + px + q = 0$ was taken as 17 in place of 13 its roots were found to be -2 and -15. The roots of the original equation are

The number of values of a for which equations $x^3 + ax + 1 = 0$ and $x^4 + ax^2 + 1 = 0$ have a common root is

Put g(x) = y = x2 − 2. If the roots of y5 + 20y4 + 40y3 + 79y2 + 74y + 23 = 0 are g(x1), g(x2), g(x3), g(x4), g(x5), find the value of g(x1) · g(x2) · g(x3) · g(x4) · g(x5) − 30g(x1x2x3x4x5).

Quadratic Equations Questions (527)