Quadratic Equations and Inequalities — JEE Main PYQs

The number of real solutions of the equation, $x^2-|x|-12=0$ is :

If $a+b+c=1$, $ab+bc+ca=2$ and $abc=3$, then the value of $a^4+b^4+c^4$ is equal to $\dots$...

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

Let $f(x)$ be a polynomial of degree 3 such that $f(k)=-\frac{2}{k}$ for $k=2,3,4,5$. Then the value of $52-10f(10)$ is equal to

Let $P(x)=x^2+bx+c$ be a quadratic polynomial with real coefficients such that ${\int }_0^{1}P(x)dx=1$ and $P(x)$ leaves remainder 5 when it is divided by $(x-2)$. Then the value of $9(b+c)$ is equal to :

Let $\alpha$ and $\beta$ be the roots of the equation $x^2-x-1=0$. If $P_k=(\alpha {)}^k+(\beta {)}^k$, $k\ge 1$, then which one of the following statements is not true?

The least positive value of ' $a$ ' for which the equation, $2x^2+(a-10)x+\frac{33}{2}=2a$ has real root is

Let S be the set of all real roots of the equation, $3^x(3^x-1)+2=|3^x-1|+|3^x-2|$. Then S:

Let $a,b\in R,a\ne 0$ be such that the equation $a{x}^2-2bx+5=0$ has a repeated root $\alpha$, which is also a root of the equation, $x^2-2bx-10=0$. If $\beta$ is the other root of this equation, then ${\alpha }^2+{\beta }^2$ is equal to:

The number of real roots of the equation, $e^{4x}+e^{3x}-4e^{2x}+e^x+1=0$ is:

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

If $\alpha$ and $\beta$ are the roots of the equation $x^2+px+2=0$ and $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ are the roots of the equation $2x^2+2qx+1=0$, then …

Consider the two sets:
$A=\{m\in R$ : both the roots of $x^2-(m+1)x+m+4=0$ are real $\}$ and $B=[-3,5)$. Which of the following is not true?

The set of all real values of $\lambda$ for which the quadratic equations, $\left({\lambda }^2+1\right){\mathrm{x}}^2-4\lambda \mathrm{x}+2=0$ always have exactly one root in the interval $(0,1)$ is:

If $\alpha$ and $\beta$ are the roots of the equation $2x(2x+1)=1$, then $\beta$ is equal to:

The product of the roots of the equation $9x^2-18|x|+5=0$, is:

If $\alpha$ and $\beta$ be two roots of the equation $x^2-64x+256=0$. Then the value of ${\left(\frac{{\alpha }^3}{{\beta }^5}\right)}^{1/8}+{\left(\frac{{\beta }^3}{{\alpha }^5}\right)}^{1/8}$ is:

Let $\alpha$ and $\beta$ be roots of $x^2-3x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^2-6x+q=0$. If $\alpha ,\beta$, $\gamma ,\delta$, form a geometric progression. Then ratio $(2q+p):(2q-p)$ is:

Let $\lambda \ne 0$ be in $\mathbb{R}$. If $\alpha$ and $\beta$ are the roots of the equation, $x^2-x+2\lambda =0$ and $\alpha$ and $\gamma$ are the roots of the equation, $3x^2-10x+27\lambda =0$, then $\frac{\beta \gamma }{\lambda }$ is equal to:

The number of integral values of $m$ for which the equation $\left(1+m^2\right)x^2-2(1+3m)x+(1+8m)=0$ has no real root is:

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

Consider the quadratic equation $(c-5)x^2-2cx+(c-4)=0,c\ne 5$. Let $S$ be the set of all integral values of c for which one root of the equation lies in the interval $(0,2)$ and its other root lies in the interval $(2,3)$. Then the number of elements in $S$ is:

If one real root of the quadratic equation $81x^2+kx+256=0$ is cube of the other root, then a value of k is:

If $\lambda$ be the ratio of the roots of the quadratic equation in $x,3m^2{x}^2+m(m-4)x+2=0$, then the least value of $m$ for which $\lambda +\frac{1}{\lambda }=1$ is:

The number of integral values of $m$ for which the quadratic expression, $(1+2m)x^2-2(1+3m)x+4(1+m),x\in R$, is always positive, is :

If $m$ is chosen in the quadratic equation $\left(m^2+1\right)x^2-3x+{\left(m^2+1\right)}^2=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:

If both the roots of the quadratic equation $x^2-mx+4=0$ are real and distinct and they lie in the interval $[1,5]$, then $m$ lies in the interval.

The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation, $6x^2-11x+\alpha =0$ are rational numbers is:

If $\alpha$ and $\beta$ are the roots of the quadratic equation, $x^2+x\sin \theta -2\sin \theta =0,\theta \in \left(0,\frac{\pi }{2}\right)$ then $\frac{{\alpha }^{12}+{\beta }^{12}}{\left({\alpha }^{-12}+{\beta }^{-12}\right)(\alpha -\beta {)}^{24}}$ is equal to:

Let $\mathrm{S}=\{x\in \mathbf{R}:x⩾0$ and
$2|\sqrt{x}-3|+\sqrt{x}(\sqrt{x}-6)+6=0\}$. Then S :

If, for a positive integer n , the quadratic equation,
$x(x+1)+(x+1)(x+2)+\dots$.
$+(x+\overline{\mathrm{n}-1})(x+\mathrm{n})=10\mathrm{n}$
has two consecutive integral solutions, then n is equal to :

The sum of all real values of x satisfying the equation ${\left(x^2-5x+5\right)}^{x^2+4x-60}=1$ is

Let $\alpha$ and $\beta$ be the roots of equation $p{x}^2+qx+r=0,p\ne 0$. If $p,q,r$ are in A.P. and $\frac{1}{\alpha }+\frac{1}{\beta }=4$, then the value of $|\alpha -\beta |$ is

If the equations ${\mathrm{x}}^2+2\mathrm{x}+3=0$ and ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0,\mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{R}$, have a common root, then $\mathrm{a}:\mathrm{b}:\mathrm{c}$ is

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

If the roots of the equation $b{x}^2+cx+a=0$ be imaginary, then for all real values of $x$, the expression $3b^2{x}^2+6bcx+2c^2$ is

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

If the difference between the roots of the equation $x^2+ax+1=0$ is less than $\sqrt{5}$, then the set of possible values of $a$ is

If the roots of the quadratic equation $x^2+px+q=0$ are $\tan 30^{\circ }$ and $\tan 15^{\circ }$, respectively then the value of $2+q-p$ is

All the values of $m$ for which both roots of the equations $x^2-2mx+m^2-1=0$ are greater than -2 but less than 4, lie in the interval

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

The value of $\alpha$ for which the sum of the squares of the roots of the equation $x^2-(a-2)x-a-1=0$ assume the least value is

If roots of the equation $x^2-bx+c=0$ be two consecutive integers, then $b^2-4c$ equals

If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than 5, then k lies in the interval

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

If $(1-p)$ is a root of quadratic equation $x^2+px+(1-p)=0$, then its roots are

If one root of the equation $x^2+px+12=0$ is 4, while the equation $x^2+px+q=0$ has equal roots, then the value of '$q$' is

If $2a+3b+6c=0$, then at least one root of the equation $a{x}^2+bx+c=0$ lies in the interval