Quadratic Equations Questions - JEE

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 $f(f(f(x))) = 0$?

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) 5Q664. Which of the following is incorrect combination?

Let $m$ and $n$ be the numbers of real roots of $x^2-12x+[x]+31=0$ and $x^2-5|x+2|-4=0$ respectively. Then $m^2+mn+n^2$ is equal to

The sum of all the roots of $|x^2-8x+15|-2x+7=0$ is

The number of real solutions of the equation $x|x+5|+2|x+7|-2=0$ is

The number of real solutions of the equation $x(x^2+3|x|+5|x-1|+6|x-2|)=0$ is

In $x^2 - nz + 2014 = 0$, where $\alpha + \beta = n$ and $\alpha\beta = 2014$. Find the sum $\alpha + \beta + \gamma$.

Since $2x^2 + 3x + 4 > 0$ (as discriminant \(= 9 - 32 = -23

The sum of the squares of all the roots of the equation $x^{2}+|2x-3|-4=0$ is:

If $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$ have a common root $\alpha \neq 0$, then $\frac{a^3 + b^3 + c^3}{abc}$ is equal to

If $ a + b + c = 0 $, then the value of $(x-1)^3 + (2x-1)^3 + (2-3x)^3$ equals:

70. The set of values of $a$ for which $(a-1)x^2 - (a+1)x + a - 1 \geq 0$ is true for all $x \geq 2$ is

If a, b, c, d are four consecutive terms of an increasing AP, then the roots of the equation $(x-a)(x-c) + 2(x-b)(x-d) = 0$ are

$P(x)$ is a polynomial with integral coefficients such that for four distinct integers $a, b, c, d$, $P(a) = P(b) = P(c) = P(d) = 3$. If $P(e) = 5$ ($e$ is an integer), then

If $\alpha, \beta, \gamma, \sigma$ are the roots of the equation $x^4 + 4x^3 - 6x^2 + 7x - 9 = 0$, then the value of $(1 - \alpha^2)(1 + \beta^2)(1 + \gamma^2)(1 + \sigma^2)$ is

Set of all real values of $a$ such that $f(x) = \dfrac{(2a-1)x^2 + 2(a+1)x + (2a-1)}{x^2 - 2x + 40}$ is always negative is

For Problems 28–30: The numbers $a$, $b$, and $c$ are between 2 and 18, such that (i) their sum is 25, (ii) the numbers 2, $a$, and $b$ are consecutive terms of an A.P., (iii) the numbers $b$, $c$, 18 are consecutive terms of a G.P.Roots of the equation $ax^2 + bx + c = 0$ are

If $\alpha$, $\beta$, $\gamma$ are the roots of the equation $x^3 - px + q = 0$, then find the cubic equation whose roots are $\alpha(1+\alpha)$, $\beta(1+\beta)$, $\gamma(1+\gamma)$.

If α, β are the roots of $x^2 + px + q = 0$ and γ, δ are the roots of $x^2 + px + r = 0$, then $\dfrac{(\alpha-\gamma)(\alpha-\delta)}{(\beta-\gamma)(\beta-\delta)} =$

If one root of $x^2 - x - k = 0$ is square of the other, then $k =$

If α and β be the roots of the equation $x^2 + px - 1/(2p^2) = 0$, where $p \in \mathbb{R}$. Then the minimum value of $\alpha^4 + \beta^4$ is

A quadratic equation with integral coefficients has two different prime numbers as its roots. If the sum of the coefficients of the equation is prime, then the sum of the roots is

Let $a, b$, and $c$ be real numbers such that $4a + 2b + c = 0$ and $ab > 0$. Then the equation $ax^2 + bx + c = 0$ has

If the roots of equation $(a-1)(x^2 + x + 1)^2 = (a+1)(x^4 + x^2 + 1)$ are real and distinct, then the value of $a$ $\in$

For Problems 33 and 34The real numbers $x_1, x_2, x_3$ satisfying the equation $x^3 - x^2 + \beta x + \gamma = 0$ are in A.P.All possible values of $\gamma$ are

Determine the values of m for which the equations 3x2 + 4mx + 2 = 0 and 2x2 + 3x − 2 = 0 may have a common root.

