Quadratic Equations Questions (527)

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 _______.$

The discriminant of a quadratic equation is

878. 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 \(x^2 - (3k+1)x + 3k^2 - k + 2 = 0\).

Consider the equation$x + 4x - n = 0$, where n$\i_n$[20, 100] is a natural number. Then the number of all distinct 2 values of n, for which the given equation has integral roots, is equal to

The number of real roots of the equation x|$x - 2$| + 3|$x - 3$| +$1 = 0$is:

Let $\alpha,\beta$ be roots of $x^2-3x-\sqrt7=0$ and $\alpha^2+\frac1\beta,\beta^2+\frac1\alpha$ be roots of $x^2+px+q=0$. For $x^2+p^2qx-(11q+p)=0$, roots are

If the parabola $y=ax^2+bx+c$ has vertex at $(4,2)$ and $\lambda\in[1,3]$, and the maximum value of product $abc$ is $\lambda$, then $\dfrac{|\lambda|}{24}$ is

Find the number of positive integral values of $k$ for which $kx^2+(k-3)x+1<0$ for at least one positive $x$.

If \((ax^2 + c)y + (a'x^2 + c') = 0\) and \(x\) is a rational function of \(y\) and \(ac\) is negative, then

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in:

If \(\alpha^2 - \alpha + 2 = 0\), then find the value of \(\dfrac{6(-\alpha^3 + 2\alpha^2 - \alpha)}{\alpha^5 - 3\alpha^4 + 3\alpha^3 - \alpha^2}\).

The equation $e^{4x} + 8e^{3x} + 13e^{2x} - 8e^x + 1 = 0$, $x \in \mathbb{R}$ has:

The number of real roots of the equation $\sqrt{x^2 - 4x + 3} + \sqrt{x^2 - 9} = \sqrt{4x^2 - 14x + 6}$, is:

The sum of all real values of $z$ satisfying the equation $(z^2 + 5z + 5)^{z^2+4z-60} = 1$ is

The sum of all real $x$ such that $\frac{4x^2 + 15x + 17}{x^2 + 4x + 12} = \frac{5x^2 + 16x + 18}{2x^2 + 5x + 13}$ is:

For $x \in \mathbb{R}$, the maximum value of $\sqrt{x^4 - 3x^2 - 6x + 13} - \sqrt{x^4 - x^2 + 1}$ is:

The number of polynomials of the form $a^3 + a^2 + b + c (\forall a, b, c \in \{1, 2, 3, \ldots, 10\})$ which are divisible by $x^2 + 1$ is equal to

The number of real solutions of the equation $3\left(x^2 + \dfrac{1}{x^2}\right) - 2\left(x + \dfrac{1}{x}\right) + 5 = 0$, is

Let $\alpha$ be a root of the equation $(a - c)x^2 + (b - a)x + (c - b) = 0$ where $a, b, c$ are distinct real numbers such that the matrix $\begin{bmatrix} \alpha^2 & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c \end{bmatrix}$ is singular. Then the value of $\dfrac{(a-c)^2}{(b-a)(c-b)} + \dfrac{(b-a)^2}{(a-c)(c-b)} + \dfrac{(c-b)^2}{(a-c)(b-a)}$ is

Given that $a > 0$, $|ax^2 + bx + c| \leq 1$ if $-1 \leq x \leq 1$, $a, b, c \in \mathbb{R}$ and $ax + b$ has its maximum value $2$ when $-1 \leq x \leq 1$. Then $a =$

If $\alpha$, $\beta$ and $\gamma$ are the roots of the equation $x^3 + x + 2 = 0$, then the equation whose roots are $(\alpha - \beta)(\alpha - \gamma)$, $(\beta - \alpha)(\beta - \gamma)$ and $(\gamma - \alpha)(\gamma - \beta)$ is

Let the $n$ real roots of the equation $x^n - 2nx^{n-1} + 2n(n-1)x^{n-2} + ax^{n-3} + bx^{n-4} + \ldots + c = 0$ be $a_1, a_2, \ldots, a_n$ then $\sum_{k=1}^{n} (-1)^{k-1} a_k$ is:

$0 < 3 - 2\sqrt{2}$

If $x^2 + xy = 12$ and $2xy + 3y^2 + 5 = 0$ then $x + 4y$ is equal to: