RELATIONS AND FUNCTIONS-Objective Questions for Entrance Exams
The following set of multiple-choice questions (MCQs) on Relations and Functions has been meticulously curated to reflect the conceptual depth, analytical rigor, and recurring patterns of questions asked in major competitive examinations such as IIT-JEE (old), JEE Main, JEE Advanced, NEET (AIPMT), AIIMS, BITSAT, KVPY, Olympiads, and various state engineering entrance exams. Each question is aligned with the NCERT Class XI Mathematics syllabus, ensuring strong foundational coverage while simultaneously addressing higher-order thinking skills required for competitive success. Emphasis has been placed on core ideas such as types of relations, domain and range, injective/surjective/bijective functions, inverse functions, composition of functions, and equivalence relations. The included explanations are concise yet rigorous, enabling learners to identify common traps, refine logical reasoning, and strengthen mathematical maturity. This collection is intended for concept reinforcement, exam-oriented practice, and self-assessment, making it a valuable resource for aspirants targeting top engineering and medical entrance examinations.
Actual questions from IIT-JEE, NEET, AIIMS, BITSAT & KVPY —
filter by exam, attempt each option, then reveal the detailed answer.
Score: 0 / 50
· Click an option to attempt · Then reveal answer
😕 No questions found for that exam filter.
Q1
Let \(R=\{(x,y)\in \mathbb{R}^2 : x^2+y^2=1\}\). Then \(R\) is (Exam: IIT-JEE Year: 1998)
(A) a function
(B) a one–one function
(C) a relation but not a function
(D) a many–one function
✅ Correct: (C)
Q2
Let \(f:\mathbb{R}\to\mathbb{R}\) be defined by \(f(x)=x^2\). Then \(f\) is (Exam: AIEEE Year: 2003)
(A) one–one and onto
(B) one–one but not onto
(C) onto but not one–one
(D) neither one–one nor onto
✅ Correct: (D)
Q3
If \(f(x)=|x|\), then \(f\) is (Exam: NEET Year: 2015)
(A) injective
(B) surjective
(C) bijective
(D) not injective
✅ Correct: (D)
Q4
The domain of \(f(x)=\sqrt{2x-1}\) is (Exam: JEE Main Year: 2014)
(A) \((-\infty,\infty)\)
(B) \([0,\infty)\)
(C) \([1/2,\infty)\)
(D) \((1/2,\infty)\)
✅ Correct: (C)
Q5
If \(A=\{1,2\}\), the number of relations on \(A\) is (Exam: IIT-JEE Year: 2001)
(A) 4
(B) 8
(C) 16
(D) 32
✅ Correct: (C)
Q6
The number of equivalence relations on a set with two elements is (Exam: KVPY Year: 2012)
(A) 1
(B) 2
(C) 3
(D) 4
✅ Correct: (C)
Q7
If \(f(x)=\frac{1}{x}\), domain is (Exam: AIIMS Year: 2009)
(A) \(\mathbb{R}\)
(B) \(\mathbb{R}-\{0\}\)
(C) \((0,\infty)\)
(D) \((-\infty,0)\)
✅ Correct: (B)
Q8
Let \(f(x)=\sin x\). Then \(f\) is (Exam: BITSAT Year: 2016)
(A) injective on \(\mathbb{R}\)
(B) injective on \([-\pi/2,\pi/2]\)
(C) surjective on \(\mathbb{R}\)
