Quadratic Equations Questions - JEE

Which of the following equations has maximum number of real roots?

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:

The quadratic equation $p(x) = 0$ with real coefficients has purely imaginary roots. Then the equation $p(p(x)) = 0$ has

If roots of an equation $x^n - 1 = 0$ are $1, a_1, a_2, \ldots, a_{n-1}$, then the value of $(1 - a_1)(1 - a_2)(1 - a_3)\cdots(1 - a_{n-1})$ will be

For Problems 35–37Consider the equation $x^4 - \lambda x^2 + 9 = 0$.If the equation has four real and distinct roots, then $\lambda$ lies in the interval

If the two roots of the equation, $(a-1)(x^4 + x^2 + 1) + (a+1)(x^2 + x + 1)^2 = 0$ are real and distinct, then the set of all values of $a$ is

If α, β are the roots of $ax^2 + bx + c = 0$ and $\alpha+h$, $\beta+h$ are the roots of $px^2 + qx + r = 0$, then h =

If $\alpha$ and $\beta$ are the roots of the equation $8x^2 - 3x + 27 = 0$, then the value of $\left(\frac{\alpha}{\beta}\right)^{1/2} + \left(\frac{\beta}{\alpha}\right)^{1/2}$ is

Find the number of pairs (a, b) of real numbers such that whenever $\alpha$ is a root of $x^2 + ax + b = 0$, $\alpha^2 - 2$ is also a root of the equation.

A value of b for which the equations $x^2 + bx - 1 = 0$ and $x^2 + x + b = 0$ have one root in common is

a, b, c are real numbers with $a^2 + b^2 + c^2 > 0$. Then the equation $x^2 + (a+b+c)x + (a^2+b^2+c^2) = 0$ has

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) 5Q663. Which of the following is correct combination?

If $a, b \in \mathbb{R}$, $a \neq 0$ and the quadratic equation $ax^2 - bx + 1 = 0$ has imaginary roots, then $(a + b + 1)$ is

Let α, β be the roots of x2 + bx + 1 = 0. Then find the equation whose roots are −(α + 1/β) and −(β + 1/α).

If $\alpha, \beta$ are the roots of the equation $ax^2 + bx + c = 0$, then the value of $\dfrac{(a\alpha^2 + c)}{(a\alpha + b)} + \dfrac{(a\beta^2 + c)}{(a\beta + b)}$ is

Consider the equation $x^2 + 2x - n = 0$, where $n \in \mathbb{N}$ and $n \in [5, 100]$. The number of different values of $n$ so that the given equation has integral roots, is

If $x = \alpha$ is the common root of both the equations, then find $P(1)$ where $P(x) = x^2 - 2x + 5$.

Given equation, $4(z^2 + \frac{1}{z^2}) + 16(z + \frac{1}{z}) - 57 = 0$. Find $x$ such that $z$ is rational.

If $\alpha \neq \beta$ but, $\alpha^2 = 4\alpha - 2$ and $\beta^2 = 4\beta - 2$, then the quadratic equation with roots $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ is

If $f(x) = 4x^2 - 20kx + (25k^2 + 15k - 66) = 0$ has roots $\alpha, \beta$ that are real, and the parabola opens upward with $\alpha, \beta < -1$, find the range of $k$.

If $f(x) = ax^2 + bx + c$ has an upward-opening parabola with roots between which $f(2) < 0$, and $f(-1) = 4(2k + 4) < 0$, find the range of $k$.

If $2 + 3i$ is one of the roots of the equation $2x^3 - 9x^2 + kx - 13 = 0$, $k \in \mathbb{R}$, then the real root of this equation

The least non-negative integral value of $\lambda$ for which the equation $2x^2 - 2(2\lambda + 1)x + \lambda(\lambda + 1) = 0$ has one root less than $\lambda$ and other root greater than $\lambda$, is equal to:

Solve $\sqrt{5x^2-6x+8}-\sqrt{5x^2-6x-7}=1$.

