Class 12 Math Ch-1 Relations and Functions MCQs Exam 2027

Topic 1: Types of Relations (Reflexive, Symmetric, Transitive, and Equivalence)

1. The Identity Relation on set $A = \{1, 2, 3\}$ is: [2020 A]

   (A) $R = \{(1, 1), (2, 2)\}$

   (B) $R = \{(1, 1), (2, 2), (3, 3)\}$

   (C) $R = \{(1, 1), (2, 2), (3, 3), (1, 2)\}$

   (D) None of these

2. Let $R$ be the relation in the set $N$ given by $R = \{(a, b) : a = b – 2, b > 6\}$. The correct answer is: [2020 A, 2024 A]

   (A) $(6, 8) \in R$

   (B) $(2, 4) \in R$

   (C) $(3, 8) \in R$

   (D) $(8, 7) \in R$

3. The relation $R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\}$ on set $A = \{1, 2, 3, 4\}$ is: [2011]

   (A) Reflexive

   (B) Transitive

   (C) Symmetric

   (D) None of these

4. In the set of real numbers, the relation ‘is less than’ ($<$) is: [2018 A]

   (A) Only Symmetric

   (B) Only Transitive

   (C) Only Reflexive

   (D) Equivalence Relation

5. The smallest reflexive relation containing $(a, a)$ in the set $A = \{a, b, c\}$ is: [2019 C]

   (A) {(a, a), (b, b), (c, c)}

   (B) $\{(a, a)\}$

   (C) $\{(a, a), (b, b)\}$

   (D) None of these

6. In the set of all straight lines in a plane, let $R$ be the relation ‘is perpendicular to’. Then $R$ is: [2019 A]

   (A) Reflexive and Transitive

   (B) Symmetric and Transitive

   (C) Symmetric

   (D) None of these

7. On the set of integers $Z$, the relation $R$ defined by $xRy \iff x – y$ is an even integer, is: [2024 A]

   (A) Only Reflexive

   (B) Only Symmetric

   (C) Equivalence Relation

   (D) Only Transitive

8. Let $A = \{1, 2, 3\}$. The number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive is: [2021 A]

   (A) 1

   (B) 2

   (C) 3

   (D) 4

9. The relation $R = \{(2, 2), (3, 3), (2, 3), (3, 2), (3, 1), (2, 1)\}$ on $A = \{1, 2, 3\}$ is: [2009]

   (A) Reflexive

   (B) Symmetric

   (C) Equivalence

   (D) Transitive

10. Let $A = \{1, 2, 3\}$. The total number of equivalence relations defined on $A$ is: [2018 A, 2021 A]

    (A) 2

    (B) 3

    (C) 5

    (D) 8

11. If $A = \{1, 2, 3\}$, then the number of equivalence relations containing $(1, 2)$ and $(2, 1)$ is: [2017 A]

    (A) 3

    (B) 1

    (C) 2

    (D) 4

12. The relation $R = \{(a, a), (b, b), (c, c), (a, b)\}$ on the set $\{a, b, c\}$ is:

    (A) Reflexive but not symmetric

    (B) Symmetric

    (C) Equivalence

    (D) Only Transitive

13. In the set of natural numbers $N$, the relation ‘is greater than’ ($>$) is:

    (A) Reflexive

    (B) Symmetric

    (C) Transitive

    (D) Equivalence

14. On a non-empty set $A$, the empty relation is:

    (A) Only Reflexive

    (B) Only Symmetric

    (C) Symmetric and Transitive

    (D) Equivalence

15. If $A = \{1, 2, 3, 4\}$, then the number of elements in the Universal Relation on $A$ is:

    (A) 4

    (B) 8

    (C) 16

    (D) 12

16. If $R = \{(a, b) : (a – b) \text{ is divisible by 5} \}$, then the relation $R$ is:

    (A) Reflexive

    (B) Symmetric

    (C) Transitive

    (D) All of the above

17. Let $L$ be the relation ‘is parallel to’ in a plane of lines, then it is:

    (A) Equivalence Relation

    (B) Only Reflexive

    (C) Only Symmetric

    (D) Only Transitive

Topic 2: Types of Functions (One-one, Onto, Into, and Inverse)

