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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
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
Q3
If \(f(x)=|x|\), then \(f\) is
(Exam: NEET Year: 2015)
(A) injective
(B) surjective
(C) bijective
(D) not injective
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) \((1/2,\infty)\)
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
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
Q7
If \(f(x)=\frac{1}{x}\), domain is
(Exam: AIIMS Year: 2009)
(A) \(\mathbb{R}\)
(B) \(\mathbb{R}-\{0\}\)
(C) \((0,\infty)\)
(D) \((-\infty,0)\)
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 \(\mathbb{R}\)
Q9
Range of \(f(x)=x^2+1\) is
(Exam: NEET Year: 2017)
(A) \(\mathbb{R}\)
(B) \((1,\infty)\)
(C) \([1,\infty)\)
(D) \((-\infty,1]\)
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) \(x/2-3\)
Q11
If \(f(x)=\log x\), domain is
(Exam: JEE Main Year: 2019)
(A) \(\mathbb{R}\)
(B) \((0,\infty)\)
(C) \([0,\infty)\)
(D) \((-\infty,0)\)
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
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
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
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
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
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) \((0,1)\)
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
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
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) \(\log x^2\)
Q21
The relation \(xRy \iff x-y\) is even is
(Exam: Olympiad Year: 2010)
(A) reflexive
(B) symmetric
(C) transitive
(D) equivalence
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) \(\mathbb{R}\)
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) \(c=d\)
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) \(n^2\)
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
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) \(\sqrt{x}\)
Q27
A function with inverse must be
(Exam: IIT-JEE Year: 1996)
(A) injective
(B) surjective
(C) bijective
(D) constant
Q28
Range of \(f(x)=|x-2|\) is
(Exam: NEET Year: 2014)
(A) \((-\infty,\infty)\)
(B) \([0,\infty)\)
(C) \((0,\infty)\)
(D) \((-\infty,0]\)
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) \(n!\)
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) \((-\infty,0)\)
Q31
Let \(f(x)=\sin^{-1}x\). Domain is
(Exam: NEET Year: 2018)
(A) \(\mathbb{R}\)
(B) \([-1,1]\)
(C) \((-\infty,\infty)\)
(D) \((0,\infty)\)
Q32
The relation “\(\le\)” on \(\mathbb{R}\) is
(Exam: Olympiad Year: 2009)
(A) equivalence
(B) partial order
(C) symmetric
(D) none
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
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) \(\sqrt{x}\)
Q35
A relation which is reflexive and symmetric but not transitive is
(Exam: Olympiad Year: 2012)
(A) equivalence
(B) possible
(C) impossible
(D) identity
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) \((1,\infty)\)
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
Q38
The inverse of a decreasing bijection is
(Exam: IIT-JEE Year: 2007)
(A) increasing
(B) decreasing
(C) constant
(D) not defined
Q39
If \(f(x)=x+|x|\), then domain is
(Exam: NEET Year: 2011)
(A) \((0,\infty)\)
(B) \((-\infty,0)\)
(C) \(\mathbb{R}\)
(D) \([0,\infty)\)
Q40
Range of \(f(x)=x+|x|\) is
(Exam: NEET Year: 2011)
(A) \(\mathbb{R}\)
(B) \([0,\infty)\)
(C) \((-\infty,0]\)
(D) \((0,\infty)\)
Q41
Let \(R\) be transitive and symmetric. Then \(R\) need not be
(Exam: Olympiad Year: 2008)
(A) reflexive
(B) symmetric
(C) transitive
(D) equivalence
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) \((-\infty,0)\)
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
Q44
Let \(f(x)=\sin x+\cos x\). Maximum value is
(Exam: NEET Year: 2018)
(A) 1
(B) \(\sqrt{2}\)
(C) 2
(D) 0
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) \(|x|\)
Q46
The relation “parallel to” among lines is
(Exam: Olympiad Year: 2007)
(A) reflexive
(B) symmetric
(C) transitive
(D) equivalence
Q47
If \(f(x)=\ln(x+1)\), domain is
(Exam: NEET Year: 2021)
(A) \((-1,\infty)\)
(B) \((0,\infty)\)
(C) \(\mathbb{R}\)
(D) \([0,\infty)\)
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
Q49
A relation which is antisymmetric and transitive is
(Exam: Olympiad Year: 2013)
(A) equivalence
(B) partial order
(C) symmetric
(D) identity only
Q50
If \(f(x)=\frac{x}{|x|}\), domain is
(Exam: NEET Year: 2012)
(A) \(\mathbb{R}\)
(B) \(\mathbb{R}-\{0\}\)
(C) \((0,\infty)\)
(D) \((-\infty,0)\)

RELATIONS AND FUNCTIONS – Learning Resources

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