Quadratic Equations Questions - JEE

If ax2 + bx + c = 0 and bx2 + cx + a = 0 have a common root and a, b, and c are nonzero real numbers, then find the value of $\dfrac{a^3 + b^3 + c^3}{abc}$.

Let $ p(x) = 51x^2 + mx + c $ and $ q(x) = 3x^2 + bx + a $ are two quadratic polynomials with integer coefficients such that $ p(r) = q(r) = 0 $. If $ r $ is an irrational number, then the value of $ \dfrac{c}{a} $ is:

Let $r$, $s$, and $t$ be the roots of equation $8x^3 + 1001x + 2008 = 0$. Then find the value of $(r+s)^3 + (s+t)^3 + (t+r)^3$.

The second degree polynomial $f(x)$, satisfying $f(0) = 0$, $f(1) = 1$, $f'(x) \geq 0; x \in (0, 1)$ is

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:

If $\alpha,\beta$, where $\alpha<\beta$, are the roots of the equation $\lambda x^2-(\lambda+3)x+3=0$ such that $\dfrac{1}{\alpha}-\dfrac{1}{\beta}=\dfrac{1}{3}$, then the sum of all possible values of $\lambda$ is

If $x^2 + ax - 3x - (a+2) = 0$ has real and distinct roots, then the minimum value of $(a^2+1)/(a^2+2)$ is

If $(m_r, 1/m_r)$, $r = 1, 2, 3, 4$, are four pairs of values of $x$ and $y$ that satisfy the equation $x^2 + y^2 + 2yx + 2fy + c = 0$, then the value of $m_1 \cdot m_2 \cdot m_3 \cdot m_4$ is

Let $f(x) = x^2 - 2px + p^2 - 1$, where $p \in R - \{-1, 1\}$. If $\alpha$ and $\beta$ are distinct real roots of the equation $f(x) = 0$ such that $\left|\dfrac{\alpha^2 + \beta^2 + 3\alpha\beta}{\alpha\beta}\right| \leq 5$, then set of values of $p \in [a, b]$. The value of $[2(a^2 + b^2)]$ is:[Note: $[k]$ denotes greatest integer less than or equal to $k$.]

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - x + 1 = 0$, then $\alpha^{2009} + \beta^{2009} =$

327. 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:

If G and L are the greatest and least values of the expression $\frac{x^2 - x + 1}{x^2 + x + 1}$, $x \in \mathbb{R}$ respectively, then the least value of $G^5 + L^5$ is

Find the sum of all integral values of $a$ for which all the roots of the equation $x^4 - 4x^3 - 8x^2 + a = 0$ are real.

Let $\alpha$ and $\beta$ be the roots of equation $x^2 - 6x - 2 = 0$. If $a_n = \alpha^n - \beta^n$, for $n \geq 1$, then the value of $\dfrac{a_{10} - 2a_8}{2a_9}$ is equal to

Let $p(x) = 0$ be a polynomial equation of the least possible degree, with rational coefficients, having $\sqrt[3]{7} + \sqrt[3]{49}$ as one of its roots. Then the product of all the roots of $p(x) = 0$ is

For Problems 35–37Consider the equation $x^4 - \lambda x^2 + 9 = 0$.If the equation has only two real roots, then the set of values of $\lambda$ is

Number of integral points (x, y) in the 1st quadrant that satisfy the equation $y^4 + 6xy^2 - 8x = 0$ is equal to

Find the values of the parameter a such that the roots $\alpha$ and $\beta$ of the equation $2x^2 + 6x + a = 0$ satisfy the inequality \(\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha}

The number of real solutions of the equation $(9/10)^x = -3 - x - x^2$ is

Let x₁, x₂, x₃ be the roots of the equation x³ + 3x + 5 = 0. Then the value of the expression is equal to

Question nos. 690 to 692Column-1 represents a quadratic equation with some given conditions. Column-2 represents number of non-positive integral values of 'k' and column-3 represents number of prime values of 'k'. Then match the following.Column-1Column-2Column-3(I) Let α and β are real roots of $x^2 - 8x + k^2 - 6k = 0$ such that $\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha} = 2$.(i) 0(P) 0(II) If one root of the equation $(k-2)x^2 - (8-2k)x + (3k+8) = 0$ is negative and other is positive.(ii) 1(Q) 1(III) If difference between the real roots of equation $4x^2 - 2kx + 1 = 0$ is less than $\sqrt{3}$.(iii) 2(R) 2(IV) If quadratic expression $2kx^2 - (4k-5)x - 10$ is negative for exactly three distinct integral values of $x$.(iv) 3(S) 3Which of the following options is the only correct combination?

