Quadratic Equations and Inequalities — JEE Advanced PYQs

Match each entry in List-I to the correct entry in List-II and choose the correct option.
| (P) If $\alpha$ and $\beta$ are the distinct roots of the equation ${\mathrm{x}}^2+\mathrm{x}+1=0$, then the quadratic equation with roots …

Let a,b,c be positive integers in arithmetic progression such that the equation
${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$
has only integer solutions.
Then which of the following statements is (are) TRUE ?
(A) $\mathrm{c}-\mathrm{b}$ is an integer multiple of a
(B) Both the roots …

Let ${\mathbb{R}}^2$ denote $\mathbb{R}\times \mathbb{R}$. Let
$\mathrm{S}=\{(\mathrm{a},\mathrm{b},\mathrm{c}):\mathrm{a},\mathrm{b},\mathrm{c}\in \mathbb{R}$ and ${\mathrm{ax}}^2+2\mathrm{bxy}+{\mathrm{cy}}^2>0$ for all …

Let $f(\mathrm{x})={\mathrm{x}}^4+{\mathrm{ax}}^3+{\mathrm{bx}}^2+\mathrm{c}$ be a polynomial with real coefficients such that $f(1)=-9$. Suppose that $\mathrm{i}\sqrt{3}$ is a root of the equation $4{\mathrm{x}}^3+3{\mathrm{ax}}^2+2\mathrm{bx}=0$, where …

The product of all positive real values of x satisfying the equation

is ____.

For $x\in \mathbb{R}$, the number of real roots of the equation $3x^2-4|x^2-1|+x-1=0$ is ____.

Suppose a, b denote the distinct real roots of the quadratic polynomial $x^2+20x-2020$ and suppose c, d denote the distinct complex roots of the quadratic polynomial $x^2-20x+2020$. Then the value of

is

Let $-\frac{\pi }{6}<\theta <-\frac{\pi }{12}$. Suppose ${\alpha }_1$ and ${\beta }_1$ are the roots of the equation $x^2-2x\sec \theta +1=0$ and ${\alpha }_2$ and ${\beta }_2$ are the roots of the equation $x^2+2x\tan \theta -1=0$. If ${\alpha }_1>{\beta }_1$ and …

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha =0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $\left|x_1-x_2\right|<1$. Which of the following intervals is(are) a subse …

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

has

The number of distinct real roots of $x^4-4x^3+12x^2+x-1=0$ is

Consider the polynomial

Let s be the sum of all distinct real roots of $\mathrm{f}(\mathrm{x})$ and let $\mathrm{t}=|\mathrm{s}|$.

The real number s lies in the interval

Let $p$ and $q$ be real numbers such that $p\ne 0,p^3\ne q$ and $p^3\ne -q$. If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha +\beta =-p$ and ${\alpha }^3+{\beta }^3=q$, then a quadratic equation having $\frac{\alpha }{\beta }$ and …

The smallest value of $k$, for which both the roots of the equation

are real, distinct and have values at least 4, is

Let $a,b,c,p,q$ be real numbers. Suppose $\alpha ,\beta$ are the roots of the equation $x^2+2px+q=0$ and $\alpha ,\frac{1}{\beta }$ are the roots of the equation $a{x}^2+2bx+c=0$, where ${\beta }^2\notin \{-1,0,1\}$.

STATEMENT-1 : …

Let $\alpha ,\beta$ be the roots of the equation $x^2-px+r=0$ and $\frac{\alpha }{2},2\beta$ be the roots of the equation $x^2-qx+r=0$. Then the value of $r$ is

Let $f(x)=\frac{x^2-6x+5}{x^2-5x+6}$.

Match the expressions/statements in Column I with expressions/statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4\times 4$ matrix given in the ORS.
| Column I | Column II ||----------| …

Let a, b, c be the sides of a triangle. No two of them are equal and $\lambda \in \mathbb{R}$. If the roots of the equation $x^2+2(a+b+c)x+3\lambda (ab+bc+ca)=0$ are real, then

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