Quadratic Equations Questions - JEE

Solve the inequality: $(k-2)x^2 + 8x + (k+4) > 0$ for all $x \in \mathbb{R}$Find the least integral value of $k$.

If the roots of the equation $x^2 - 10cx - 11d = 0$ are $a, b$ and those of $x^2 - 10ax - 11b = 0$ are $c, d$, then find the value of $a + b + c + d$. ($a, b, c, d$ are distinct numbers)

If $x^2 + 2ax + 10 - 3a > 0$ for all $x \in \mathbb{R}$, then

If (1 – p) is a root of quadratic equation x2 + px + (1 – p) = 0 then its roots are

(a) The harmonic mean of the roots of the equation $x^2 - (\sqrt{8} + 2\sqrt{2})x + 8 + 2\sqrt{2} = 0$ is

A value of b for which the equationsx2 + bx – 1 = 0x2 + x + b = 0have one root in common is

Complete set of real values of k for which the inequality kx² – kx – 1 x, satisfy

If $a, b, c$ are distinct positive real numbers and $a^2 + b^2 + c^2 = 1$ then $ab + bc + ca$ is

If a, b, c, p, q, r are non-zero real numbers, such that a < b < c and\[f(x) = (x - a)(x - b)(x - c) - p^2(x - a) - q^2(x - b) - r^2(x - c),\]then $f(x) = 0$ must have

Let α1 and β1 be the roots of the equation x2 + 2x − 1 = 0, and α2 and β2 be the roots of the equation x2 + 2xtanθ − 1 = 0. If α1 > β1 and α2 > β2, then α1 + β2 equals

Three real numbers $x, y, z$ are such that $x^2 + 6y = -17$, $y^2 + 4z = 11$ and $z^2 + 2x = 2$.Then the value of $x^3 + y^3 + z^3$ is equal to

Find the sum of the eighth powers of all roots of the equation $x^3 - x + 1 = 0$

Find the set of all possible real values of a such that the inequality $(x - (a-1))(x - (a^2 + 2)) < 0$ holds for all $x \in (-1, 3)$.

If the roots of the equation $ax^2 + bx + c = 0$ are of the form $\dfrac{k+1}{k}$ and $\dfrac{k+2}{k+1}$, then $(a+b+c)^2$ is equal to

If $\alpha$ and $\beta$ ($\alpha<\beta$) are the roots of the equation $(-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6\sqrt{x})+(9-2\sqrt{3})=0$, $x\geq0$, then $\sqrt{\dfrac{\beta}{\alpha}}+\sqrt{\alpha\beta}$ is equal to:

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:

The curve $y = (\lambda + 1)x^2 + 2$ intersects the curve $y = \lambda x + 3$ in exactly one point, if $\lambda$ equals

Let $S=\{x^3+ax^2+bx+c:a,b,c\in\mathbb{N}$ and $a,b,c\leq20\}$ be a set of polynomials. Then the number of polynomials in $S$, which are divisible by $x^2+2$, is

If the equations $x^2 + 2x + 3 = 0$ and $ax^2 + bx + c = 0$, $a, b, c \in \mathbb{R}$, have a common root, then $a : b : c$ is

Question nos. 663 to 665Match the condition of column-I with corresponding number of real roots of $f(x) = 0$ in column-II and number of points of non-derivability of $y = f(|x|)$ in column-III, where $f(x) = ax^2 + bx + c$.Column-I(I) $a^2 + b^2 + c^2 - ab - bc - ca \le 0$(II) $a^2 + b^2 + c^2 + ab + bc + ca \le 0$(III) $3(a^2 + b^2 + c^2 + 1) \le 2(a + b + c + ab + bc + ca)$(IV) $a^2 + b^2 + c^2 \le 2a + 6b + 4c + 14$Column-II(i) 0 (ii) 1 (iii) 2 (iv) $\infty$Column-III(P) 0 (Q) 1 (R) 3 (S) 5Q. 663. Which of the following is correct combination?

If the roots of the equation $a(b-c)x^2 + b(c-a)x + c(a-b) = 0$ are equal, show that $2/b = 1/a + 1/c$.

If $\alpha, \beta$ are the roots of $x^2 + px + q = 0$ and $\alpha^{2n} + p^n\alpha^n + q^n = 0$ and if $(\alpha/\beta), (\beta/\alpha)$ are the roots of $x^n + 1 + (x+1)^n = 0$, then $n$ ($n \in \mathbb{N}$)

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:

Which of the following equations has maximum number of real roots?