18. If $f: R \to R$ where $f(x) = \cos x$, then function $f$ is: [2017 C]

    (A) One-one onto

    (B) Many-one into

    (C) Many-one onto

    (D) One-one into

19. If $f: A \to B$ is a one-one onto (bijective) function, then: [2018 C]

    (A) $n(A) > n(B)$

    (B) $n(A) < n(B)$

    (C) $n(A) = n(B)$

    (D) None of these

20. $f: A \to B$ will be an onto function if: [2019 A, 2025 A]

    (A) $f(A) \subseteq B$

    (B) $f(A) = B$

    (C) $B \subseteq f(A)$

    (D) $f(B) \subseteq A$

21. If $f: R \to R$ is a function, then $f^{-1}$ exists if $f$ is: [2021 A]

    (A) One-one into

    (B) Onto

    (C) One-one onto

    (D) Many-one onto

22. If $A = \{1, 2, 3\}$, $B = \{2, 5, 10, 17\}$ and $f(x) = x^2 + 1$, then $f: A \to B$ is: [2018 C]

    (A) One-one into

    (B) One-one onto

    (C) Many-one into

    (D) Many-one onto

23. If $f: R \to R$ is defined as $f(x) = \frac{1}{x}$, then $f$ is: [2015 A]

    (A) One-one onto

    (B) Many-one onto

    (C) One-one into

    (D) Not defined (at $x=0$)

24. If $f: A \to B$ is an Into function, then: [2022 A]

    (A) $f(A) = B$

    (B) $f(A) \subset B$

    (C) $n(A) = n(B)$

    (D) None of these

25. If $f(x_1) = f(x_2) \implies x_1 = x_2$, then the function is: [2019 A]

    (A) One-one

    (B) Constant

    (C) Onto

    (D) Many-one

26. If $f: R \to R$ where $f(x) = x^2$, then the function is: [2009]

    (A) One-one

    (B) Onto

    (C) One-one onto

    (D) Neither one-one nor onto

27. If $A = \{a, b, c\}$, $B = \{1, 2, 3\}$ and $f = \{(a, 1), (b, 2), (c, 2)\}$, then the function is: [2021 A]

    (A) One-one

    (B) Many-one into

    (C) Many-one onto

    (D) One-one onto

28. If $f: R \to R$ is defined by $f(x) = x^4$: [2021 A]

    (A) One-one onto

    (B) Many-one onto

    (C) One-one into

    (D) Neither one-one nor onto

29. If function $f: N \to N$ is defined by $f(n) = n+1$, then it is:

    (A) One-one onto

    (B) One-one into

    (C) Many-one onto

    (D) Many-one into

30. If $f: A \to B$ is a one-one into function, then:

    (A) $n(A) \leq n(B)$

    (B) $n(A) > n(B)$

    (C) $n(A) = n(B)$

    (D) None of these

31. The function $f(x) = |x|$, $f: R \to R$ is:

    (A) One-one onto

    (B) Many-one into

    (C) One-one into

    (D) Many-one onto

32. If $f(x) = x^3$, then $f: R \to R$ is:

    (A) One-one onto

    (B) Many-one onto

    (C) One-one into

    (D) None of these

33. The domain of the Greatest Integer Function $f(x) = [x]$ is:

    (A) $R$

    (B) $Z$

    (C) $N$

    (D) $(0, 1)$

34. $f: A \to B$ will be Onto if:

    (A) Range = Codomain

    (B) Range $\subset$ Codomain

    (C) Codomain $\subset$ Range

    (D) None

35. The Range of $f(x) = \sqrt{9 – x^2}$ is: [2024 A]

    (A) [0, 3]

