Quadratic Equations and Inequalities — JEE Main PYQs

The sum of all the roots of the equation $(\mathrm{x}-1{)}^2-5|\mathrm{x}-1|+6=0$, is:

Let $\alpha$ and $\beta$ be the roots of equation $x^2+2ax+(3a+10)=0$ such that $\alpha <1<\beta$. Then the set of all possible values of $a$ is :

Let $\alpha ,\beta$ be the roots of the quadratic equation $12{\mathrm{x}}^2-20\mathrm{x}+3\lambda =0,\lambda \in \mathbb{Z}$. If $\frac{1}{2}\le |\beta -\alpha |\le \frac{3}{2}$, then the sum of all possible values of $\lambda$ is :

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

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in:

The smallest positive integral value of $a$, for which all the roots of ${\mathrm{x}}^4-{\mathrm{ax}}^2+9=0$ are real and distinct, is equal to

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

Let the arithmetic mean of $\frac{1}{\mathrm{a}}$ and $\frac{1}{\mathrm{b}}$ be $\frac{5}{16},\mathrm{a}>2$.

If $\alpha$ is such that $\mathrm{a},4,\alpha ,\mathrm{b}$ are in A.P., then the equation $\alpha x^2-ax+2(\alpha -2b)=0$ has:

Let $\alpha ,\alpha +2,\alpha \in \mathbf{Z}$, be the roots of the quadratic equation $x(x+2)+(x+1)(x+3)+(x+2)(x+4)+\dots +(x+n-1)(x+n+1)=4n$ for some $n\in \mathbf{N}$.

Then $(n+\alpha )$ is equal to :

Let $a,b,c\in \{1,2,3,4\}$. If the probability, that $a{x}^2+2\sqrt{2}bx+c>0$ for all $x\in \mathbf{R}$, is $\frac{m}{n}$, $\gcd (m,n)=1$, then $m+n$ is equal to ______.

If the quadratic equation $(\lambda +2){\mathrm{x}}^2-3\lambda \mathrm{x}+4\lambda =0$, $\lambda \ne -2$, has two positive roots, then the number of possible integral values of $\lambda$ is :

If the set of all solutions of $\left|x^2+x-9\right|=|x|+\left|x^2-9\right|$ is $[\alpha ,\beta ]\cup [\gamma ,\infty )$, then $\left({\alpha }^2+{\beta }^2+{\gamma }^2\right)$ is equal to :

Let $\mathrm{a},\mathrm{b}\in \mathrm{C}$. Let $\alpha ,\beta$ be the roots of the equation ${\mathrm{x}}^2+\mathrm{ax}+\mathrm{b}=0$. If $\beta -\alpha =\sqrt{11}$ and ${\beta }^2-{\alpha }^2=3\mathrm{i}\sqrt{11}$, then ${\left({\beta }^3-{\alpha }^3\right)}^2$ is equa …

Let $\alpha ,\beta$ be the roots of the equation ${\mathrm{x}}^2-\mathrm{x}+\mathrm{p}=0$ and $\gamma ,\delta$ be the roots the equation ${\mathrm{x}}^2-4\mathrm{x}+\mathrm{q}=0$; $\mathrm{p},\mathrm{q}\in \mathrm{Z}$. If $\alpha ,\beta ,\gamma ,\delta$ are in G.P. …

Consider the quadratic equation $({\mathrm{n}}^2-2\mathrm{n}+2){\mathrm{x}}^2-3\mathrm{x}+({\mathrm{n}}^2-2\mathrm{n}+2{)}^2=0,\mathrm{n}\in \mathrm{R}$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its r …

Let one root of the quadratic equation in x:
$(k^2-15k+27)x^2+9(k-1)x+18=0$
be twice the other. Then the length of the latus rectum of the parabola $y^2=6kx$ is equal to:

Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2-ax+2=0$. Let the sets
$\{a\in \mathbb{R}:e_1$ and $e_2$ are the eccentricities of hyperbolas $\}=(\alpha ,\beta )$, and $\{a\in \mathbb{R}:e_1$ and $e_2$ are the eccentricities of an ellipse and a hyp …

The sum of squares of all real solution of the equation
${\log }_{(x+1)}\left(2x^2+5x+3\right)=4-{\log }_{(2x+3)}\left(x^2+2x+1\right)$ is equal to ____.

