Relations and Functions JEE Main PYQs (2020–2026)

Solved Relations and Functions PYQs

3 recent questions with answers. The full set of 131 is free to practise.

1. JEE Main 2026EasyTypes of Functions: One-One, Many-One, Onto, Into (Bijective)
   Q1

   The number of functions $f:\{1,2,3,4\}\to \{a,b,c\}$ , which are not onto, is :

   • A48
   • B45
   • C51
   • D35

   Show answer & solution

   Correct answer: (B)

   Total number of functions from $\{1,2,3,4\}$ to $\{a,b,c\}$ is
   
   $$3^4=81$$
   
   Now we find the number of onto functions.
   
   For a function to be onto, all the three elements $a,b,c$ must appear as images of elements of $\{1,2,3,4\}$ .
   
   Using inclusion-exclusion principle:
   
   Total functions: $3^4=81$
   
   Let
   
   $A$ = functions missing $a$
   
   $B$ = functions missing $b$
   
   $C$ = functions missing $c$
   
   If one element is missing, then the function goes into only $2$ elements.
   
   So,
   
   $$|A|=|B|=|C|=2^4=16$$
   
   Hence,
   
   $$|A|+|B|+|C|=3\times 16=48$$
   
   If two elements are missing, then all $4$ elements of domain must go to only one element of codomain.
   
   So,
   
   $$|A\cap B|=|B\cap C|=|C\cap A|=1^4=1$$
   
   Thus,
   
   $$|A\cap B|+|B\cap C|+|C\cap A|=3$$
   
   If all three are missing, that is impossible, so
   
   $$|A\cap B\cap C|=0$$
   
   Therefore, number of functions which are not onto is
   
   $$|A\cup B\cup C|=48-3=45$$
   
   So the correct answer is
   
   $$\boxed{45}$$
   
   Hence, Option B is correct.
2. JEE Main 2026HardOdd, Even, Periodic Functions & Functional Equations
   Q2

   Let $f$ be a function such that $3f(x)+2f\left(\frac{m}{19x}\right)=5x,x\ne 0$ , where $m={\sum }_{i=1}^9(i{)}^2$ . Then $f(5)-f(2)$ is equal to

   • A36
   • B9
   • C-9
   • D18

   Show answer & solution

   Correct answer: (D)

   Given $m={\sum }_{i=1}^9(i{)}^2=1^2+2^2+\cdots 9$
   
   We know, $1^2+2^2+3^2+\cdots n^2=\frac{n(n+1)(2n+1)}{6}$
   
   so, $m=\frac{9\times 10\times 19}{6}=3\times 5\times 19=285$
   
   Given equation
   
   $$3f(x)+2f\left(\frac{m}{19x}\right)=5x,x\ne 0$$
   
   put $m=285,3f(x)+2f\left(\frac{285}{19x}\right)=5x$
   
   $$3f(x)+2f\left(\frac{15}{x}\right)=5x-(1)$$
   
   put $x=\frac{15}{x}$
   
   $$\begin{array}{rl}& 3f\left(\frac{15}{x}\right)+2f\left(\frac{15}{15/x}\right)=5\times \frac{15}{x}\\ & 3f\left(\frac{15}{x}\right)+2f(x)=\frac{75}{x}-(2)\end{array}$$
   
   from (1) $f\left(\frac{15}{x}\right)=\frac{5x-3f(x)}{2}$ substitute in equation (2)
   
   $$\begin{array}{rl}& 3\left[\frac{5x-3f(x)}{2}\right]+2f(x)=\frac{75}{x}\\ & 15x-9f(x)+4f(x)=\frac{150}{x}\\ & 5f(x)=15x-\frac{150}{x}\\ & f(x)=3x-\frac{30}{x}\end{array}$$
   
   to find $f(5)-f(2)$
   
   $$\begin{array}{rl}& f(5)-f(2)=\left(3(5)-\frac{30}{5}\right)-\left(3\times 2-\frac{30}{2}\right)\\ & =(15-6)-(6-15)\end{array}$$
   
   = $9-(-9)=18$
3. JEE Main 2026EasyComposition of Functions
   Q3

   Let $f$ and $g$ be functions satisfying $f(x+y)=f(x)f(y),f(1)=7$ and $g(x+y)=g(xy),g(1)=1$ , for all $x,y\in \mathbf{N}$ . If ${\sum }_{x=1}^n\left(\frac{f(x)}{g(x)}\right)=19607$ , then n is equal to :

   • A6
   • B7
   • C4
   • D5

   Show answer & solution

   Correct answer: (D)

   $$\begin{array}{rl}& f(x+y)=f(x)\cdot f(y)\Rightarrow f(x)=a^x\\ & \left(\because f(1)=7\Rightarrow a^1=7\right)\end{array}$$
   
   So $f(x)=7^x$
   
   Now
   
   $$g(x+y)=g(xy)(\text{ put }y=1)\Rightarrow g(x+1)=g(x)$$
   
   So $g(1)=g(2)=g(3)=\dots \dots \dots =g(n)=1$
   
   $$\begin{array}{rl}& \text{ Given }\sum _{x=1}^n\frac{f(x)}{g(x)}=19607\\ & \sum _{x=1}^n\frac{7^x}{1}=19607\Rightarrow 7\left(\frac{7^n-1}{7-1}\right)=19607\\ & 7^n-1=\frac{6}{7}\times 19607\\ & 7^n=16807\Rightarrow n=5\end{array}$$

Topic-wise question count

Where JEE Main has asked from Relations and Functions, most frequent first.

• Types of Functions: One-One, Many-One, Onto, Into (Bijective)33
• Odd, Even, Periodic Functions & Functional Equations25
• Domain and Range of Algebraic Functions22
• Composition of Functions19
• Domain and Range of Transcendental & Composite Functions17
• Special Functions: Step, Fractional Part & Signum Functions10
• Inverse of a Function5

Year-wise Relations and Functions questions in JEE Main

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