(D) bijective on \(\mathbb{R}\)
✅ Correct: (B)
Q9
Range of \(f(x)=x^2+1\) is (Exam: NEET Year: 2017)
(A) \(\mathbb{R}\)
(B) \((1,\infty)\)
(C) \([1,\infty)\)
(D) \((-\infty,1]\)
✅ Correct: (C)
Q10
If \(f(x)=2x+3\), then \(f^{-1}(x)\) is (Exam: IIT-JEE Year: 1997)
(A) \((x-3)/2\)
(B) \(2x-3\)
(C) \((x+3)/2\)
(D) \(x/2-3\)
✅ Correct: (A)
Q11
If \(f(x)=\log x\), domain is (Exam: JEE Main Year: 2019)
(A) \(\mathbb{R}\)
(B) \((0,\infty)\)
(C) \([0,\infty)\)
(D) \((-\infty,0)\)
✅ Correct: (B)
Q12
Let \(R=\{(a,a),(b,b),(c,c)\}\). Then \(R\) is (Exam: Olympiad Year: 2011)
(A) reflexive only
(B) symmetric only
(C) transitive only
(D) equivalence relation
✅ Correct: (D)
Q13
The number of functions from a 3-element set to a 2-element set is (Exam: IIT-JEE Year: 2000)
(A) 6
(B) 8
(C) 9
(D) 12
✅ Correct: (B)
Q14
If \(f(x)=x^3\), then \(f\) is (Exam: JEE Advanced Year: 2018)
(A) one–one only
(B) onto only
(C) bijective
(D) neither
✅ Correct: (C)
Q15
If \(f(x)=\tan x\), \(x\in(-\pi/2,\pi/2)\), then \(f\) is (Exam: IIT-JEE Year: 1999)
(A) injective
(B) surjective
(C) bijective
(D) not a function
✅ Correct: (C)
Q16
Let \(R=\{(x,y):x-y=0\}\). Then \(R\) is (Exam: NEET Year: 2013)
(A) reflexive
(B) symmetric
(C) transitive
(D) all of these
✅ Correct: (D)
Q17
The range of \(f(x)=\frac{x}{1+x^2}\) is (Exam: IIT-JEE Year: 2004)
(A) \((-1,1)\)
(B) \([-1,1]\)
(C) \((-\infty,\infty)\)
(D) \((0,1)\)
✅ Correct: (A)
Q18
Let \(f(x)=x^2\) with domain \([0,\infty)\). Then \(f\) is (Exam: JEE Main Year: 2020)
(A) injective
(B) surjective
(C) bijective
(D) not injective
✅ Correct: (A)
Q19
If \(f\circ g\) is defined, then (Exam: BITSAT Year: 2014)
(A) range of \(g\subseteq\) domain of \(f\)
(B) domain of \(f\subseteq\) range of \(g\)
(C) both
(D) none
✅ Correct: (A)
Q20
If \(f(x)=e^x\), then \(f^{-1}(x)\) is (Exam: NEET Year: 2016)
(A) \(e^x\)
(B) \(\ln x\)
(C) \(1/x\)
(D) \(\log x^2\)
✅ Correct: (B)
Q21
The relation \(xRy \iff x-y\) is even is (Exam: Olympiad Year: 2010)
(A) reflexive
(B) symmetric
(C) transitive
(D) equivalence
✅ Correct: (D)
Q22
Domain of \(f(x)=\sqrt{x^2-4}\) is (Exam: JEE Main Year: 2018)
(A) \((-2,2)\)
(B) \((-\infty,-2]\cup[2,\infty)\)
(C) \([0,\infty)\)
(D) \(\mathbb{R}\)
✅ Correct: (B)
Q23
Let \(f(x)=\frac{ax+b}{cx+d}\). For invertibility, (Exam: IIT-JEE Year: 2006)
(A) \(ad-bc=0\)
(B) \(ad-bc\neq0\)
(C) \(a=b\)
(D) \(c=d\)
✅ Correct: (B)
Q24
The number of reflexive relations on a set with \(n\) elements is (Exam: IIT-JEE Year: 2002)
(A) \(2^{n^2}\)
(B) \(2^{n(n-1)}\)
(C) \(2^{n^2-n}\)
(D) \(n^2\)
✅ Correct: (C)
Q25
Let \(f(x)=\cos x\) on \([0,\pi]\). Then \(f\) is (Exam: NEET Year: 2019)
(A) increasing
(B) decreasing
(C) constant
(D) not defined
✅ Correct: (B)
Q26