    (B) $(-3, 3)$

    (C) $[0, 9]$

    (D) None of these

Topic 3: Inverse & Composition of Functions

36. If $f(x) = 3x – 4$ then $f^{-1}(x) =$ [2025 A]

    (A) $\frac{x + 4}{3}$

    (B) $\frac{x – 4}{3}$

    (C) $3x – 4$

    (D) undefined

37. If $f(x) = (x^2 – 1)$ and $g(x) = (2x + 3)$ then $(g \circ f)(x) = ?$ [2020 A]

    (A) $(2x^2 + 1)$

    (B) $(2x^2 + 3)$

    (C) $(3x^2 + 2)$

    (D) None of these

38. If $f(x) = 5x + 4$, then $f^{-1}(x) =$ [2011]

    (A) $\frac{x-5}{4}$

    (B) $\frac{x-y}{5}$

    (C) $\frac{x-4}{5}$

    (D) $\frac{x-5}{4}$

39. If $f(x) = 8x^3$ and $g(x) = x^{1/3}$, then $(f \circ g)(x) =$ [2016 A]

    (A) $8x$

    (B) $8x^3$

    (C) $x$

    (D) $2x$

40. If $f(x) = \frac{x-1}{x+1}$, then $f(f(x)) =$ [2021 A]

    (A) $x$

    (B) $1/x$

    (C) $-1/x$

    (D) $-x$

41. If $f(x) = 2x + 3$ then $f^{-1}(x) =$ [2009]

    (A) $2x – 3$

    (B) $\frac{x-3}{2}$

    (C) $\frac{x+3}{2}$

    (D) None of these

42. If $f: A \to B$ and $g: B \to C$ are one-one, then $g \circ f: A \to C$ is: [2019 C]

    (A) One-one

    (B) Many-one

    (C) Both

    (D) None of these

43. If $f(x) = \sin x$ and $g(x) = x^2$, then $(f \circ g)(x) =$ [2018 A]

    (A) $\sin^2 x$

    (B) $\sin x^2$

    (C) $x^2 \sin x$

    (D) $\sin \sqrt{x}$

44. If $f(x) = (3 – x^3)^{1/3}$ then $f(f(x)) =$ [2013]

    (A) $x^{1/3}$

    (B) $x^3$

    (C) $(3 – x^3)$

    (D) $x$

45. If $f(x) + 2f(1-x) = x^2 + 2$, then $f(x) =$ [2010]

    (A) $x^2 – 2$

    (B) 1

    (C) $\frac{1}{3}(x-2)^2$

    (D) $\frac{(x-2)^2}{3}$

46. If $f(x) = x^2 – 3x + 2$ then $(f \circ f)(1) =$

    (A) 1

    (B) 0

    (C) 2

    (D) $-1$

47. If $f(x) = \frac{x}{x-1}$, then $(f \circ f)(x) =$ [2017 A]

    (A) $x$

    (B) $1$

    (C) $-x$

    (D) $x-1$

48. If $f(x) = \frac{3-x}{4}$, then $f^{-1}(x) =$

    (A) $4x – 3$

    (B) $3 – 4x$

    (C) $\frac{4}{3-x}$

    (D) None

49. If $f(x) = \frac{1}{1-x}$ and $g(x) = \frac{x-1}{x}$, then $(f \circ g)(x) =$

    (A) $x$

    (B) $1/x$

    (C) $-x$

    (D) $1-x$

50. If $g \circ f$ is one-one, then:

    (A) $f$ must be one-one

    (B) $g$ must be one-one

    (C) $f$ must be onto

    (D) None

51. If $f(x) = 2x$ and $g(x) = x^2$, then $(g \circ f)(2) =$

    (A) 8

    (B) 16

    (C) 4

    (D) 32

52. If $f(x) = \frac{x-1}{x+1}$, then $f^{-1}(0) =$

    (A) 0

    (B) 1

    (C) $-1$

    (D) 2

Topic 4: Binary Operations

53. For operation $a * b = a + 3b^2$, the value of $2 * 4$ is: [2023 A]

    (A) 50

    (B) 48

    (C) 52

    (D) 14

54. If $(a \circ b) = a^3 + b^3$, then $4 \circ (1 \circ 2) =$ [2025 A]

    (A) 729

    (B) 793

    (C) 783

    (D) 792

55. For $a * b = a + b + 1$, the identity element is: [2019 A]

    (A) 1

    (B) -1

    (C) 0

    (D) 2

56. For $a * b = a^2 + b^2$, then $(1 * 2) * 6$ is: [2010]

    (A) 12

    (B) 28

    (C) 61

    (D) None of these

57. If $a * b = 2a + b$, then $(2 * 3) * 4$ is: [2024 A]

    (A) 18

    (B) 17

    (C) 19

    (D) 21

58. $a \circ b = a^3 + b^3$ in $N$ is: [2022 A]

