Quadratic Equations Questions - JEE

For Problems 35–37Consider the equation $x^4 - \lambda x^2 + 9 = 0$.If the equation has no real root, then $\lambda$ lies in the interval

The solution set of the equation $pqx^2 - (p + q)^2 x + (p + q)^2 = 0$ is

\((\lambda^2 + \lambda - 2)x^2 + (\lambda + 2)x - 1

Given $p, q, r$ are real numbers ($p \neq q,\ r \neq 0$) and \[\frac{1}{x+p} + \frac{1}{x+q} = \frac{1}{r}\] The roots $\alpha$ and $\beta$ of the resulting quadratic satisfy which general form?

If equations $ax^2 + bx + c = 0$, $(a, b, c \in R,\ a \neq 0)$ and $2x^2 + 3x + 4 = 0$ have a common root then $a : b : c$ equals

One of the roots of $ax^2 + bx + c = 0$ is greater than 2 and the other is less than −1. If the roots of $cx^2 + bx + a = 0$ are α and β, then

For how many values of $n$ in the range $n \in [5, 100]$ is $D$ a perfect square, where $D$ is the discriminant of $x^2 + 2x - n = 0$?

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

For the quadratic $ x^2 + (n+1)x + (n+2) = 0 $ (where n is an integer), the discriminant must be a perfect square and positive. Given $ D = (n+1)^2 - 4n(n+2) = -3n^2 - 6n + 1 = 4 - 3(n+1)^2 $, find the possible values of n.

If $p^3 + q^3 + q(1 - 3p) = 0$, find $(\alpha + \alpha^3)^2$ where $\alpha$ and $\alpha^3$ are roots of $x^2 + px - q = 0$.

Given the quadratic equation $81x^2 + kx + 256 = 0$ and one real root is the cube of the other root. Find the value of $k$.

Find the value of a for which one root of the quadratic equation $(a^2 - 5a + 3)x^2 + (3a-1)x + 2 = 0$ is twice as large as the other.

Given the equation $x^2 - 1154x + 1 = 0$ with roots $a$ and $b$, find $a + b$ and $ab$.

If $\alpha$ and $\beta$ are roots of $x^2 - 2px + p^2 - 1 = 0$ and $\left|\frac{\alpha^2+\beta^2}{\alpha\beta}+3\right| \leq 5$, then $p \in [a,\,b]$. Find the value of $[2(a^2+b^2)]$.

If $a$, $b$ and $c$ are side lengths of a triangle $ABC$ such that $x^2 - 2(a+b+c)x + 3k(ab+bc+ca) = 0$, where \(k

If the roots of equation $x^2 - bx + m - 1 = 0$ are equal but opposite in sign, then the value of m will be

83. If $a, b, c$ are distinct positive numbers, then the nature of roots of the equation $\dfrac{1}{x-a} + \dfrac{1}{x-b} + \dfrac{1}{x-c} = \dfrac{1}{x}$ is

Let f(x) be a quadratic expression positive for all real x. If g(x) = f(x) − f'(x) + f''(x), then for any real x

Difference between the corresponding roots of $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ is same and $a \neq b$, then

84. For $x^2 - (a+3)|x| + 4 = 0$ to have real solutions, the range of $a$ is

The coefficient of x in the equation $x^2 + px + q = 0$ was taken as 17 in place of 13 its roots were found to be -2 and -15. The roots of the original equation are

The number of values of a for which equations $x^3 + ax + 1 = 0$ and $x^4 + ax^2 + 1 = 0$ have a common root is

Put g(x) = y = x2 − 2. If the roots of y5 + 20y4 + 40y3 + 79y2 + 74y + 23 = 0 are g(x1), g(x2), g(x3), g(x4), g(x5), find the value of g(x1) · g(x2) · g(x3) · g(x4) · g(x5) − 30g(x1x2x3x4x5).

The polynomial $f(x) = x^4 + ax^3 + bx^2 + cx + d$ has real coefficients and $f(2i) = f(2+i) = 0$. Find the value of $(a + b + c + d)$.

The quadratic equation $x^2 + bx + c = 0$ has distinct roots. If 2 is subtract from each root then result are the reciprocal of the original root. The value of $(b^2 + c^2)$ is:

Let $f(x) = (1 + b^2)x^2 + 2bx + 1$ and let $m_b()$ be the minimum value of $f(x)$. As b varies, the range of $m_b()$ 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

Let n ∈ Z and △ABC be a right triangle with right angle at C. If sin A and sin B are the roots of the quadratic equation $(5n+8)x^2 - (7n-20)x + 120 = 0$, then find the value of n.

