# MCQ Questions for Class 12 Maths Chapter 1 Relations and Functions with Answers

Question 1.

If f(x_{1}) = f (x_{2}) ⇒ x_{1} = x_{2} ∀ x_{1} x_{2} ∈ A then the function f: A → B is

(a) one-one

(b) one-one onto

(c) onto

(d) many one

## Answer

Answer: (a) one-one

Question 2.

What type of a relation is R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} on the set A – {1, 2, 3, 4}

(a) Reflexive

(b) Transitive

(c) Symmetric

(d) None of these

## Answer

Answer: (d) None of these

Question 3.

If F : R → R such that f(x) = 5x + 4 then which of the following is equal to f^{-1}(x).

(a) \(\frac{x-5}{4}\)

(b) \(\frac{x-y}{5}\)

(c) \(\frac{x-4}{5}\)

(d) \(\frac{x}{4}\) -5

## Answer

Answer: (c) \(\frac{x-4}{5}\)

Question 4.

If an operation is defined by a* b = a² + b², then (1 * 2) * 6 is

(a) 12

(b) 28

(c) 61

(d) None of these

## Answer

Answer: (c) 61

Question 5.

Consider the binary operation * on a defined by x * y = 1 + 12x + xy, ∀ x, y ∈ Q, then 2 * 3 equals

(a) 31

(b) 40

(c) 43

(d) None of these

## Answer

Answer: (a) 31

Question 6.

The range of the function f(x) = \(\sqrt{(x-1)(3-x)}\) is

(a) [1, 3]

(b) [0, 1]

(c) [-2, 2]

(d) None of these

## Answer

Answer: (a) [1, 3]

Question 7.

If f: R → R defined by f(x) = 2x + 3 then f^{-1}(x) =

(a) 2x – 3

(b) \(\frac{x-3}{2}\)

(c) \(\frac{x+3}{2}\)

(d) None of these

## Answer

Answer: (b) \(\frac{x-3}{2}\)

Question 8.

The function f(x) = log (x² + \(\sqrt{x^2+1}\) ) is

(a) even function

(b) odd function

(c) Both

(d) None of these

## Answer

Answer: (a) even function

Question 9.

Let E = {1, 2, 3, 4} and F = {1, 2} Then, the number of onto functions from E to F is

(a) 14

(b) 16

(c) 12

(d) 8

## Answer

Answer: (a) 14

Question 10.

If A, B and C are three sets such that A ∩ B = A ∩ C and A ∪ B = A ∪ C. then

(a) A = B

(b) A = C

(c) B = C

(d) A ∩ B = d

## Answer

Answer: (c) B = C

Question 11.

Let A = {1, 2}, how many binary operations can be defined on this set?

(a) 8

(b) 10

(c) 16

(d) 20

## Answer

Answer: (c) 16

Question 12.

Let A = {1, 2, 3, 4,…. n} How many bijective function f : A → B can be defined?

(a) \(\frac{1}{2}\)n

(d) n

## Answer

Answer: (c) [n

Question 13.

If A = (1, 2, 3}, B = {6, 7, 8} is a function such that f(x) = x + 5 then what type of a function is f?

(a) Many-one onto

(b) Constant function

(c) one-one onto

(d) into

## Answer

Answer: (c) one-one onto

Question 14.

Let function R → R is defined as f(x) = 2x³ – 1, then ‘f’ is

(a) 2x³ + 1

(b) (2x)³ + 1

(c) (1 – 2x)³

(d) (\(\frac{1+x}{2}\))^{1/3}

## Answer

Answer: (d) (\(\frac{1+x}{2}\))^{1/3}

Question 15.

Let the functioin ‘f’ be defined by f (x) = 5x² + 2 ∀ x ∈ R, then ‘f’ is

(a) onto function

(b) one-one, onto function

(c) one-one, into function

(d) many-one into function

## Answer

Answer: (d) many-one into function

Question 16.

A relation R in human being defined as, R = {{a, b) : a, b ∈ human beings : a loves A} is-

(a) reflexive

(b) symmetric and transitive

(c) equivalence

(d) None of these

## Answer

Answer: (c) equivalence

Question 17.

If f(x) + 2f (1 – x) = x² + 2 ∀ x ∈ R, then f(x) =

(a) x² – 2

(b) 1

(c) \(\frac{1}{3}\) (x – 2)²

(d) None of these

## Answer

Answer: (c) \(\frac{1}{3}\) (x – 2)²

Question 18.

The period of sin² θ is

(a) π²

(b) π

(c) 2π

(d) \(\frac{π}{2}\)

## Answer

Answer: (b) π

Question 19.

The domain of sin-1 (log (x/3)] is. .