    (A) Associative and Commutative

    (B) Commutative but not Associative

    (C) Only Associative

    (D) None

59. If $a * b = 3a + 4b – 2$, then $4 * 5$ is: [2020 A]

    (A) 30

    (B) 20

    (C) 10

    (D) 15

60. For $x * y = 1 + 12x + xy$, the value of $2 * 3$: [2009]

    (A) 31

    (B) 41

    (C) 43

    (D) 51

61. If $a \circ b = 3a + b$, then $(2 \circ 3) \circ 5 =$ [2017 C]

    (A) 28

    (B) 32

    (C) 36

    (D) 22

62. For $a * b = |a – b|$, the value of $(2 * 5) * 8$ is:

    (A) 5

    (B) 3

    (C) 11

    (D) 1

63. For $a * b = a + b – ab$, then $2 * (3 * 4) =$

    (A) 11

    (B) 15

    (C) 21

    (D) 23

64. For $a * b = \text{L.C.M. of } a, b$, the value of $5 * 7$ is: [2020 A]

    (A) 12

    (B) 35

    (C) 1

    (D) 2

65. For $a \ast b = a + b – 5$, the identity element is:

    (A) 0

    (B) 1

    (C) 5

    (D) $-5$

Topic 5: Counting Relations and Functions

66. If $n(A) = 4$ and $n(B) = 2$, then $n(A \times B) =$ [2025 A]

    (A) 6

    (B) 8

    (C) 16

    (D) None of these

67. Total number of distinct relations defined on $A = \{1, 2, 3\}$ is: [2022 A]

    (A) $2^9$

    (B) $2^3$

    (C) 9

    (D) $2^6$

68. If $A$ has $n$ elements, total number of relations from $A$ to $A$: [2021 A]

    (A) $2^n$

    (B) $n^2$

    (C) $2^{n^2}$

    (D) $n^n$

69. If $A = \{a, b\}$, $B = \{1, 2, 3\}$, the number of one-one functions from $A$ to $B$: [2023 A]

    (A) 6

    (B) 8

    (C) 9

    (D) None of these

70. Number of binary operations on $A = \{a, b\}$ is: [2021 A]

    (A) 4

    (B) 8

    (C) 16

    (D) 20

71. Total number of one-one functions defined on $A = \{1, 2, 3\}$ is: [2019 C]

    (A) 2

    (B) 3

    (C) 4

    (D) 6

72. If $n(A) = 3$ and $n(B) = 2$, then $n(A \times B) =$ [2017 A]

    (A) 6

    (B) 4

    (C) 2

    (D) 0

73. Total number of binary operations on $A = \{1, 2, 3\}$ is: [2022 A]

    (A) $3^6$

    (B) $3^9$

    (C) $2^9$

    (D) $3^{3^2}$

74. If $n(A) = m$, total reflexive relations on $A$ is:

    (A) $2^m$

    (B) $2^{m^2-m}$

    (C) $2^{m^2}$

    (D) $m^2$

75. If $A = \{1, 2\}$, $B = \{a, b, c\}$, total functions from $A$ to $B$ is: [2023 A]

    (A) 9

    (B) 12

    (C) 64

    (D) None

76. Total bijective functions $f: A \to A$ where $A = \{1, 2, \dots, n\}$ is:

    (A) $n$

    (B) $n!$

    (C) $\frac{1}{2}n$

    (D) $n^n$

77. If $A = \{1, 2, 3\}$ and $B = \{a, b\}$, total onto functions from $A$ to $B$: [2022 A]

    (A) 6

    (B) 8

    (C) 2

    (D) 9

78. Number of non-commutative binary operations on $A = \{a, b\}$ is:

    (A) 2

    (B) 4

    (C) 8

    (D) 12

79. Total symmetric relations on $A = \{a, b, c\}$ is:

    (A) $2^3$

    (B) $2^6$

