Quadratic Equations Questions - JEE

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.

Solve the inequality: $(k-2)x^2 + 8x + (k+4) > 0$ for all $x \in \mathbb{R}$Find the least integral value of $k$.

If the roots of the equation $x^2 - 10cx - 11d = 0$ are $a, b$ and those of $x^2 - 10ax - 11b = 0$ are $c, d$, then find the value of $a + b + c + d$. ($a, b, c, d$ are distinct numbers)

If $x^2 + 2ax + 10 - 3a > 0$ for all $x \in \mathbb{R}$, then

If (1 – p) is a root of quadratic equation x2 + px + (1 – p) = 0 then its roots are

(a) The harmonic mean of the roots of the equation $x^2 - (\sqrt{8} + 2\sqrt{2})x + 8 + 2\sqrt{2} = 0$ is

A value of b for which the equationsx2 + bx – 1 = 0x2 + x + b = 0have one root in common is

Complete set of real values of k for which the inequality kx² – kx – 1 x, satisfy

If $a, b, c$ are distinct positive real numbers and $a^2 + b^2 + c^2 = 1$ then $ab + bc + ca$ is

If a, b, c, p, q, r are non-zero real numbers, such that a < b < c and\[f(x) = (x - a)(x - b)(x - c) - p^2(x - a) - q^2(x - b) - r^2(x - c),\]then $f(x) = 0$ must have

Let α1 and β1 be the roots of the equation x2 + 2x − 1 = 0, and α2 and β2 be the roots of the equation x2 + 2xtanθ − 1 = 0. If α1 > β1 and α2 > β2, then α1 + β2 equals

Three real numbers $x, y, z$ are such that $x^2 + 6y = -17$, $y^2 + 4z = 11$ and $z^2 + 2x = 2$.Then the value of $x^3 + y^3 + z^3$ is equal to

Find the sum of the eighth powers of all roots of the equation $x^3 - x + 1 = 0$

Find the set of all possible real values of a such that the inequality $(x - (a-1))(x - (a^2 + 2)) < 0$ holds for all $x \in (-1, 3)$.

If the roots of the equation $ax^2 + bx + c = 0$ are of the form $\dfrac{k+1}{k}$ and $\dfrac{k+2}{k+1}$, then $(a+b+c)^2$ is equal to

If $\alpha$ and $\beta$ ($\alpha<\beta$) are the roots of the equation $(-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6\sqrt{x})+(9-2\sqrt{3})=0$, $x\geq0$, then $\sqrt{\dfrac{\beta}{\alpha}}+\sqrt{\alpha\beta}$ is equal to:

If roots of the equation $\dfrac{1}{x-a} + \dfrac{1}{x-b} + \dfrac{1}{x-c} + \dfrac{1}{x-d} + \dfrac{(x-2)(x^2+2x+4)}{(x-a)(x-b)(x-c)(x-d)} = 0$ are $\alpha$, $\beta$ and $\gamma$, then sum of the roots of the equation $5(x-\alpha)(x-\beta)(x-\gamma) + 8 - x^3 = 0$ is:

The curve $y = (\lambda + 1)x^2 + 2$ intersects the curve $y = \lambda x + 3$ in exactly one point, if $\lambda$ equals

Let $S=\{x^3+ax^2+bx+c:a,b,c\in\mathbb{N}$ and $a,b,c\leq20\}$ be a set of polynomials. Then the number of polynomials in $S$, which are divisible by $x^2+2$, is

If the equations $x^2 + 2x + 3 = 0$ and $ax^2 + bx + c = 0$, $a, b, c \in \mathbb{R}$, have a common root, then $a : b : c$ 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) $a^2 + b^2 + c^2 - ab - bc - ca \le 0$(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. 663. Which of the following is correct combination?

If the roots of the equation $a(b-c)x^2 + b(c-a)x + c(a-b) = 0$ are equal, show that $2/b = 1/a + 1/c$.

If $\alpha, \beta$ are the roots of $x^2 + px + q = 0$ and $\alpha^{2n} + p^n\alpha^n + q^n = 0$ and if $(\alpha/\beta), (\beta/\alpha)$ are the roots of $x^n + 1 + (x+1)^n = 0$, then $n$ ($n \in \mathbb{N}$)

Let $f(x) = (k-3)x^2 - 2kx + 3k - 6$ where $x \in R$. If the range of $f(x)$ is $[0, \infty)$, then the value of $k$ can be:

Quadratic Equations Questions (527)