(a) [1, 9]

(b) [-1, 9]

(c) [-9, 1]

(d) [-9, -1]

## Answer

Answer: (a) [1, 9]

Question 20.

f(x) = \(\frac{log_2(x+3)}{x^2+3x+2}\) is the domain of

(a) R – {-1, -2}

(b) (- 2, ∞) .

(c) R- {- 1,-2, -3}

(d) (-3, + ∞) – {-1, -2}

## Answer

Answer: (d) (-3, + ∞) – {-1, -2}

Question 21.

If the function f(x) = x³ + e^{x/2} and g (x) = f^{n}(x), then the value of g'(1) is

(a) 1

(b) 2

(c) 3

(d) 4

## Answer

Answer: (b) 2

Question 22.

What type of relation is ‘less than’ in the set of real numbers?

(a) only symmetric

(b) only transitive

(c) only reflexive

(d) equivalence

## Answer

Answer: (b) only transitive

Question 23.

If A = [1, 2, 3}, B = {5, 6, 7} and f: A → B is a function such that f(x) = x + 4 then what type of function is f?

(a) into

(b) one-one onto

(c) many-onto

(d) constant function

## Answer

Answer: (b) one-one onto

Question 24.

f: A → B will be an into function if

(a) range (f) ⊂ B

(b) f(a) = B

(c) B ⊂ f(a)

(d) f(b) ⊂ A

## Answer

Answer: (a) range (f) ⊂ B

Question 25.

If f : R → R such that f(x) = 3x then what type of a function is f?

(a) one-one onto

(b) many one onto

(c) one-one into

(d) many-one into

## Answer

Answer: (c) one-one into

Question 26.

If f: R → R such that f(x) = 3x – 4 then which of the following is f^{-1}(x)?

(a) \(\frac{1}{3}\) (x + 4)

(b) \(\frac{1}{3}\) (x – 4)

(c) 3x – 4

(d) undefined

## Answer

Answer: (a) \(\frac{1}{3}\) (x + 4)

Question 27.

A = {1, 2, 3} which of the following function f: A → A does not have an inverse function

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

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

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

(d) {(1, 2), (2, 3), (3, 1)

## Answer

Answer: (b) {(1, 2), (2, 1), (3, 1)}

Question 28.

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a congruent to b ∀ a, b ∈ T. Then R is

(a) reflexive but-not transitive

(b) transitive but not symmetric

(c) equivalence

(d) None of these

## Answer

Answer: (c) equivalence

Question 29.

Consider the non-empty set consisting of children is a family and a relation R defined as aRb If a is brother of b. Then R is

(a) symmetric but not transitive

(b) transitive but not symmetric

(c) neither symmetric nor transitive

(d) both symmetric and transitive

## Answer

Answer: (b) transitive but not symmetric

Question 30.

The maximum number of equivalence relations on the set A = {1, 2, 3} are

(a) 1

(b) 2

(c) 3

(d) 5

## Answer

Answer: (d) 5

Question 31.

If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is

(a) reflexive

(b) transitive

(c) symmetric

(d) None of these

## Answer

Answer: (b) transitive

Question 32.

Let us define a relation R in R as aRb if a ≥ b. Then R is

(a) an equivalence relation

(b) reflexive, transitive but not symmetric

(c) neither transitive nor reflexive but symmetric

(d) symmetric, transitive but not reflexive

## Answer

Answer: (b) reflexive, transitive but not symmetric

Question 33.

Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Then R is

(a) reflexive but not symmetric

(b) reflexive-but not transitive.

(c) symmetric and transitive

(d) neither symmetric, nor transitive

## Answer

Answer: (a) reflexive but not symmetric

Question 34.

The identity element for the binary operation * defined on Q ~ {0} as

a * b = \(\frac{ab}{2}\) ∀ a, b ∈ Q ~ {0} is

(a) 1

(b) 0

(c) 2

(d) None of these

## Answer

Answer: (c) 2

Question 35.

If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

(a) 720

(b) 120

(c) 0

(d) None of these

## Answer

Answer: (c) 0

Question 36.

Let A = {1, 2,3,…. n} and B = { a, b}. Then the number of surjections from A into B is

(a) ^{n}P_{2}

(b) 2^{n} – 2

(c) 2^{n} – 1

(d) None of these

## Answer

Answer: (b) 2^{n} – 2

Question 37.

Let f : R → R be defined by f (x) = \(\frac{1}{x}\) ∀ x ∈ R. Then f is

(a) one-one

(b) onto

(c) bijective

(d) f is not defined

## Answer

Answer: (d) f is not defined

Question 38.

Let f: R → R. be defined by f (x) = 3x² – 5 and g : R → R by g (x) = \(\frac{x}{x^2+1}\). Then g o f is

## Answer

Answer: (a)

Question 39.

Which of the following functions from Z into Z are bijective?

(a) f(x) = x³

(b) f(x) = x + 2

(c) f(x) = 2x + 1

(d) f{x) = x² + 1

## Answer

Answer: (b) f(x) = x + 2

Question 40.

