Quadratic Equations Questions - JEE

The polynomial $f(x) = x^4 + ax^3 + bx^2 + cx + d$ has real coefficients and $f(2i) = f(2+i) = 0$. Find the value of $(a + b + c + d)$.

The quadratic equation $x^2 + bx + c = 0$ has distinct roots. If 2 is subtract from each root then result are the reciprocal of the original root. The value of $(b^2 + c^2)$ is:

Let $f(x) = (1 + b^2)x^2 + 2bx + 1$ and let $m_b()$ be the minimum value of $f(x)$. As b varies, the range of $m_b()$ is

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 2 distinct real roots is

Let n ∈ Z and △ABC be a right triangle with right angle at C. If sin A and sin B are the roots of the quadratic equation $(5n+8)x^2 - (7n-20)x + 120 = 0$, then find the value of n.

If $(a+1)(b+1)(c+1)(d+1) = 1$, $(a+2)(b+2)(c+2)(d+2) = 2$, $(a+3)(b+3)(c+3)(d+3) = 3$, $(a+4)(b+4)(c+4)(d+4) = 4$. Then the value of $(a+5)(b+5)(c+5)(d+5)$ is

Given that the equation $x^2 - ax + 3 - b = 0$ has two distinct real roots, $x^2 + (6-a)x + 6 - b = 0$ has two equal real roots, and $x^2 + (4-a)x + 5 - b = 0$ has no real roots. Then the ranges of a and b are

If \tan\theta_i; i = 1, 2, 3, 4\ are the roots of equation x^4 - x^3\sin 2\beta + x^2\cos 2\beta - x\cos\beta - \sin\beta = 0\, then \tan(\theta_1 + \theta_2 + \theta_3 + \theta_4) =\

If $ f(x) \geq -3 $ for all $ x \in \mathbb{R} $, and the inequality $ x^2 + 2px + 4p + f(x) \geq 0 $ holds for all $ x \in \mathbb{R} $, find the range of $ p $ such that $ p^2 - 4p + 3 , i.e., \( 1 \leq p \leq 3 $. How many integer values does $ p $ take?

If $p$ and $q$ are the roots of the equation $x^2 + px + q = 0$, then

For polynomials of the form $a_n x^n + a_{n-1}x^{n-1} + \ldots + a_1 x + a_0$ with $a_i \in \{-1, 1\}$, $(i = 0, 1, 2, \ldots, n)$ which has all roots real, find the maximum value of $n$.

Consider the inequality $9 - x^2 > |x| - a$, where $a$ is a real number. The complete set of values of $a$ for which the given inequality has at least one negative solution is

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^4 - Kx^3 + Kx^2 + Lx + M = 0$, where $K, L,$ and $M$ are real numbers, then the minimum value of $\alpha^2 + \beta^2 + \gamma^2 + \delta^2$ is

If $x$ is real, then $x/(x^2 - 5x + 9)$ lies between

If a and b are the roots of the equation 7x2 – 3x – 2 = 0, then the value of $\frac{a}{1-a^2} + \frac{b}{1-b^2}$ is equal to

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

If the inequality $x^2 + ax + a^2 + 6a < 0$ is satisfied for all $x \in (1,2)$, then the sum of all the integral values of $a$ must be equal to

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. 665. Which of the following is correct combination?

The sum of the solutions of the equation $\left|\sqrt{x} - 2\right| + \sqrt{x}(\sqrt{x} - 4) + 2 = 0$, $(x > 0)$ is equal to __________.

If $2b^2 + 27d = 9bc$, find the relationship using roots.

If $\alpha, \beta$ are the roots of $x^2 - px + q = 0$ and $\alpha', \beta'$ are the roots of $x^2 - p'x + q' = 0$, then the value of $(\alpha - \alpha')^2 + (\beta - \alpha')^2 + (\alpha - \beta')^2 + (\beta - \beta')^2$ 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) $a^2 + b^2 + c^2 - ab - bc - ca \leq 0$(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) 5Q665. Which of the following is correct combination?

If $a(b-c)x^2 + b(c-a)x + c(a-b) = 0$ has equal roots and $a+c=15$, $b = \dfrac{36}{5}$, then $a^2+c^2$ is equal to

Let $A = [-2, 4]$, $B = \{x: x^2 - ax - 4 \leq 0\}$. If $B \subseteq A$, then the range of real $a$ is:

If $(x+1)$ is a factor of $x^3 + kx^2 - 3x + k + 2$, then $k$ is equal to:

The number of solutions of $\left(\dfrac{9}{x} - \dfrac{9}{\sqrt{x}} + 2\right)\left(\dfrac{2}{x} - \dfrac{7}{\sqrt{x}} + 3\right) = 0$ is

The number of solutions of the equation $\left(\dfrac{9}{x}-\dfrac{9}{\sqrt{x}}+2\right)\!\left(\dfrac{2}{x}-\dfrac{7}{\sqrt{x}}+3\right)=0$ is:

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

Quadratic Equations Questions (527)