If \(f(x)=x^2\) and \(g(x)=\sqrt{x}\), then \(f\circ g(x)\) is (Exam: JEE Main Year: 2015)
(A) \(x\)
(B) \(|x|\)
(C) \(x^2\)
(D) \(\sqrt{x}\)
✅ Correct: (A)
Q27
A function with inverse must be (Exam: IIT-JEE Year: 1996)
(A) injective
(B) surjective
(C) bijective
(D) constant
✅ Correct: (C)
Q28
Range of \(f(x)=|x-2|\) is (Exam: NEET Year: 2014)
(A) \((-\infty,\infty)\)
(B) \([0,\infty)\)
(C) \((0,\infty)\)
(D) \((-\infty,0]\)
✅ Correct: (B)
Q29
The number of symmetric relations on a set of \(n\) elements is (Exam: IIT-JEE Year: 2005)
(A) \(2^{n^2}\)
(B) \(2^{n(n+1)/2}\)
(C) \(2^{n(n-1)}\)
(D) \(n!\)
✅ Correct: (B)
Q30
If \(f(x)=\ln(x^2)\), domain is (Exam: JEE Main Year: 2021)
(A) \(\mathbb{R}\)
(B) \(\mathbb{R}-\{0\}\)
(C) \((0,\infty)\)
(D) \((-\infty,0)\)
✅ Correct: (B)
Q31
Let \(f(x)=\sin^{-1}x\). Domain is (Exam: NEET Year: 2018)
(A) \(\mathbb{R}\)
(B) \([-1,1]\)
(C) \((-\infty,\infty)\)
(D) \((0,\infty)\)
✅ Correct: (B)
Q32
The relation “\(\le\)” on \(\mathbb{R}\) is (Exam: Olympiad Year: 2009)
(A) equivalence
(B) partial order
(C) symmetric
(D) none
✅ Correct: (B)
Q33
If \(f(x)=x^3+1\), then \(f\) is (Exam: JEE Main Year: 2017)
(A) many–one
(B) injective
(C) not a function
(D) periodic
✅ Correct: (B)
Q34
If \(f(x)=\sqrt{x}\) and \(g(x)=x^2\), then \(g\circ f(x)\) equals (Exam: IIT-JEE Year: 2003)
(A) \(x\)
(B) \(x^2\)
(C) \(|x|\)
(D) \(\sqrt{x}\)
✅ Correct: (A)
Q35
A relation which is reflexive and symmetric but not transitive is (Exam: Olympiad Year: 2012)
(A) equivalence
(B) possible
(C) impossible
(D) identity
✅ Correct: (B)
Q36
Range of \(f(x)=\frac{1}{1+e^{-x}}\) is (Exam: NEET Year: 2020)
(A) \((0,1)\)
(B) \([0,1]\)
(C) \((-\infty,\infty)\)
(D) \((1,\infty)\)
✅ Correct: (A)
Q37
If \(f\) is odd, then (Exam: JEE Advanced Year: 2019)
(A) \(f(-x)=f(x)\)
(B) \(f(-x)=-f(x)\)
(C) \(f(x)=0\)
(D) none
✅ Correct: (B)
Q38
The inverse of a decreasing bijection is (Exam: IIT-JEE Year: 2007)
(A) increasing
(B) decreasing
(C) constant
(D) not defined
✅ Correct: (B)
Q39
If \(f(x)=x+|x|\), then domain is (Exam: NEET Year: 2011)
(A) \((0,\infty)\)
(B) \((-\infty,0)\)
(C) \(\mathbb{R}\)
(D) \([0,\infty)\)
✅ Correct: (C)
Q40
Range of \(f(x)=x+|x|\) is (Exam: NEET Year: 2011)
(A) \(\mathbb{R}\)
(B) \([0,\infty)\)
(C) \((-\infty,0]\)
(D) \((0,\infty)\)
✅ Correct: (B)
Q41
Let \(R\) be transitive and symmetric. Then \(R\) need not be (Exam: Olympiad Year: 2008)
(A) reflexive
(B) symmetric
(C) transitive
(D) equivalence
✅ Correct: (A)
Q42
If \(f(x)=\frac{1}{x^2}\), range is (Exam: JEE Main Year: 2016)
(A) \((-\infty,\infty)\)
(B) \((0,\infty)\)
(C) \([0,\infty)\)
(D) \((-\infty,0)\)
✅ Correct: (B)
Q43
The number of onto functions from a 2-element set to itself is (Exam: IIT-JEE Year: 1995)
(A) 1
(B) 2
(C) 3
(D) 4
✅ Correct: (B)
Q44
