Quadratic Equations Questions (527)

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

138. Let a polynomial \(P(x)\), when divided by \(x-1\), \(x-2\), \(x-3\) leaves the remainder 4, 5, 6 respectively. When \(P(x)\) is divided by \((x-1)(x-2)(x-3)\), the remainder is \(ax^2+bx+c\), then \(3a+2b+c\) is equal to:

If \(x^2 + px + q = 0\) is the quadratic equation whose roots are \(a + 2\) and \(b + 2\), where a and b are the roots of \(x^2 - 3x + 1 = 0\), then

913. Let \(f\) be monic cubic polynomial such that \(f(1)=1^4-1\), \(f(2)=2^4-2\) and \(f(3)=3^4-3\). If \(f(4)=N\), then find the number of prime factors of \(N\).

174. Solution set of the equation \(\sqrt{4^x - 2^{x+1} + 1} + \sqrt{4^x - 2^{x+3} + 16} = 3\) is:

71. If the equation \(ax^2 + bx + c = x\) has no real roots, then the equation \(a(ax^2 + bx + c)^2 + b(ax^2 + bx + c) + c = x\) will have

78. If the roots of the quadratic equation \((4p - p^2 - 5)x^2 - (2p-1)x + 3p = 0\) lie on either side of unity, then the number of integral values of \(p\) is

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:

75. If \(a, b, c, d\) are four consecutive terms of an increasing A.P., then the roots of the equation \((x - a)(x - c) + 2(x - b)(x - d) = 0\) are

Let the set of all values of $p \in \mathbb{R}$, for which both the roots of the equation $x^2 - (p + 2)x + (2p + 9) = 0$ are negative real numbers, be the interval $(\alpha, \beta]$. Then $\beta - 2\alpha$ is equal to

Let $\alpha$ and $\beta$ be the roots of the equation $px^2+qx-r=0$, where $p\neq0$. If $p,q$ and $r$ be the consecutive terms of a non-constant G.P. and $\dfrac{1}{\alpha}+\dfrac{1}{\beta}=\dfrac{3}{4}$, then the value of $(\alpha-\beta)^2$ is:

If 2 and 6 are the roots of the equation $ax^2+bx+1=0$, then the quadratic equation, whose roots are $\dfrac{1}{2a+b}$ and $\dfrac{1}{6a+b}$, is:

Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3}x-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2+3x-1=0$. If $P_n=\alpha^n+\beta^n$ and $Q_n=\gamma^n+\delta^n$, then $\dfrac{P_{25}+\sqrt{3}P_{24}}{2P_{23}}+\dfrac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to

Let $x_1,x_2,x_3,x_4$ be the solutions of the equation $4x^4+8x^3-17x^2-12x+9=0$ and $(4+x_1^2)(4+x_2^2)(4+x_3^2)(4+x_4^2)=\dfrac{125}{16}m$. Then the value of $m$ is

Let $\alpha,\beta$ be the distinct roots of the equation $x^2-(t^2-5t+6)x+1=0$, $t\in\mathbb{R}$ and $a_n=\alpha^n+\beta^n$. Then the minimum value of $\dfrac{a_{2023}+a_{2025}}{a_{2024}}$ is

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:

The product of all the rational roots of the equation $\bigl(x^{2}-9x+11\bigr)^{2}-(x-4)(x-5)=3$, is equal to:

Let $\alpha_{\theta}$ and $\beta_{\theta}$ be the distinct roots of $2x^{2}+(\cos\theta)x-1=0,\ \theta\in(0,2\pi)$. If $m$ and $M$ are the minimum and the maximum values of $\alpha_{\theta}^{4}+\beta_{\theta}^{4}$, then $16(M+m)$ equals:

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

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

138. Let a polynomial \(P(x)\), when divided by \(x-1,\, x-2,\, x-3\) leaves the remainder \(4,\,5,\,6\) respectively. When \(P(x)\) is divided by \((x-1)(x-2)(x-3)\), the remainder is \(ax^2+bx+c\), then \(3a+2b+c\) is equal to:

Let $S$ be the set of positive integral values of $a$ for which $\dfrac{ax^2+2(a+1)x+9a+4}{x^2-8x+32}<0$, $\forall x\in\mathbb{R}$. Then, the number of elements in $S$ is:

Let $a,b,c$ be the lengths of three sides of a triangle satisfying the condition $(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0$. If the set of all possible values of $x$ is the interval $(\alpha,\beta)$, then $12(\alpha^2+\beta^2)$ is equal to

The product of all the rational roots of the equation $(x^2-9x+11)^2-(x-4)(x-5) = 3$ is

Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2x^2 + (\cos\theta)x - 1 = 0$, $\theta \in [0,2\pi)$. If $m$ and $M$ are respectively the minimum and maximum values of $\alpha_\theta^4 + \beta_\theta^4$, then $16(M+m)$ is equal to

The number of irrational roots of the equation \(\dfrac{4x}{x^2 + x + 3} + \dfrac{5x}{x^2 - 5x + 3} = -\dfrac{3}{2}\) is

Let P n =$\alpha n +$$\beta n,n$$\i_n N. If P$10 = 123$,$$P_{9}$= 76$,$P = 47$and$P = 1$, then the quadratic equation having roots 8 1 1$$\alpha and 1$$\beta is$:

If the set of all a$\i_n$R - {1}$, for which the roots of the equation ($1 - a)x + 2$(a - 3)$x + 9 = 0$are positive is 2 (-$$\infty, -$$\alpha$]$\cup$[$\beta,$$\gamma), then 2$$\alpha +$$\beta +$$\gamma is equal to _______.$