Sets JEE Main PYQs (2020–2026)

Solved Sets PYQs

3 recent questions with answers. The full set of 70 is free to practise.

1. JEE Main 2026HardSubsets and Power Set
   Q1

   Let $S$ be the set of the first 11 natural numbers. Then the number of elements in $A=\{B\subseteq S:n(B)\ge 2\text{ and the product of all elements of }B\text{ is even}\}$ is ____.
   
   [JEE Main 2026 (Online) 22nd January Evening Shift]

   Numerical answer type

   Show answer & solution

   Correct answer: 1979

   Let $S=\{1,2,\dots ,11\}$. The product of all elements of $B$ is even iff $B$ contains at least one even number.
   
   Evens in $S$: $5$ ($2,4,6,8,10$). Odds in $S$: $6$.
   
   Number of subsets containing at least one even number $=2^{11}-2^6=2048-64=1984$ (subtract the subsets made only of odd numbers).
   
   Now enforce $n(B)\ge 2$: subtract the $5$ singleton subsets $\{2\},\{4\},\{6\},\{8\},\{10\}$.
   
   $$1984-5=1979$$
2. JEE Main 2026MediumCartesian Product and Relations
   Q2

   Let $A=\{2,3,4,5,6\}$. Let $R$ be a relation on the set $A\times A$ given by $(x,y)R(z,w)$ if and only if $x$ divides $z$ and $y\le w$. Then the number of elements in $R$ is ____.
   
   [JEE Main 2026 (Online) 2nd April Evening Shift]

   Numerical answer type

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   Correct answer: 120

   $R$ is defined on $A\times A$, so its elements are of the form $\left((x,y),(z,w)\right)$ with $x,y,z,w\in A$.
   
   The condition on $(x,z)$ is independent of the condition on $(y,w)$, so by the fundamental principle of counting, $n(R)=($number of pairs $(x,z)$ with $x\mid z)\times ($number of pairs $(y,w)$ with $y\le w)$.
   
   Pairs $(x,z)$ with $x\mid z$: $x=2$: $z=2,4,6$ (3); $x=3$: $z=3,6$ (2); $x=4$: $z=4$ (1); $x=5$: $z=5$ (1); $x=6$: $z=6$ (1). Total $=8$.
   
   Pairs $(y,w)$ with $y\le w$: $5+4+3+2+1=15$.
   
   $$n(R)=8\times 15=120$$
3. JEE Main 2026HardTypes of Relations and Equivalence Relations
   Q3

   Let $R=\{(x,y)\in \mathbb{N}\times \mathbb{N}:{\log }_e(x+y)\le 2\}$. Then the minimum number of elements, required to be added in $R$ to make it a transitive relation, is ____.
   
   [JEE Main 2026 (Online) 6th April Evening Shift]

   Numerical answer type

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   Correct answer: 15

   ${\log }_e(x+y)\le 2\iff x+y\le e^2\approx 7.389$, so for natural numbers $R=\{(x,y):x+y\le 7\}$.
   
   Take $(a,1)\in R$ and $(1,c)\in R$ (i.e. $a\le 6$, $c\le 6$). Transitivity forces $(a,c)\in R$ for all $a,c\in \{1,2,\dots ,6\}$. So the relation must contain $\{1,\dots ,6\}\times \{1,\dots ,6\}$, which has $36$ pairs.
   
   Pairs of this set already in $R$ (those with $a+c\le 7$): $6+5+4+3+2+1=21$.
   
   Pairs to be added: $36-21=15$.
   
   Sufficiency: the relation $\{1,\dots ,6\}\times \{1,\dots ,6\}$ is clearly transitive.

Topic-wise question count

Where JEE Main has asked from Sets, most frequent first.

• Types of Relations and Equivalence Relations31
• Operations on Sets and Algebra of Sets13
• Subsets and Power Set9
• Practical Problems on Sets and Venn Diagrams9
• Cartesian Product and Relations8

Year-wise Sets questions in JEE Main

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