    (C) $2^9$

    (D) $2^4$

80. If $n(A) = 3, n(B) = 3$, number of one-one onto functions from $A$ to $B$:

    (A) 3

    (B) 9

    (C) 6

    (D) 27

81. Number of constant functions from $A = \{1, 2, 3, 4\}$ to itself:

    (A) 4

    (B) 16

    (C) 256

    (D) 1

82. If $A = \{a, b, c\}$ and $B = \{d\}$, total functions from $A$ to $B$:

    (A) 3

    (B) 1

    (C) 4

    (D) 0

Topic 6: Miscellaneous

83. Range of $f(x) = \sqrt{(x-1)(3-x)}$ is: [2009]

    (A) $[1, 3]$

    (B) [0, 1]

    (C) $[-2, 2]$

    (D) None of these

84. Let $A = \{1, 2, 3\}$. Which function does not have an inverse? [2018 A]

    (A) $\{(1,1), (2,2), (3,3)\}$

    (B) $\{(1,2), (2,1), (3,1)\}$

    (C) $\{(1,3), (3,2), (2,1)\}$

    (D) None

85. Domain of $f(x) = \sqrt{x^2 – 1}$ is: [2014 A]

    (A) $(-\infty, -1] \cup [1, \infty)$

    (B) $[-1, 1]$

    (C) $(1, \infty)$

    (D) None

86. Range of $f(x) = \frac{x^2 + x + 2}{x^2 + x + 1}$ is:

    (A) $(1, \infty)$

    (B) (1, 7/3]

    (C) $(1, 7/5)$

    (D) $(1, 11/7)$

87. If $A = \{1, 2, 3, 4\}$ and $R = \{(1, 2), (1, 1), (1, 3), (2, 2), (2, 1)\}$, then $R$ is: [2018 A]

    (A) Transitive

    (B) Reflexive

    (C) Symmetric

    (D) None

88. Domain of $f(x) = \frac{1}{\sqrt{x – [x]}}$ is:

    (A) $R$

    (B) $Z$

    (C) $R – Z$

    (D) $\phi$

89. The smallest equivalence relation on $A = \{1, 2, 3\}$ is:

    (A) $\phi$

    (B) {(1,1), (2,2), (3,3)}

    (C) $A \times A$

    (D) $\{(1,2), (2,1)\}$

90. $A = \{1, 2, 3\}$, $B = \{6, 7, 8\}$ and $f(x) = x + 5$. Then $f$ is: [2018 A]

    (A) Into

    (B) One-one onto

    (C) Many-one onto

    (D) Constant

91. Number of relations on an empty set $\phi$:

    (A) 0

    (B) 1

    (C) 2

    (D) $2^n$

92. Inverse of $a$ in $a * b = a + b – ab$ is:

    (A) $1/a$

    (B) $a/(a-1)$

    (C) $0$

    (D) Not defined

93. Domain of $f(x) = \sqrt{x}$ is:

    (A) $R$

    (B) $[0, \infty)$

    (C) $(0, \infty)$

    (D) $Z$

94. $R$ on $A$ is equivalence if $R$ is:

    (A) Reflexive

    (B) Symmetric

    (C) Transitive

    (D) All three

95. If $f: R \to R$ is a constant function, its Range is:

    (A) $R$

    (B) {c}

    (C) $\phi$

    (D) $N$

96. Period of $f(x) = \cos x$ is:

    (A) $\pi$

    (B) $2\pi$

    (C) $\pi/2$

    (D) $4\pi$

97. Identity for $a * b = a + b$ is:

    (A) 1

    (B) 0

    (C) $-1$

    (D) $e$

98. Number of subsets of $A \times A$ if $A = \{1, 2\}$ is:

    (A) 4

    (B) 16

    (C) 8

    (D) 32

99. Domain of $f(x) = \log x$ is:

    (A) $R$

    (B) (0, \infty)

    (C) $[0, \infty)$

    (D) $R – \{0\}$

100. Maximum number of relations on $A = \{1, 2, 3\}$ is:

     (A) $2^3$

     (B) $2^6$

     (C) $2^9$

     (D) $3^3$