Quadratic Equation and Inequalities JEE Main PYQs (2020–2026)

Solved Quadratic Equation and Inequalities PYQs

3 recent questions with answers. The full set of 138 is free to practise.

1. JEE Main 2026MediumTransformation of Equations & Higher Degree Polynomials
   Q1

   Let $S=\{x^3+a{x}^2+bx+c:a,b,c\in N$ and $a,b,c\le 20\}$ be a set of polynomials. Then the number of polynomials in S , which are divisible by $x^2+2$ , is

   • A6
   • B120
   • C20
   • D10

   Show answer & solution

   Correct answer: (D)

   $$S=\left\{x^3+a{x}^2+bx+c:a,b,c\in \mathbb{N}\text{ and }a,b,c\le 20\right\}\text{ so }a,b,c\in \{1,2,3,\dots 20\}$$
   
   $x^3+a{x}^2+bx+c$ is divisible by $x^2+2$
   
   $$\therefore x^3+a{x}^2+bx+c=\left(x^2+2\right)L(x)$$
   
   Let $L(x)=x+n$ because L.H.S is cubic have leading coefficient 1 .
   
   $$\begin{array}{rl}{x}^3+a{x}^2+bx+c& =\left(x^2+2\right)(x+n)\\ & =x^3+x^{2}n+2x+2n\end{array}$$
   
   Compare $n=a,b=2,2n=c\Rightarrow c=2a$
   
   so, $c=2a$ and $b=2$
   
   Finding the Number of Possible Polynomials
   
   For $b$ : There is only $1$ choice $(b=2)$ , which is within the range $[1,20]$ .
   
   For $a$ and $c$ : Since $c=2a$ , we must ensure $c\le 20$ .
   
   If $a=1,c=2$
   
   If $a=2,c=4$
   
   …
   
   If $a=10,c=20$
   
   There are 10 possible values for the pair ( $a,c$ ).
   
   Total polynomials $=($ Choices for $b)\times ($ Choices for $a,c$ pairs $)=1\times 10=10$
2. JEE Main 2026EasyNature of Roots, Discriminant & Symmetric Functions of Roots
   Q2

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

   • A7
   • B3
   • C4
   • D9

   Show answer & solution

   Correct answer: (A)

   Given equation
   
   $$x^4-a{x}^2+9=0---\text{ (1) }$$
   
   let $x^2=t$, equation (1) become
   
   $$t^2-at+9=0---\text{ (2) }$$
   
   If $t_1$ & $t_2$ are roots of equation (2) then four distinct roots of equation (1) is $\left\{-\sqrt{t_1},-\sqrt{t_2},\sqrt{t_1},\sqrt{t_2}\right\}$
   
   roots of this equation are real and distinct if $D>0$
   
   $$\begin{array}{rl}& (-a{)}^2-4\times 9>0\\ & \\ & \Rightarrow a^2-36>0\end{array}$$
   
   so $a\in (-\infty ,-6)\cup (6,\infty )$
   
   so smallest positive integral value of $a$ is 7 as $a>6$ .
3. JEE Main 2026MediumAlgebraic Inequalities, Modulus & Wavy Curve Method
   Q3

   The number of the real solutions of the equation: $x|x+3|+|x-1|-2=0$ is

   • A3
   • B2
   • C5
   • D4

   Show answer & solution

   Correct answer: (A)

   given $x|x+3|+|x-1|-2=0$.
   
   critical points $x=-3,1$.
   
   two critical points means 3 cases.
   
   Case 1 : $x<-3,|x+3|=-(x+3)$ and $|x-1|=-(x-1)$. equation become
   
   $$\begin{array}{rl}& -x(x+3)-(x-1)-2=0\\ & -x^2-3x-x+1-2=0\\ & -x^2-4x-1=0\Rightarrow x^2+4x+1=0\\ & x=\frac{-4\pm \sqrt{16-4}}{2}=\frac{-4\pm 2\sqrt{3}}{2}=-2\pm \sqrt{3}\end{array}$$
   
   since $x<-3$ so $x=-2-\sqrt{3}$ is only solution.
   
   Case 2 : $-3\le x\le 1,|x+3|=x+3$ and $|x-1|=-(x-1)$. equation become
   
   $$\begin{array}{rl}& x(x+3)-(x-1)-2=0.\\ & x^2+3x-x+1-2=0.\\ & x^2+2x-1=0.\\ & x=\frac{-2\pm \sqrt{4+4}}{2}=\frac{-2\pm 2\sqrt{2}}{2}=-1\pm \sqrt{2}.\end{array}$$
   
   since $-3\le x\le 1$ so $x=-1+\sqrt{2},-1-\sqrt{2}$ both accepted.
   
   Case 3 : $x>1,|x+3|=x+3\text{ and }|x-1|=x-1\text{. }$
   
   equation become
   
   $$x(x+3)+x-1-2=0.$$
   
   $$x^2+4x-3=0.$$
   
   $x=\frac{-4\pm \sqrt{16+12}}{2}=\frac{-4\pm 2\sqrt{7}}{2}=-2\pm \sqrt{7}$. Checking positive solution. since $x>1$, so $x=\sqrt{7}-2\approx 0.645$, so no solution.
   
   so final solution is
   
   $$x=-2-\sqrt{3},-1+\sqrt{2},-1-\sqrt{2}.$$
   
   Number of solution = 3

Topic-wise question count

Where JEE Main has asked from Quadratic Equation and Inequalities, most frequent first.

• Roots of Quadratic Equation & Relation between Roots and Coefficients51
• Transformation of Equations & Higher Degree Polynomials25
• Nature of Roots, Discriminant & Symmetric Functions of Roots24
• Algebraic Inequalities, Modulus & Wavy Curve Method21
• Maximum and Minimum Values of Quadratic Expressions7
• Location of Roots & Signs of Roots6
• Common Roots of Quadratic Equations4

Year-wise Quadratic Equation and Inequalities questions in JEE Main

Number of questions from this chapter in each year's paper(s).