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:

The quadratic equation $p(x) = 0$ with real coefficients has purely imaginary roots. Then the equation $p(p(x)) = 0$ has

If roots of an equation $x^n - 1 = 0$ are $1, a_1, a_2, \ldots, a_{n-1}$, then the value of $(1 - a_1)(1 - a_2)(1 - a_3)\cdots(1 - a_{n-1})$ will be

For Problems 35–37Consider the equation $x^4 - \lambda x^2 + 9 = 0$.If the equation has four real and distinct roots, then $\lambda$ lies in the interval

If the two roots of the equation, $(a-1)(x^4 + x^2 + 1) + (a+1)(x^2 + x + 1)^2 = 0$ are real and distinct, then the set of all values of $a$ is

If α, β are the roots of $ax^2 + bx + c = 0$ and $\alpha+h$, $\beta+h$ are the roots of $px^2 + qx + r = 0$, then h =

If $\alpha$ and $\beta$ are the roots of the equation $8x^2 - 3x + 27 = 0$, then the value of $\left(\frac{\alpha}{\beta}\right)^{1/2} + \left(\frac{\beta}{\alpha}\right)^{1/2}$ is

Find the number of pairs (a, b) of real numbers such that whenever $\alpha$ is a root of $x^2 + ax + b = 0$, $\alpha^2 - 2$ is also a root of the equation.

A value of b for which the equations $x^2 + bx - 1 = 0$ and $x^2 + x + b = 0$ have one root in common is

a, b, c are real numbers with $a^2 + b^2 + c^2 > 0$. Then the equation $x^2 + (a+b+c)x + (a^2+b^2+c^2) = 0$ has

Question nos. 663 to 665Match the condition of column-I with corresponding number of real roots of $f(x) = 0$ in column-II and number of points of non-derivability of $y = f(|x|)$ in column-III, where $f(x) = ax^2 + bx + c$.Column-I(I) $a^2 + b^2 + c^2 - ab - bc - ca \leq 0$(II) $a^2 + b^2 + c^2 + ab + bc + ca \leq 0$(III) $3(a^2 + b^2 + c^2 + 1) \leq 2(a + b + c + ab + bc + ca)$(IV) $a^2 + b^2 + c^2 \leq 2a + 6b + 4c + 14$Column-II(i) 0 (ii) 1 (iii) 2 (iv) $\infty$Column-III(P) 0 (Q) 1 (R) 3 (S) 5Q663. Which of the following is correct combination?

If $a, b \in \mathbb{R}$, $a \neq 0$ and the quadratic equation $ax^2 - bx + 1 = 0$ has imaginary roots, then $(a + b + 1)$ is

Let α, β be the roots of x2 + bx + 1 = 0. Then find the equation whose roots are −(α + 1/β) and −(β + 1/α).

If $\alpha, \beta$ are the roots of the equation $ax^2 + bx + c = 0$, then the value of $\dfrac{(a\alpha^2 + c)}{(a\alpha + b)} + \dfrac{(a\beta^2 + c)}{(a\beta + b)}$ is

Consider the equation $x^2 + 2x - n = 0$, where $n \in \mathbb{N}$ and $n \in [5, 100]$. The number of different values of $n$ so that the given equation has integral roots, is

If $x = \alpha$ is the common root of both the equations, then find $P(1)$ where $P(x) = x^2 - 2x + 5$.

Given equation, $4(z^2 + \frac{1}{z^2}) + 16(z + \frac{1}{z}) - 57 = 0$. Find $x$ such that $z$ is rational.

If $\alpha \neq \beta$ but, $\alpha^2 = 4\alpha - 2$ and $\beta^2 = 4\beta - 2$, then the quadratic equation with roots $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ is

If $f(x) = 4x^2 - 20kx + (25k^2 + 15k - 66) = 0$ has roots $\alpha, \beta$ that are real, and the parabola opens upward with $\alpha, \beta < -1$, find the range of $k$.

If $f(x) = ax^2 + bx + c$ has an upward-opening parabola with roots between which $f(2) < 0$, and $f(-1) = 4(2k + 4) < 0$, find the range of $k$.

If $2 + 3i$ is one of the roots of the equation $2x^3 - 9x^2 + kx - 13 = 0$, $k \in \mathbb{R}$, then the real root of this equation

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:

Solve $\sqrt{5x^2-6x+8}-\sqrt{5x^2-6x-7}=1$.

Let p and q be real numbers such that p ≠ 0, p3 ≠ q, and p3 ≠ −q. If α and β are nonzero complex numbers satisfying α + β = −p and α3 + β3 = q, then a quadratic equation having α/β and β/α as its roots is

Find all real numbers $a$ for which the equation $x^2 + (a-2)x + 1 = 3|x|$ has exactly three distinct real solutions in x.

250. If $\alpha$ and $\beta$ are the roots of the equation $x^2 - x\sin 2\theta + 2\cos^2\theta = 0$, $\theta \in R$ and the maximum value of $(2-\alpha)(2-\beta)$ is $(a + \sqrt{a})$, then $a$ is equal to:

Quadratic Equations Questions (527)