Quadratic Equations Questions - JEE

Let $\alpha,\beta$ be roots of $x^2+\sqrt{6}x+3=0$. Then $\dfrac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}$ is equal to

If $a$ and $b$ are roots of $x^2-7x-1=0$, then $\dfrac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is equal to

If $\dfrac{1}{\sqrt{\alpha}}$ and $\dfrac{1}{\sqrt{\beta}}$ are the roots of the equation $ax^2 + bx + 1 = 0$ $(a \neq 0,\ a, b \in R)$, then the equation $x(x + b^3) + (a^3 - 3abx) = 0$ has roots

Number of real values of $\lambda$ such that $(\lambda^2 - 4\lambda + 3)x^2 + (\lambda^2 - 5\lambda + 6)x + (\lambda^2 - 9) = 0$ has more than 2 roots is:

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

The number of real roots of $x|x|-5|x+2|+6=0$ is

85. In the quadratic equation $4x^2 - 2(a+c-1)x + ac - b = 0$ ($a > b > c$),

Let the set of all $a \in \mathbb{R}$ for which the equation $2x^2 + (a-5)x + 15 = 3a$ has no real root be $(\alpha,\beta)$. If $X = \{x \in \mathbb{Z} : \alpha < x < \beta\}$, then $\displaystyle\sum_{x \in X} x^2$ is

164. If the roots of $x^4 + qx^2 + kx + 225 = 0$ are in arithmetic progression, then the value of $q$ is:

We have, $(5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0$Find the harmonic mean of the roots.

If $\alpha$ and $\beta$ are roots of the equation, $x^2 - 4\sqrt{2}\,kx + 2e^{4\ln k} - 1 = 0$ for some $k$, and $\alpha^2 + \beta^2 = 66$ then $\alpha^3 + \beta^3$ is equal to

If $2 < \lambda < 4$

The number of points where $f(x)=e^{8x}-e^{6x}-3e^{4x}-e^{2x}+1=0$ cuts the $x$-axis is equal to............

Let f be a quadratic function such that: f(x) = 0 has 2 real solutions and f(f(x)) = 0 has 3 real solutions. What is the maximum number of solutions for f(f(f(x))) = 0?

If $8\alpha^3 + \beta^3 - \gamma^3 + 6\alpha\beta\gamma = 0$ and $\alpha^2 + 3\gamma = 2\beta$ where $\alpha,\ \beta,\ \gamma \in R$ and $\beta + \gamma \neq 0$, then find the largest integral value of $\gamma$.

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 + 4x - 5 = 0$, then the equation whose roots are $\frac{1}{2\alpha}$ and $\frac{1}{2\beta}$ is

Let $\alpha,\beta\in\mathbb{N}$ be roots of equation $x^2-70x+\lambda=0$, where $\dfrac{\lambda}{2},\dfrac{\lambda}{3}\notin\mathbb{N}$. If $\lambda$ assumes the minimum possible value, then $\dfrac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$ is equal to:

Let $\alpha,\beta,\gamma$ be roots of $x^3+bx+c=0$ with $\beta\gamma=1=-\alpha$. Then $b^3+2c^3-3\alpha^3-6\beta^3-8\gamma^3$ is equal to

If $a$ and $b$ are distinct zeroes of the polynomial $x^3 - 2x + c$ and $a^2\left(2a^2 + 4ab + 3b^2\right) = 3$ then $b^2\left(3a^2 + 4ab + 2b^2\right)$ is equal to:

The number of solutions of the equation $e^{\sin x}-2e^{-\sin x}=2$ is

Let $\alpha,\beta$ be roots of $x^2-\sqrt{2}x+2=0$. Then $\alpha^{14}+\beta^{14}$ is equal to

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:

Given that the solution set of the quadratic inequality ax² + bx + c > 0 is (2, 3).Then the solution set of the inequality cx² + bx + a

Let $f$ be a quadratic function such that: $f(x) = 0$ has 2 real solutions and $f(f(x)) = 0$ has 3 real solutions. What is the maximum number of solutions for $f(f(f(x))) = 0$?

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) 5Q664. Which of the following is incorrect combination?

