Quadratic Equations Questions - JEE

Consider two quadratic expressions f(x) = ax^2 + bx + c and g(x) = ax^2 + px + q (a, b, c, p, q \in \mathbb{R}, b \neq p) such that their discriminants are equal. If f(x) = g(x) has a root x = \alpha, then

Let $a, b, c \in \mathbb{R}$ with $a > 0$ such that the equation $ax^2 + bcx + b^3 + c^3 - 4abc = 0$ has non-real roots. If $P(x) = ax^2 + bx + c$ and $Q(x) = ax^2 + cx + b$, then

If $\cos A$, $\cos B$ and $\cos C$ are the roots of cubic $x^3 + ax^2 + bx + c = 0$, where A, B, C are the angles of a triangle then find the value of $a^2 - 2b - 2c$.

Let $f(x) = ax^2 - bx + c^2$, $b \neq 0$ and $f(x) \neq 0$ for all $x \in \mathbb{R}$. Then

The graph of a quadratic polynomial y = ax^2 + bx + c; a, b, c \in \mathbb{R} is as shown. Which one of the following is not correct?

If $\alpha,\beta$ are the roots of the equation $x^2-x-1=0$ and $S_n=2023\alpha^n+2024\beta^n$, then

Let the set $C=\{(x,y)\mid x^2-2^y=2023,\,x,y\in\mathbb{N}\}$. Then $\displaystyle\sum_{(x,y)\in C}(x+y)$ is equal to

The number of real roots of the equation $5 + |2^x - 1| = 2^x(2^x - 2)$ is __________.

If x_1 and x_2 are the arithmetic and harmonic means of the roots of the equation ax^2 + bx + c = 0, the quadratic equation whose roots are x_1 and x_2, is

If one root of equation $x^2 + ax + 12 = 0$ is 4 while the equation $x^2 + ax + b = 0$ has equal roots, then the value of b is

The number of ordered pairs (a, b), where a, b are integers satisfying the inequality $\min\left(x^2 + (a-b)x + (1-a-b)\right) > \max\left(-x^2 + (a+b)x - (1+a+b)\right)$ for all $x \in \mathbb{R}$, is:

If a, b, c are real and a \neq b, the roots of the equation 2(a - b)x^2 - 11(a + b + c)x - 3(a - b) = 0 are

If the equations $ax^2 + bx + c = 0$ and $cx^2 + bx + a = 0$, where $a, b, c \in \mathbb{R}$ and $ac \neq 0$, have a common non-real root, then

If the roots of the equation $x^2 - 8x + a^2 - 6a = 0$ are real and distinct, then find all possible values of $a$.

If $k \in ?$

Let f(x) = 0 be an equation of degree six, having integer coefficients and whose one root is $2\cos\dfrac{\pi}{18}$. Then, the sum of all the roots of f'(x) = 0, is

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:

If $-3 \leq \frac{x^2 + bx + 1}{x^2 - bx + 1} \leq 2$ for all $x \in \mathbb{R}$, then the value of $b$ belongs to

For the equation $|x^2 - 2x - 3| = b$, which of the following statements is true?

$f(t) = 4t^2 - 16t + t$. Opening upward parabola. Find the number of integer values of $t$ for which $f(1) > 0$, $f(2) < 0$, $f(3) > 0$.

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

Let $\alpha,\beta$; $\alpha>\beta$, be the roots of the equation $x^2-\sqrt{2}x-\sqrt{3}=0$. Let $P_n=\alpha^n-\beta^n$, $n\in\mathbb{N}$. Then $(11\sqrt{3}-10\sqrt{2})P_{10}+(11\sqrt{2}+10)P_{11}-11P_{12}$ is equal to

If the absolute value of the expression \[\frac{\alpha - 1}{\alpha + 2} + \frac{\beta - 1}{\beta + 2} + \frac{\gamma - 1}{\gamma + 2}\] can be expressed as m/n, where m and n are co-prime, the value of m2/[(m − n)(m + n)] is

$7z^2 + 1 = 0$

If $1, \alpha + \beta, \alpha\beta$ are in A.P. and $1, \frac{1}{\alpha}, \frac{1}{\beta}$ are in A.P., find the value.

Find the number of integral values of parameter 'a' so that the inequality $(2a - a^2) \leq x^2 - 3x + 2 \leq 3 - a^2$ holds for any real x in the interval $[0, 2]$.

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:

Let $\lambda \in \mathbb{R}$ and let the equation $E$ be $|x|^2 - 2|x| + |\lambda - 3| = 0$. Then the largest element in the set $S = \{x + \lambda : x \text{ is an integer solution of } E\}$ is ______.