Let $\alpha ,\beta$ be the roots of the equation $x^2-3x+r=0$, and $\frac{\alpha }{2},2\beta$ be the roots of the equation ${\mathrm{x}}^2+3\mathrm{x}+\mathrm{r}=0$. If the roots of the equation ${\mathrm{x}}^2+6\mathrm{x}=\mathrm{m}$ are …

Let ${\alpha }_{\theta }$ and ${\beta }_{\theta }$ be the distinct roots of $2{\mathrm{x}}^2+(\cos \theta )\mathrm{x}-1=0,\theta \in (0,2\pi )$. If m and M are the minimum and the maximum values of ${\alpha }_{\theta }^4+{\beta }_{\theta }^4$, then $16(\mathrm{M}+\mathrm{m})$ equals :

If the equation $\mathrm{a}(\mathrm{b}-\mathrm{c}){\mathrm{x}}^2+\mathrm{b}(\mathrm{c}-\mathrm{a})\mathrm{x}+\mathrm{c}(\mathrm{a}-\mathrm{b})=0$ has equal roots, where $\mathrm{a}+\mathrm{c}=15$ and $\mathrm{b}=\frac{36}{5}$, then ${\mathrm{a}}^2+{\mathrm{c}}^2$ is equal to __ …

If the set of all $\mathrm{a}\in \mathbf{R}$, for which the equation $2{\mathrm{x}}^2+(\mathrm{a}-5)\mathrm{x}+15=3\mathrm{a}$ has no real root, is the interval $(\alpha ,\beta )$, and $X=\{x\in Z:\alpha <x<\beta \}$, then ${\sum }_{x\in X}{x}^2$ is equal to

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

The product of all the rational roots of the equation ${\left(x^2-9x+11\right)}^2-(x-4)(x-5)=3$, is equal to :

If the set of all $a\in \mathbb{R}-\{1\}$, for which the roots of the equation $(1-a)x^2+2(a-3)x+9=0$ are positive is $(-\infty ,-\alpha ]\cup [\beta ,\gamma )$, then $2\alpha +\beta +\gamma$ is equal to ______.

let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $\left(\mathrm{k},\frac{\mathrm{k}}{2}\right)$ from the line $3x+4y+5=0$ is

Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3}x-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2+3x-1=0$. If $P_n={\alpha }^n+{\beta }^n$ and $Q_n={\gamma }^n+{\delta }^n$, then …

Consider the equation ${\mathrm{x}}^2+4\mathrm{x}-\mathrm{n}=0$, where $n\in [20,100]$ is a natural number. Then the number of all distinct values of n , for which the given equation has integral roots, is equal to

If the range of the function $f(x)=\frac{5-x}{x^2-3x+2}$, $x\ne 1,2$, is $(-\infty ,\alpha ]\cup [\beta ,\infty )$, then ${\alpha }^2+{\beta }^2$ is equal to :

Let the set of all values of $p\in \mathbb{R}$, for which both the roots of the equation $x^2-(p+2)x+(2p+9)=0$ are negative real numbers, be the interval $(\alpha ,\beta ]$. Then $\beta -2\alpha$ is equal to

The sum of the squares of the roots of $|x-2{|}^2+|x-2|-2=0$ and the squares of the roots of $x^2-2|x-3|-5=0$, is

The sum of the squares of all the roots of the equation x² + |2x − 3| − 4 = 0 is:

If 2 and 6 are the roots of the equation $a{x}^2+bx+1=0$, then the quadratic equation, whose roots are $\frac{1}{2a+b}$ and $\frac{1}{6a+b}$, is :

Let $S=\{{\sin }^{2}2\theta :({\sin }^4\theta +{\cos }^4\theta )x^2+(\sin 2\theta )x+({\sin }^6\theta +{\cos }^6\theta )=0\text{ has real roots}\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3((\alpha -2{)}^2+(\beta -1{)}^2)$ …

The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is ____.

The number of real solutions of the equation $x|x+5|+2|x+7|-2=0$ is ____ .

