Sets, relations and functions: JEE Main and CUET-style questions
Original JEE Main and CUET-style questions on sets, relations and functions, with a worked solution and the usual trap for each. Tap an option, or type a numerical answer, to check it.
Let \(A = \{1, 2, 3, 4\}\). How many relations on \(A\) are both reflexive and symmetric?
Show answer
Answer: (a) \(64\)
Reflexive: all four pairs \((a, a)\) must be in the relation, so there is no choice there.
Symmetric: the other pairs come in unordered couples \(\{(a, b), (b, a)\}\) with \(a \ne b\). There are \(\binom{4}{2} = 6\) couples, and each couple is either in (both pairs) or out.
Number of relations \(= 2^6 = 64\).
Trap: Counting the 12 off-diagonal pairs one by one (giving \(2^{12}\)) forgets that symmetry ties each pair to its mirror image.
Q2
·CUET style·Multiple choiceInverse of a function
\(f : \mathbb{R} \setminus \{1\} \to \mathbb{R} \setminus \{2\}\) is given by \(f(x) = \dfrac{2x + 3}{x - 1}\). Then \(f^{-1}(x)\) is
Show answer
Answer: (b) \(\dfrac{x + 3}{x - 2}\)
Put \(y = \dfrac{2x + 3}{x - 1}\). Then \(y(x - 1) = 2x + 3\), so \(xy - 2x = y + 3\) and \(x = \dfrac{y + 3}{y - 2}\).
Swap the letters: \(f^{-1}(x) = \dfrac{x + 3}{x - 2}\). Check: \(f(0) = -3\) and \(f^{-1}(-3) = 0\).
Trap: The reciprocal \(\dfrac{x-1}{2x+3}\) is \(1/f(x)\), not the inverse function.
Q3
·JEE Main style·Numerical valueOnto functions
Find the number of onto (surjective) functions from \(\{1, 2, 3, 4, 5\}\) to \(\{a, b, c\}\).
Show answer
Answer: 150
All functions: \(3^5 = 243\). Remove those that miss at least one value (inclusion–exclusion):
miss a named value: \(3 \times 2^5 = 96\); miss two named values: \(3 \times 1^5 = 3\) (added back).
Onto functions \(= 243 - 96 + 3 = 150\).
Trap: Subtracting \(3 \times 2^5\) and stopping gives 147: the three constant functions were removed twice.
3 more questions with CBSE Essentials
CBSE Essentials opens them with worked solutions, along with every chapter question and marking scheme for Class 10, 11 and 12. Try it free for 7 days, no card.
All questions are original, written by CBSE Math Revision in the style of each exam; none are taken from NTA, IIT, CBSE or NCERT papers. Every answer was checked twice: by a computer re-solve and by hand. No calculator needed. Spotted a mistake? Tell us.