32. The range of value of $\lambda$ for which the expression $\dfrac{2x^2 - 5x + 3}{4x - \lambda}$ can take all real values for $x \in R - \left\{\dfrac{\lambda}{4}\right\}$, is:

Find the number of integral values of $k$ for which $e^{\lambda^2 - 2\lambda + 1 + \ln 3}$ and $e^{-(\lambda^2 - 2\lambda + 1) + \ln 2}$, where $\lambda \in R - \{1\}$ are the roots of the equation $x^2 - (3k+1)x + 3k^2 - k + 2 = 0$.

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

Let $f(x) = (e^x - a)(3ax + 1)$. Number of possible values of $a$ satisfying $f(x) \geq 0$ for $\forall\, x \in R$.

Let $\alpha,\beta$ be roots of $x^2+\sqrt{2}x-8=0$. If $U_n=\alpha^n+\beta^n$, then $\dfrac{U_{10}+\sqrt{2}U_9}{2U_8}$ is equal to

The sum of all the solutions of the equation $(8)^{2x}-16\cdot(8)^x+48=0$ is:

The number of integral values of $m$ for which the quadratic expression, $(1 + 2m)x^2 - 2(1 + 3m)x + 4(1 + m)$, $x \in R$, is always positive, is __________.

The number of distinct real roots of the equation $|x+1||x+3|-4|x+2|+5=0$, is

If the equation $|2x + \sin^2 a| + |2x + 3 + 2\sin a| = 0$ has exactly one solution $x = \lambda$ (where $a$ is a constant) then find the value of $4\lambda^2$.

If both roots of the quadratic $ax^2 + bx + c = 0$ lie in $(0, 2)$ then $25ac + 20bc + 16c^2$ is always

The equation $x^3 - 6x^2 + 9x + \lambda = 0$ have exactly one root in (1, 3) then find the number of integral values of $[\lambda + 1]$ (where [.] denotes the greatest integer function)

For t > -1, let$\alpha and$$\beta be the roots of the equation t t 1 1 1 2 (($t + 2)$$7 - 1)$$x + ((t + 2)$$6 - 1)$$x + ((t + 2)$$21 - 1) = 0$If lim t$$\$to-1$+$$\$alphat = a$and lim t$$\$to-1$+$$$$\$betat = b$, then 72($a + b)$is equal to ________. 2$

Let $S = \left\{\alpha : \log_2(9^{2\alpha-4}+13) - \log_2\left(\dfrac{5}{2} \cdot 3^{2\alpha-4}+1\right) = 2\right\}$. Then the maximum value of $\beta$ for which the equation $x^2 - 2\left(\displaystyle\sum_{\alpha \in S} \alpha\right)^2 x + \displaystyle\sum_{\alpha \in S}(\alpha+1)^2 \beta = 0$ has real roots, is ______.

79. The interval of $a$ for which the equation $\tan^2 x - (a-4)\tan x + 4 - 2a = 0$ has at least one solution $\forall x \in [0, \pi/4]$ is

Let a, b are two real roots of equation $x^2 + px + q = 0$, p, q ∈ ℝ, q ≠ 0. If the quadratic equation g(x) = 0 has two roots $a + \frac{1}{a}$, $b + \frac{1}{b}$ such that sum of its roots is equal to product of roots, then find the number of integral values q can attain.

Let $a \in \mathbb{R}$ and let $\alpha, \beta$ be the roots of the equation $x^2 + 60^{\frac{1}{4}}x + a = 0$. If $\alpha^4 + \beta^4 = -30$, then the product of all possible values of $a$ is ______.

Let $\lambda \neq 0$ be a real number. Let $\alpha, \beta$ be the roots of the equation $14x^2 - 31x + 3\lambda = 0$ and $\alpha, \gamma$ be the roots of the equation $35x^2 - 53x + 4\lambda = 0$. Then $\dfrac{3\alpha}{\beta}$ and $\dfrac{4\alpha}{\gamma}$ are the roots of the equation:

If the value of real number $a > 0$ for which $x^2 - 5ax + 1 = 0$ and $x^2 - ax - 5 = 0$ have a common real root is $\dfrac{3}{\sqrt{2\beta}}$, then $\beta$ is equal to ______.

Two dice are thrown independently. Let A be the event that the number appeared on the 1st die is less than the number appeared on the 2nd die, B be the event that the number appeared on the 1st die is even and that on the second die is odd, and C be the event that the number appeared on the 1st die is odd and that on the 2nd is even. Then:

177. The polynomials $P(x) = kx^3 + 3x^2 - 3$ and $Q(x) = 2x^3 - 5x + k$, when divided by $(x-4)$ leave the same remainder, then $k$ is equal to:

For $x\in\mathbb{R}$, the expression $\dfrac{x^2+2x+c}{x^2+4x+3c}$ can take all real values if $c\in$

82. If the equation $\cot^4 x - 2\csc^2 x + a^2 = 0$ has at least one solution, then the sum of all possible integral values of $a$ is equal to

Quadratic Equations Questions (527)