If $\alpha$, $\beta$ and $\gamma$ are the roots of the equation $x^3 + 3x^2 - 4x - 2 = 0$, then find the values of the following expressions:(i) $\alpha^2 + \beta^2 + \gamma^2$(ii) $\alpha^3 + \beta^3 + \gamma^3$(iii) $\dfrac{1}{\alpha} + \dfrac{1}{\beta} + \dfrac{1}{\gamma}$

If α, β are the roots of $x^2 - 3x + 1 = 0$, then the equation whose roots are $\frac{1}{\alpha + 2}$, $\frac{1}{\beta + 2}$, is

If the roots of $(a^2 + b^2)x^2 - 2(bc + ad)x + c^2 + d^2 = 0$ are equal, then

Solve the equation $x(x+2)(x^2-1)=-1$.

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:

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

If $a(p+q)^2 + 2bpq + c = 0$ and $a(p+r)^2 + 2bpr + c = 0$ $(a \neq 0)$, then

If $\alpha, \beta, \gamma$ are the roots of $x^3 - x^2 - 1 = 0$, then the value of $\dfrac{1+\alpha}{1-\alpha} + \dfrac{1+\beta}{1-\beta} + \dfrac{1+\gamma}{1-\gamma}$ is equal to

The smallest positive integral value of a for which the greater root of the equation $x^2 - (a^2 + a + 1)x + a(a^2 + 1) = 0$ lies between the roots of the equation $x^2 - a^2x - 2(a^2 - 2) = 0$, is less than:

73. Given $x, y \in \mathbb{R}$, $x^2 + y^2 > 0$. Then the range of $\dfrac{x^2 + y^2}{x^2 + xy + 4y^2}$ is

For Problems 13–15Suppose $f(x)$ is a function satisfying the following conditions:(i) $f(0) = 2,\ f(1) = 1$,(ii) $f$ has a minimum value at $x = 5/2$,(iii) For all $x$,\[f'(x) = \begin{vmatrix} 2ax & 2ax-1 & 2ax+b+1 \\ b & b+1 & -1 \\ 2(ax+b) & 2ax+2b+1 & 2ax+b \end{vmatrix}\]$f(x) = 0$ has

If $x^2 - x\sin 2\theta + 2\cos^2\theta = (x-\alpha)(x-\beta)$, then the maximum value of $(2-\alpha)(2-\beta)$ is $5 + \sqrt{a}$. Find $a$.

The equation $(x^2 + x + 1)^2 + 1 = (x^2 + x + 1)(x^2 - x - 5)$ for $x \in (-2, 3)$ will have number of solutions.

If $x^2 + 3x + 1 + \lambda(x+1) > -10$ for all $x \in \mathbb{R}$, and $\lambda$ is an integer, find the sum of all integer values of $\lambda$.

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - mx + 2 = 0$ and $\alpha + \dfrac{1}{\beta}$, $\beta + \dfrac{1}{\alpha}$ are the roots of the equation $x^2 - px + q = 0$, then the value of $2q$ equals:

If α, β are the roots of the equation $x^2 - 2x + 3 = 0$. Then the equation whose roots are $P = \alpha^3 - 3\alpha^2 + 5\alpha - 2$ and $Q = \beta^3 - \beta^2 + \beta + 5$ is

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

The number of values of k for which $[x^2 - (k-2)x + k^2] \times [x^2 + kx + (2k-1)]$ is a perfect square is

If $b_1 b_2 = 2(c_1 + c_2)$, then at least one of the equations $x^2 + b_1 x + c_1 = 0$ and $x^2 + b_2 x + c_2 = 0$ has

86. If the equation $x^2 + ax + b = 0$ has distinct real roots and $x^2 + a|x| + b = 0$ has only one real root, then which of the following is true?

Find the number of quadratic equations with real roots which remain unchanged even after squaring their roots.

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

What is the minimum height of any point on the curve $y = x^2 - 4x + 6$ above the $x$-axis?

The value of the expression $x^4 - 8x^3 + 18x^2 - 8x + 2$, when $x = 2 + \sqrt{3}$, is

Quadratic Equations Questions (527)