Let $\alpha ,\beta$ be the distinct roots of the equation $x^2-(t^2-5t+6)x+1=0$, $t\in R$ and $a_n={\alpha }^n+{\beta }^n$. Then the minimum value of $\frac{a_{2023}+a_{2025}}{a_{2024}}$ is

Let $x_1,x_2,x_3,x_4$ be the solution of the equation $4x^4+8x^3-17x^2-12x+9=0$ and $(4+x_1^2)(4+x_2^2)(4+x_3^2)(4+x_4^2)=\frac{125}{16}m$. Then the value of $m$ is ____.

Let $\alpha ,\beta$ be roots of $x^2+\sqrt{2}x-8=0$. If $U_n={\alpha }^n+{\beta }^n$ then $\frac{U_{10}+\sqrt{2}{U}_9}{2U_8}$ is equal to ______.

The sum of all the solutions of the equation $(8{)}^{2\mathrm{x}}-16\cdot (8{)}^{\mathrm{x}}+48=0$ is :

Let $\alpha ,\beta$ be the roots of the equation
$x^2+2\sqrt{2}x-1=0$. The quadratic equation, whose roots are ${\alpha }^4+{\beta }^4$ and $\frac{1}{10}\left({\alpha }^6+{\beta }^6\right)$, is :

Let $\mathrm{a},\mathrm{b},\mathrm{c}$ be the length of three sides of a triangle satisfying the condition $\left({\mathrm{a}}^2+{\mathrm{b}}^2\right){\mathrm{x}}^2-2\mathrm{b}(\mathrm{a}+\mathrm{c})$. $\mathrm{x}+\left({\mathrm{b}}^2+{\mathrm{c}}^2\right)=0$. If the s …

Let $\alpha ,\beta \in \mathbb{N}$ be roots of equation $x^2-70x+\lambda =0$, where $\frac{\lambda }{2},\frac{\lambda }{3}\notin \mathbb{N}$. If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha -1}+\sqrt{\beta -1})(\lambda +35)}{|\alpha -\beta |}$ is equal to _ …

Let $\alpha$ and $\beta$ be the roots of the equation ${\mathrm{px}}^2+\mathrm{qx}-\mathrm{r}=0$, where $\mathrm{p}\ne 0$. If $\mathrm{p},\mathrm{q}$ and r be the consecutive terms of a non-constant G.P and $\frac{1}{\alpha }+\frac{1}{\beta }=\frac{3}{4}$, then the …

For $0<\mathrm{c}<\mathrm{b}<\mathrm{a}$, let $(\mathrm{a}+\mathrm{b}-2\mathrm{c}){\mathrm{x}}^2+(\mathrm{b}+\mathrm{c}-2\mathrm{a})\mathrm{x}+(c+a-2b)=0$ and $\alpha \ne 1$ be one of its root. Then, among the two statements
(I) If $\alpha \in (-1,0)$, then $b$ ca …

Let a be the sum of all coefficients in the expansion of ${\left(1-2x+2x^2\right)}^{2023}{\left(3-4x^2+2x^3\right)}^{2024}$ and $b={\lim }_{x\to 0}\left(\frac{{\int }_0^x\frac{\log (1+t)}{t^{2024}+1}dt}{x^2}\right)$. If the equations $c{x}^2+dx+e=0$ and …

Let $S$ be the set of positive integral values of $a$ for which $\frac{a{x}^2+2(a+1)x+9a+4}{x^2-8x+32}<0,\forall x\in \mathbb{R}$. Then, the number of elements in $S$ is:

Let $S=\left\{x\in R:(\sqrt{3}+\sqrt{2}{)}^x+(\sqrt{3}-\sqrt{2}{)}^x=10\right\}$.

Then the number of elements in S is :

Let $\alpha ,\beta ,\gamma$ be the three roots of the equation ${\mathrm{x}}^3+\mathrm{bx}+\mathrm{c}=0$. If $\beta \gamma =1=-\alpha$, then $b^3+2c^3-3{\alpha }^3-6{\beta }^3-8{\gamma }^3$ is equal to

The sum of all the roots of the equation $\left|{\mathrm{x}}^2-8\mathrm{x}+15\right|-2\mathrm{x}+7=0$ is: