Quadratic Equations Questions - JEE

Let $g(x_2) \leq f(x_1)$ for all $x \in [0,1]$, where $f(x)$ is an increasing function on $[0,1]$ with minimum value $f(0) = -1$. Given $h(x) = x^2 - 2ax + 5$, find the minimum value of $a$ such that $h(x) \leq 0$ for at least one $x \in [1, 2]$.

If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 + 8 = 0$, then find the equation whose roots are $\alpha^2$, $\beta^2$, and $\gamma^2$.

The number of solutions to the equation $2\sqrt{1 + \sqrt{1 + (x+1)\sqrt{1 + (x+2)\sqrt{1 + (x+3)(x+5)}}}} = x$ 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 $b =$

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 four distinct real roots is:

The quadratic $x^2 + ax + b + 1 = 0$ has roots which are positive integers, then $(a^2 + b^2)$ can be equal to

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 3 distinct real roots 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:

If $\frac{1}{\sqrt{\alpha}} + \frac{1}{\sqrt{\beta}} = -\frac{b}{a}$ and $\frac{1}{\sqrt{\alpha}} \cdot \frac{1}{\sqrt{\beta}} = \frac{1}{a}$, and $\alpha_1, \beta_1$ are roots of $x^2 + (b^3 - 3ab)x + a^3 = 0$, then the required roots of the new equation are:

If x2 + px + q = 0 and x2 + qx + p = 0, (p ≠ q) have a common root, show that 1 + p + q = 0. Also, show that their other roots are the roots of the equation x2 + x + pq = 0.

Let $\alpha$ and $\beta$ be the real roots of the equation $x^2 - x(k - 2) + \left(k^2 + 3k + 5\right) = 0$. The maximum value of $\alpha^2 + \beta^2$ is:

For the equation $4x^2 - 16x + c = 0$, find the range of $c$ such that both roots are real and lie in the interval $(1, 3)$.

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

If x and y are positive integers such that xy + x + y = 71 and x2y + xy2 = 880, then x2 + y2 is equal to

Let $-\frac{\pi}{6} \leq I \leq \frac{\pi}{12}$. Suppose $r_1$ and $s_1$ are the roots of equation $x^2 - 2x\sec I + 1 = 0$ and $r_2$ and $s_2$ are the roots of equation $x^2 + 2x\tan I - 1 = 0$. If $r_1 > s_1$ and $r_2 > s_2$, then $r_1 + s_2$ equals

If α and β (α ≠ β) are the roots of the equation x2 + bx + c = 0, where c ≠ 0 ≠ b, then

The solution of the equation $\frac{8}{[x]} - \frac{9}{x} + \frac{10}{\{x\}} = 2$ is of the form $\frac{k+1}{k}, k \in \mathbb{N}$ then $k = $ _______. ($[x]$ denotes largest integer less than or equal to $x$, and $\{x\}$ denotes fractional part of $x$)

The value of 'a' so that the equation $x^3 - 6x^2 + 11x + a - 6 = 0$ has exactly three integer solutions is _______.

If f(x) = ax2 − bx + c has two distinct roots α and β, and f(0) and f(1) are of the same sign, with the constraint α(1 − α) ≤ 1/4, find the least value of b.

Remainder when $P(x^5)$ is divided by $P(x) = x^4 + x^3 + x^2 + x + 1$ is _______.

If $a,b,c \in \mathbb{R}, a > 10$ and $(x-a)(x-12) + 2 = (x+b)(x+c)$ for all $x \in \mathbb{R}$ then $|b-c| = $ _______.

For real $a,b,c, a+b+c = 2, a^2 + b^2 + c^2 = 6$ and $a^3 + b^3 + c^3 = 8$ then $(1-a)(1-b)(1-c) = $ _______.

Let $f(x) = x^2 + bx + c, b,c \in \mathbb{R}$. If $f(x)$ is a factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^2 + 28x + 5$, then the minimum value of $f(x)$ is _______.

Let $p$ be an integer such that both roots of the equation $5x^2 - 5px + (66p-1) = 0$ are positive integers. Then the value of $\left\lfloor\frac{p}{10}\right\rfloor$ is equal to ($\lfloor . \rfloor$ denotes greatest integer function)

Suppose $A = \{x: 5x - a \leq 0\}$, $B = \{x: 6x - b > 0\}$, $a, b \in \mathbb{N}$ and $A \cap B \cap \mathbb{N} = \{2, 3, 4\}$. The number of such pairs $(a, b)$ is:

The number of real solutions to the equation $\sqrt{3x^2 - 18x + 52} + \sqrt{2x^2 - 12x + 162} = \sqrt{-x^2 + 6x + 280}$ is(are):

The number of monic quadratic polynomials of the form $x^2 + ax + b$ with integer roots, where $1, a, b$ are in AP is(are):

A quadratic equation is chosen from the set of all quadratic equations which are unchanged by squaring their roots. The chance that the chosen equation has equal roots, is

If α, β, γ are the roots of $x^3 + 2x^2 - 3x - 1 = 0$, then $α^{-2} + β^{-2} + γ^{-2}$ is equal to

Given a and b are the roots of the equation $x^2 - 6x - 2 = 0$. Let $a_n = a^n - b^n$ for $n \geq 1$. Find $\frac{a_{10} - 2a_8}{2a_9}$.

Given $p + q = 2$ and $p^4 + q^4 = 272$, find the value of $pq$.

If $a \neq b$, then the roots of the equation $2(a^2 + b^2)x^2 + 2(a + b)x + 1 = 0$ are

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+2ax+(3a+10)=0$ such that $\alpha<1<\beta$. Then the set of all possible values of $a$ is:

Let $p(x) = x^6 + ax^5 + bx^4 + x^3 + bx^2 + ax + 1$. Given that 1 is a root of $p(x) = 0$ and –1 is not. What is the maximum number of distinct real roots that $p$ could have

The sum of all the real solutions of the equation $\log_{(x+3)}(6x^2+28x+30)=5-2\log_{(6x+10)}(x^2+6x+9)$ is equal to

Let $x^2 - ax + 30 = y$ and $y = 2\sqrt{y} + 15$. Given that $\lambda = \frac{\alpha + \beta}{2}$ where $\alpha, \beta$ are roots of $x^2 - ax + 20 = 0$. Find the minimum value of $\lambda$.

The least value of 'b' is equal to

The given equation is $|x - 2|^2 + |x - 2| - 2 = 0$. Solve for $x$ when $x \geq 2$.

One root of the equation $x^4 - 5x^3 + ax^2 + bx + c = 0$ is $3 + \sqrt{...}$. If all the roots of the equation are real given that $a, b, c$ are rational parameters, then the greatest value of 'a' is equal to

The quadratic equation $3x^2 + 6x + a = 0$ must have equal roots. Find the value of $a$.

If the equation E2 has equal roots, then b + q is equal to

If $\alpha$ is the root of the equation $x^2 - x + 2 = 0$, then the value of $\dfrac{6(-\alpha^3 + 2\alpha^2 - \alpha)}{\alpha^5 - 3\alpha^4 + 3\alpha^3 - \alpha^2}$ is equal to:

Let $S=\left\{x\in\mathbb{R}:(\sqrt{3}+\sqrt{2})^x+(\sqrt{3}-\sqrt{2})^x=10\right\}$. Then the number of elements in $S$ is:

If both the roots of $(6x^2 + 3) - rx + 2x^2 - 1 = 0$ and $6(2x^2 + 1) + px + 4x^2 - 2 = 0$ are common, then $2r - p$ is equal to

If a root of the equation a_1x^2 + b_1x + c_1 = 0\ is the reciprocal of a root of the equation a_2x^2 + b_2x + c_2 = 0\, then:

If a \neq b\ but a^2 = 5a - 3\ and b^2 = 5b - 3\, then the equation with roots \frac{a}{b}, \frac{b}{a}\ is:

Paragraph (Questions 11-13): $a, b, c$ are the lengths of sides $BC, CA, AB$ respectively of $\triangle ABC$ satisfying \[\log\left(1 + \frac{c}{a}\right) + \log a - \log b = \log 2\] Also, the quadratic equation $a(1-x^2) + 2bx + c(1+x^2) = 0$ has two equal roots.

For $0<c<b<a$, let $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ and $\alpha\neq1$ be one of its roots. Then, among the two statements: (I) If $\alpha\in(-1,0)$, then $b$ cannot be the geometric mean of $a$ and $c$. (II) If $\alpha\in(0,1)$, then $b$ may be the geometric mean of $a$ and $c$.

If the difference between the roots of x^2 + ax + b = 0\ is same as that of x^2 + bx + a = 0\, a \neq b\, then:

If 0 , find the range of a.

Quadratic Equations Questions (527)