Let \(f(x)=\sin x+\cos x\). Maximum value is (Exam: NEET Year: 2018)
(A) 1
(B) \(\sqrt{2}\)
(C) 2
(D) 0
✅ Correct: (B)
Q45
If \(f(x)=x^2\), then \(f\circ f(x)\) is (Exam: JEE Main Year: 2014)
(A) \(x^2\)
(B) \(x^4\)
(C) \(2x^2\)
(D) \(|x|\)
✅ Correct: (B)
Q46
The relation “parallel to” among lines is (Exam: Olympiad Year: 2007)
(A) reflexive
(B) symmetric
(C) transitive
(D) equivalence
✅ Correct: (D)
Q47
If \(f(x)=\ln(x+1)\), domain is (Exam: NEET Year: 2021)
(A) \((-1,\infty)\)
(B) \((0,\infty)\)
(C) \(\mathbb{R}\)
(D) \([0,\infty)\)
✅ Correct: (A)
Q48
If \(f(x)=x^3\) and \(g(x)=\sqrt[3]{x}\), then (Exam: IIT-JEE Year: 1994)
(A) \(f=g\)
(B) \(f=g^{-1}\)
(C) \(f\circ g\neq I\)
(D) none
✅ Correct: (B)
Q49
A relation which is antisymmetric and transitive is (Exam: Olympiad Year: 2013)
(A) equivalence
(B) partial order
(C) symmetric
(D) identity only
✅ Correct: (B)
Q50
If \(f(x)=\frac{x}{|x|}\), domain is (Exam: NEET Year: 2012)
(A) \(\mathbb{R}\)
(B) \(\mathbb{R}-\{0\}\)
(C) \((0,\infty)\)
(D) \((-\infty,0)\)
✅ Correct: (B)
Frequently Asked Questions
An ordered pair is a pair of elements written as \((a, b)\), where the order matters. Two ordered pairs are equal if and only if their corresponding elements are equal.
The Cartesian product of sets \(A\) and \(B\), denoted \(A \times B\), is the set of all ordered pairs \((a, b)\) where \(a \in A\) and \(b \in B\).
If set \(A\) has \(m\) elements and set \(B\) has \(n\) elements, then \(A \times B\) has \(m \times n\) elements.
A relation from set \(A\) to set \(B\) is any subset of the Cartesian product \(A \times B\).
The domain is the set of all first elements of the ordered pairs belonging to the relation.
The range is the set of all second elements of the ordered pairs of a relation.
The codomain is the set from which the second elements of ordered pairs are taken, regardless of whether all elements appear in the relation or not.
A relation that contains no ordered pair is called an empty relation.
A relation that contains all possible ordered pairs of a Cartesian product is called a universal relation.
An identity relation on a set \(A\) consists of all ordered pairs \((a, a)\) for every \(a \in A\).
A relation is reflexive if every element of the set is related to itself, i.e., \((a, a)\) belongs to the relation for all \(a\).
A relation is symmetric if whenever \((a, b)\) belongs to the relation, \((b, a)\) also belongs to it.
A relation is transitive if whenever \((a, b)\) and \((b, c)\) belong to the relation, then \((a, c)\) must also belong to it.
A relation that is reflexive, symmetric, and transitive is called an equivalence relation.
An equivalence class is the set of all elements related to a given element under an equivalence relation.