Solve $\dfrac{x^2+3x+2}{x^2-6x-7}=0$.

If the roots of the equation $10x^3 + cx^2 + 54x + 27 = 0$ are in harmonic progression, find the value of $c$.

$a, b, c, d$ are distinct integers such that $(x - a)(x - b)(x - c)(x - d) = 4$ has an integral root $r$. Then $a + b + c + d$ is equal to:

The quadratic equations $x^2 - 6x + a = 0$ and $x^2 - cx + 6 = 0$ have one root $\alpha$ in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then equation $x^2 - 3x - 4 = 0$ has

If complex numbers satisfying α + β = –p and α3 + β3 = q, then a quadratic equation having α3 and β3 as its roots is

If x = 1 and x = 2 are solutions of equations $x^3 + ax^2 + bx + c = 0$ and $a + b = 1$, then find the value of b.

If positive numbers $a, b, c$ are in H.P., then equation $x^2 - kx + 2b^{101} - a^{101} - c^{101} = 0$ ($k \in R$) has

For the quadratic equation 6x2 − 11x + a = 0, find the number of integer values of a for which the roots are rational.

The minimum possible value of a is

Let a, b, c ∈ ℝ such that two of them are equal and satisfy 2abcbc2ac2ab = 0, then equation 24ax2 + 4bx + c = 0 has

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 $\beta$ are

The quadratic equation $x(x+1) + (x+1)(x+2) + \cdots + [x+(n-1)](x+n) = 10n$ has two consecutive integral solutions. Find the value of $n$.

If the roots of the equation $x^2 + 2ax + b = 0$ are real and distinct and they differ by at most $2m$, then $b$ lies in the interval

The total number of integral values of $a$ so that $x^2 - (a+1)x + a - 1 = 0$ has integral roots is equal to

If α, β are the roots of the equation ax2 + bx + c = 0 and An = αn + βn, then aAn+2 + bAn+1 + cAn is equal to

If x is real, the maximum value of $\dfrac{3x^2+9x+17}{3x^2+9x+7}$ is

Let $ p(x) = 51x^2 + mx + c $ and $ q(x) = 3x^2 + bx + a $ are two quadratic polynomials with integer coefficients such that $ p(r) = q(r) = 0 $. If $ r $ is an irrational number, then the value of $ \dfrac{c}{a} $ is:

For a constant k, the two roots of the quadratic equation $3x^2 - x + k = 0$ are $\sin\theta$ and $\cos\theta$. The value of $54(\sin^3\theta + \cos 3\theta)$ is:

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

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

If $\alpha \neq \beta$ but $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$ then the equation having $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ as its roots is

If x2 + 3x + 5 = 0 and ax2 + bx + c = 0 have a common root and a, b, c ∈ ℕ, the minimum value of a + b + c is

Let the original equation have two roots $\alpha$ and $\beta$. Then $\alpha\beta = \alpha^2\beta^2$ ... (i) and $\alpha^2 + \beta^2 = \alpha + \beta$ ... (ii). Find the number of quadratic equations satisfying these conditions.

The equation formed by decreasing each root of $ax^2 + bx + c = 0$ by 1 is $2x^2 + 8x + 2 = 0$, then

Number of integral value(s) of k for which the equation 4x2 − 16x + k = 0 has one root lie between 1 and 2 and other root lies between 2 and 3, is

Solve for x, $(x^2 + 3x + 1)(x^2 + 3x - 3) \geq 5$

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 α(1 − α) ≤ 1/4 for all α ∈ (0,1), find the least value of a.

If x_1 and x_2 are the roots of x^2 + (1 - \sin \theta)x - \frac{\cos^2 \theta}{2} = 0, then the maximum value of x_1^2 + x_2^2 is

If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, then the value of q is

Quadratic Equations Questions (527)