If $(a+1)(b+1)(c+1)(d+1) = 1$, $(a+2)(b+2)(c+2)(d+2) = 2$, $(a+3)(b+3)(c+3)(d+3) = 3$, $(a+4)(b+4)(c+4)(d+4) = 4$. Then the value of $(a+5)(b+5)(c+5)(d+5)$ is

Given that the equation $x^2 - ax + 3 - b = 0$ has two distinct real roots, $x^2 + (6-a)x + 6 - b = 0$ has two equal real roots, and $x^2 + (4-a)x + 5 - b = 0$ has no real roots. Then the ranges of a and b are

If \tan\theta_i; i = 1, 2, 3, 4\ are the roots of equation x^4 - x^3\sin 2\beta + x^2\cos 2\beta - x\cos\beta - \sin\beta = 0\, then \tan(\theta_1 + \theta_2 + \theta_3 + \theta_4) =\

If $ f(x) \geq -3 $ for all $ x \in \mathbb{R} $, and the inequality $ x^2 + 2px + 4p + f(x) \geq 0 $ holds for all $ x \in \mathbb{R} $, find the range of $ p $ such that $ p^2 - 4p + 3 , i.e., \( 1 \leq p \leq 3 $. How many integer values does $ p $ take?

If $p$ and $q$ are the roots of the equation $x^2 + px + q = 0$, then

For polynomials of the form $a_n x^n + a_{n-1}x^{n-1} + \ldots + a_1 x + a_0$ with $a_i \in \{-1, 1\}$, $(i = 0, 1, 2, \ldots, n)$ which has all roots real, find the maximum value of $n$.

Consider the inequality $9 - x^2 > |x| - a$, where $a$ is a real number. The complete set of values of $a$ for which the given inequality has at least one negative solution is

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^4 - Kx^3 + Kx^2 + Lx + M = 0$, where $K, L,$ and $M$ are real numbers, then the minimum value of $\alpha^2 + \beta^2 + \gamma^2 + \delta^2$ is

If $x$ is real, then $x/(x^2 - 5x + 9)$ lies between

If a and b are the roots of the equation 7x2 – 3x – 2 = 0, then the value of $\frac{a}{1-a^2} + \frac{b}{1-b^2}$ is equal to

If $ \alpha, \beta $ and $ \gamma $ are roots of $ x^3 - 2x^2 + 6x - 1 = 0 $, find the value of the following expression:\[ \alpha\left(\dfrac{\alpha^2+\alpha+1}{\alpha^2-\alpha+1}\right) + \beta\left(\dfrac{\beta^2+\beta+1}{\beta^2-\beta+1}\right) + \gamma\left(\dfrac{\gamma^2+\gamma+1}{\gamma^2-\gamma+1}\right) \]

If the inequality $x^2 + ax + a^2 + 6a < 0$ is satisfied for all $x \in (1,2)$, then the sum of all the integral values of $a$ must be equal to

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) $a^2 + b^2 + c^2 - ab - bc - ca \le 0$(II) $a^2 + b^2 + c^2 + ab + bc + ca \le 0$(III) $3(a^2 + b^2 + c^2 + 1) \le 2(a + b + c + ab + bc + ca)$(IV) $a^2 + b^2 + c^2 \le 2a + 6b + 4c + 14$Column-II(i) 0 (ii) 1 (iii) 2 (iv) $\infty$Column-III(P) 0 (Q) 1 (R) 3 (S) 5Q. 665. Which of the following is correct combination?

The sum of the solutions of the equation $\left|\sqrt{x} - 2\right| + \sqrt{x}(\sqrt{x} - 4) + 2 = 0$, $(x > 0)$ is equal to __________.

If $2b^2 + 27d = 9bc$, find the relationship using roots.

If $\alpha, \beta$ are the roots of $x^2 - px + q = 0$ and $\alpha', \beta'$ are the roots of $x^2 - p'x + q' = 0$, then the value of $(\alpha - \alpha')^2 + (\beta - \alpha')^2 + (\alpha - \beta')^2 + (\beta - \beta')^2$ is

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) 5Q665. Which of the following is correct combination?

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

Let $A = [-2, 4]$, $B = \{x: x^2 - ax - 4 \leq 0\}$. If $B \subseteq A$, then the range of real $a$ is:

If $(x+1)$ is a factor of $x^3 + kx^2 - 3x + k + 2$, then $k$ is equal to:

The number of solutions of $\left(\dfrac{9}{x} - \dfrac{9}{\sqrt{x}} + 2\right)\left(\dfrac{2}{x} - \dfrac{7}{\sqrt{x}} + 3\right) = 0$ is

The number of solutions of the equation $\left(\dfrac{9}{x}-\dfrac{9}{\sqrt{x}}+2\right)\!\left(\dfrac{2}{x}-\dfrac{7}{\sqrt{x}}+3\right)=0$ is:

Quadratic Equations Questions (527)