Let p and q be real numbers such that p ≠ 0, p3 ≠ q, and p3 ≠ −q. If α and β are nonzero complex numbers satisfying α + β = −p and α3 + β3 = q, then a quadratic equation having α/β and β/α as its roots is

Find all real numbers $a$ for which the equation $x^2 + (a-2)x + 1 = 3|x|$ has exactly three distinct real solutions in x.

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:

Given equation is $[1 - a^2]x^2 + 2ax - 1 = 0$. Its discriminant $D = 4$ and roots are $-\frac{1}{a+1}, -\frac{1}{a-1}$. Find the range of $a$ for which this holds.

Solve the equation $(1 + i)x^2 + (1 - i)x - 2i = 0$ and find $|\alpha - \beta|^2$ where $\alpha, \beta$ are the roots.

If

Find the value of m for which the expression 12x² − 10xy + 2y² − 11x − 5y + m can be resolved into two rational linear factors.

If the roots of the equation \frac{1}{x+p} + \frac{1}{x+q} = \frac{1}{r}\ are equal in magnitude but opposite in sign, then the product of the roots is:

When $x > \frac{3}{2}$, the equation $x^2 + 2x - 7 = 0$ gives roots $x = -1 \pm 2\sqrt{2}$. For \(x

68. If \((b^2 - 4ac)^2(1 + 4a^2)

The smallest positive integral value of $a$, for which all the roots of $x^4-ax^2+9=0$ are real and distinct, is equal to

Let $f(x) = x^2 + ax + b; a, b \in \mathbb{R}$. If $f(1) + f(2) + f(3) = 0$, then the roots of the equation $f(x) = 0$

If $\alpha$ and $\beta$ are the roots of equation $2x^2 - 5x + 7 = 0$, then the equation whose roots are $2\alpha + 3\beta$ and $3\alpha + 2\beta$ is

Let $p, q \in \mathbb{R}$. If $2 - \sqrt{3}$ is a root of the quadratic equation $x^2 + px + q = 0$, then

Find the number of integral values of $\lambda$ such that \((\lambda^2 + \lambda - 2)x^2 + (\lambda + 2)x

Find the number of positive integral values of k for which \(kx^2 + (k - 3)x + 1 x.

If f(x) = ax^2 + bx + c, where a, b, c \in \mathbb{R} and the equation f(f(x)) - x = 0 has imaginary roots \alpha, \beta, and \gamma and \delta be the roots of f(f(x)) - x = 0, then \frac{\beta^2\alpha + \delta}{\gamma\beta + 1} is

The number of integral values of $a$ for which the quadratic equation $(x - a)(x - 1991) + 1 = 0$ has integral roots are

Suppose that $f(x)$ is a quadratic expression positive for all real $x$. If $g(x) = f(x) + f'(x) + f''(x)$, then for any real $x$ (where $f'(x)$ and $f''(x)$ represent 1st and 2nd derivative, respectively)

The inequality \frac{x + 3}{x^2 - x - 2} \geq \frac{1}{x - 4}\ holds for all x satisfying:

81. The set of all possible real values of $a$ such that the inequality \((x - (a-1))(x - (a^2 + 2))

Solution set of the equation 3^{2x^2} - 3^{x^2 + x + 6} + 3^{2(x + 6)} = 0 is

74. $x_1$ and $x_2$ are the roots of $ax^2 + bx + c = 0$ and \(x_1 x_2

Given that α, β, γ are all real roots of the equation x3 − 2007x + 2002 = 0, then the value of $\frac{α − 1}{α + 1} + \frac{β − 1}{β + 1} + \frac{γ − 1}{γ + 1}$ is equal to

The solution set of the equation $(\cos p - 1)x^2 + (\cos p)x + \sin p = 0$ are real, then

If $x, y$ and $z$ are real such that $x + y + z = 4$, $x^2 + y^2 + z^2 = 6$, find the range of $x$.

Quadratic Equations Questions (527)