Let $m$ and $n$ be the numbers of real roots of $x^2-12x+[x]+31=0$ and $x^2-5|x+2|-4=0$ respectively. Then $m^2+mn+n^2$ is equal to

The sum of all the roots of $|x^2-8x+15|-2x+7=0$ is

The number of real solutions of the equation $x|x+5|+2|x+7|-2=0$ is

The number of real solutions of the equation $x(x^2+3|x|+5|x-1|+6|x-2|)=0$ is

In $x^2 - nz + 2014 = 0$, where $\alpha + \beta = n$ and $\alpha\beta = 2014$. Find the sum $\alpha + \beta + \gamma$.

Since $2x^2 + 3x + 4 > 0$ (as discriminant \(= 9 - 32 = -23

The sum of the squares of all the roots of the equation $x^{2}+|2x-3|-4=0$ is:

If $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$ have a common root $\alpha \neq 0$, then $\frac{a^3 + b^3 + c^3}{abc}$ is equal to

If $ a + b + c = 0 $, then the value of $(x-1)^3 + (2x-1)^3 + (2-3x)^3$ equals:

70. The set of values of $a$ for which $(a-1)x^2 - (a+1)x + a - 1 \geq 0$ is true for all $x \geq 2$ is

If a, b, c, d are four consecutive terms of an increasing AP, then the roots of the equation $(x-a)(x-c) + 2(x-b)(x-d) = 0$ are

$P(x)$ is a polynomial with integral coefficients such that for four distinct integers $a, b, c, d$, $P(a) = P(b) = P(c) = P(d) = 3$. If $P(e) = 5$ ($e$ is an integer), then

If $\alpha, \beta, \gamma, \sigma$ are the roots of the equation $x^4 + 4x^3 - 6x^2 + 7x - 9 = 0$, then the value of $(1 - \alpha^2)(1 + \beta^2)(1 + \gamma^2)(1 + \sigma^2)$ is

Set of all real values of $a$ such that $f(x) = \dfrac{(2a-1)x^2 + 2(a+1)x + (2a-1)}{x^2 - 2x + 40}$ is always negative is

For Problems 28–30: The numbers $a$, $b$, and $c$ are between 2 and 18, such that (i) their sum is 25, (ii) the numbers 2, $a$, and $b$ are consecutive terms of an A.P., (iii) the numbers $b$, $c$, 18 are consecutive terms of a G.P.Roots of the equation $ax^2 + bx + c = 0$ are

If $\alpha$, $\beta$, $\gamma$ are the roots of the equation $x^3 - px + q = 0$, then find the cubic equation whose roots are $\alpha(1+\alpha)$, $\beta(1+\beta)$, $\gamma(1+\gamma)$.

If α, β are the roots of $x^2 + px + q = 0$ and γ, δ are the roots of $x^2 + px + r = 0$, then $\dfrac{(\alpha-\gamma)(\alpha-\delta)}{(\beta-\gamma)(\beta-\delta)} =$

If one root of $x^2 - x - k = 0$ is square of the other, then $k =$

If α and β be the roots of the equation $x^2 + px - 1/(2p^2) = 0$, where $p \in \mathbb{R}$. Then the minimum value of $\alpha^4 + \beta^4$ is

A quadratic equation with integral coefficients has two different prime numbers as its roots. If the sum of the coefficients of the equation is prime, then the sum of the roots is

Let $a, b$, and $c$ be real numbers such that $4a + 2b + c = 0$ and $ab > 0$. Then the equation $ax^2 + bx + c = 0$ has

If the roots of equation $(a-1)(x^2 + x + 1)^2 = (a+1)(x^4 + x^2 + 1)$ are real and distinct, then the value of $a$ $\in$

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 $\gamma$ are

Determine the values of m for which the equations 3x2 + 4mx + 2 = 0 and 2x2 + 3x − 2 = 0 may have a common root.

If $\alpha$, $\beta$ and $\gamma$ are the roots of the equation $x^3 + 3x^2 - 4x - 2 = 0$, then find the values of the following expressions:(i) $\alpha^2 + \beta^2 + \gamma^2$(ii) $\alpha^3 + \beta^3 + \gamma^3$(iii) $\dfrac{1}{\alpha} + \dfrac{1}{\beta} + \dfrac{1}{\gamma}$

Quadratic Equations Questions (527)