Quadratic Equations Questions - JEE

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 $\alpha$.

If ax2 + bx + c = 0 and bx2 + cx + a = 0 have a common root and a, b, and c are nonzero real numbers, then find the value of $\dfrac{a^3 + b^3 + c^3}{abc}$.

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:

Let $r$, $s$, and $t$ be the roots of equation $8x^3 + 1001x + 2008 = 0$. Then find the value of $(r+s)^3 + (s+t)^3 + (t+r)^3$.

The second degree polynomial $f(x)$, satisfying $f(0) = 0$, $f(1) = 1$, $f'(x) \geq 0; x \in (0, 1)$ is

If $\alpha$ is the root of the equation $x^2 - x + 2 = 0$, then the value of $\dfrac{6(-\alpha^3 + 2\alpha^2 - \alpha)}{\alpha^5 - 3\alpha^4 + 3\alpha^3 - \alpha^2}$ is equal to:

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

If $x^2 + ax - 3x - (a+2) = 0$ has real and distinct roots, then the minimum value of $(a^2+1)/(a^2+2)$ is

If $(m_r, 1/m_r)$, $r = 1, 2, 3, 4$, are four pairs of values of $x$ and $y$ that satisfy the equation $x^2 + y^2 + 2yx + 2fy + c = 0$, then the value of $m_1 \cdot m_2 \cdot m_3 \cdot m_4$ is

Let $f(x) = x^2 - 2px + p^2 - 1$, where $p \in R - \{-1, 1\}$. If $\alpha$ and $\beta$ are distinct real roots of the equation $f(x) = 0$ such that $\left|\dfrac{\alpha^2 + \beta^2 + 3\alpha\beta}{\alpha\beta}\right| \leq 5$, then set of values of $p \in [a, b]$. The value of $[2(a^2 + b^2)]$ is:[Note: $[k]$ denotes greatest integer less than or equal to $k$.]

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - x + 1 = 0$, then $\alpha^{2009} + \beta^{2009} =$

327. The smallest positive integral value of a for which the greater root of the equation $x^2 - (a^2 + a + 1)x + a(a^2 + 1) = 0$ lies between the roots of the equation $x^2 - a^2x - 2(a^2 - 2) = 0$, is less than:

If G and L are the greatest and least values of the expression $\frac{x^2 - x + 1}{x^2 + x + 1}$, $x \in \mathbb{R}$ respectively, then the least value of $G^5 + L^5$ is

Find the sum of all integral values of $a$ for which all the roots of the equation $x^4 - 4x^3 - 8x^2 + a = 0$ are real.

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

Let $p(x) = 0$ be a polynomial equation of the least possible degree, with rational coefficients, having $\sqrt[3]{7} + \sqrt[3]{49}$ as one of its roots. Then the product of all the roots of $p(x) = 0$ is

For Problems 35–37Consider the equation $x^4 - \lambda x^2 + 9 = 0$.If the equation has only two real roots, then the set of values of $\lambda$ is

Number of integral points (x, y) in the 1st quadrant that satisfy the equation $y^4 + 6xy^2 - 8x = 0$ is equal to

Find the values of the parameter a such that the roots $\alpha$ and $\beta$ of the equation $2x^2 + 6x + a = 0$ satisfy the inequality \(\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha}

The number of real solutions of the equation $(9/10)^x = -3 - x - x^2$ is

Let x₁, x₂, x₃ be the roots of the equation x³ + 3x + 5 = 0. Then the value of the expression is equal to

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?

Solve $\dfrac{x^2+3x+2}{x^2-6x-7}=0$.

If the roots of the equation $10x^3 + cx^2 + 54x + 27 = 0$ are in harmonic progression, find the value of $c$.

$a, b, c, d$ are distinct integers such that $(x - a)(x - b)(x - c)(x - d) = 4$ has an integral root $r$. Then $a + b + c + d$ is equal to:

The quadratic equations $x^2 - 6x + a = 0$ and $x^2 - cx + 6 = 0$ have one root $\alpha$ in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then equation $x^2 - 3x - 4 = 0$ has

If complex numbers satisfying α + β = –p and α3 + β3 = q, then a quadratic equation having α3 and β3 as its roots is

If x = 1 and x = 2 are solutions of equations $x^3 + ax^2 + bx + c = 0$ and $a + b = 1$, then find the value of b.

Quadratic Equations Questions (527)