Let f: R → R be the function defined by f(x) = x³ + 5. Then f^{-1} (x) is

(a) (x + 5)^{1/3}

(b) (x -5)^{1/3}

(c) (5 – x)^{1/3}

(d) 5 – x

## Answer

Answer: (b) (x -5)^{1/3}

Question 41.

Let f: A → B and g : B → C be the bijective functions. Then (g o f)^{-1} is,

(a) f^{-1} o g^{-1}

(b) f o g

(c ) g^{-1} o f^{-1}

(d) g o f

## Answer

Answer: (a) f^{-1} o g^{-1}

Question 42.

Let f: R – {\(\frac{3}{5}\)} → R be defined by f(x) = \(\frac{3x+2}{5x-3}\) then

(a) f^{-1}(x) = f(x)

(b) f^{-1}(x) = -f(x)

(c ) (f o f)x = -x

(d ) f^{-1}(x) = \(\frac{1}{19}\) f(x)

## Answer

Answer: (a) f^{-1}(x) = f(x)

Question 43.

Let f: [0, 1| → [0, 1| be defined by

(a) Constant

(b) 1 + x

(c) x

(d) None of these

## Answer

Answer: (c) x

Question 44.

Let f: |2, ∞) → R be the function defined by f(x) – x² – 4x + 5, then the range of f is

(a) R

(b) [1, ∞)

(c) [4, ∞)

(d) [5, ∞)

## Answer

Answer: (b) [1, ∞)

Question 45.

Let f: N → R be the function defined by f(x) = \(\frac{2x-1}{2}\) and g: Q → R be another function defined by g (x) = x + 2. Then (g 0 f) \(\frac{3}{2}\) is

(a) 1

(b) 0

(c) \(\frac{7}{2}\)

(d) None of these

## Answer

Answer: (d) None of these

Question 46.

Let f: R → R be defined by

then f(- 1) + f (2) + f (4) is

(a) 9

(b) 14

(c) 5

(d) None of these

## Answer

Answer: (a) 9

Question 47.

Let f : R → R be given by f (,v) = tan x. Then f^{-1}(1) is

(a) \(\frac{π}{4}\)

(b) {nπ + \(\frac{π}{4}\) : n ∈ Z}

(c) does not exist

(d) None of these

## Answer

Answer: (b) {nπ + \(\frac{π}{4}\) : n ∈ Z}

Question 48.

The relation R is defined on the set of natural numbers as {(a, b): a = 2b}. Then, R^{-1} is given by

(a) {(2, 1), (4, 2), (6, 3),….}

(b) {(1, 2), (2, 4), (3, 6),….}

(c) R^{-1} is not defined

(d) None of these

## Answer

Answer: (b) {(1, 2), (2, 4), (3, 6),….}

Question 49.

The relation R = {(1,1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} on set A = {1, 2, 3} is

(a) Reflexive but not symmetric

(b) Reflexive but not transitive

(c) Symmetric and transitive

(d) Neither symmetric nor transitive

## Answer

Answer: (a) Reflexive but not symmetric

Question 50.

Let P = {(x, y) | x² + y² = 1, x, y ∈ R]. Then, P is

(a) Reflexive

(b) Symmetric

(c) Transitive

(d) Anti-symmetric

## Answer

Answer: (b) Symmetric

Question 51.

Let R be an equivalence relation on a finite set A having n elements. Then, the number of ordered pairs in R is

(a) Less than n

(b) Greater than or equal to n

(c) Less than or equal to n

(d) None of these

## Answer

Answer: (b) Greater than or equal to n

Question 52.

For real numbers x and y, we write xRy ⇔ x – y + √2 is an irrational number. Then, the relational R is

(a) Reflexive

(b) Symmetric

(c) Transitive

(d) None of these

## Answer

Answer: (a) Reflexive

Question 53.

Let R be a relation on the set N be defined by {(x, y) | x, y ∈ N, 2x + y = 41}. Then R is

(a) Reflexive

(b) Symmetric

(c) Transitive

(d) None of these

## Answer

Answer: (d) None of these

Question 54.

Which one of the following relations on R is an equivalence relation?

(a) aR_{1}b ⇔ |a| = |b|

(b) aR_{2}b ⇔ a ≥ b

(c) aR_{3}b ⇔ a divides b

(d) aR_{4}b ⇔ a < b

## Answer

Answer: (a) aR_{1}b ⇔ |a| = |b|

Question 55.

Let R be a relation on the set N of natural numbers denoted by nRm ⇔ n is a factor of m (i.e. n | m). Then, R is

(a) Reflexive and symmetric

(b) Transitive and symmetric

(c) Equivalence

(d) Reflexive, transitive but not symmetric

## Answer

Answer: (d) Reflexive, transitive but not symmetric

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