CBSE Class 12 Mathematics Relations and Functions MCQs Set E

Question: Let f : R → R be defined as f(x) = x²+1/2 , then
(a) f is one-one onto
(b) f is one-one but not onto
(c) f is onto but not one-one
(d) f is neither one-one nor onto
Answer: d

Question: The relation R in R defined by R ={(a ,b ):≤b³} is
(a) reflexive
(b) symmetric
(c) transitive
(d) None of these 
Answer: d

Question: If f = {( 5,2 ), (6, 3)} and g = {(2,5 ), ( 3,6 )}, then the range of f and g respectively. 
(a) (5,3) and (2, 3)
(b) (5, 6) and (3,2)
(c) (2,3) and (5,6)
(d) (2,3) and (6,5) 
Answer: c

Question: Consider the non-empty set consisting of children in a family and a relation R defined as a R b 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: b

Question: Let A = {0,1,2,3} and define a relation R on A as follows R = {(0,0), (0,1), (0,3), (1 ,0), (1,1), (2,2), (3,0), (3,3), then R is
(a) reflexive and symmetric
(b) symmetric and transitive
(c) reflexive and transitive
(d) a equivalence relation
Answer: a

Question: Relation R in set N of natural numbers defined as R {(x ,y) :y=x+5 and x < 4} are 
(a) reflexive
(b) symmetric
(c) not reflexive not symmetric
(d) not reflexive not symmetric not transitive 
Answer: d

Question: Let R be a relation on the set N of natural numbers defined by nRm, if n divides m. Then, R is
(a) reflexive and symmetric
(b) transitive and symmetric
(c) equivalence
(d) reflexive, transitive but not symmetric
Answer: d

Question: Relation R in the set Z of all integers defined as R={( x, y ) :x-y is an integer} is/are 
(a) reflexive and symmetric
(b) only transitive
(c) symmetric but not transitive
(d) an equivalence relation 
Answer: d

Question: Relation R in the set A = {1,2,3,4,5,6} as R={(x ,y) :y is divisible by x} are
(a) reflexive and transitive
(b) reflexive and symmetric
(c) symmetric and transitive
(d) None of the above
Answer: a

Question: Let A = {1, 2, 3, 4} and let R = {(2, 2), (3, 3), (4, 4), (1, 2)} be a relation on A. Then R is:
(a) reflexive
(b) symmetric
(c) transitive
(d) None of these
Answer: c

Question: Let S = {1, 2, 3, 4, 5} and let A = S × S. Define the relation R on A as follows: (a, b) R (c, d) iff ad = cb. Then, R is
(a) reflexive only
(b) Symmetric only
(c) Transitive only
(d) Equivalence relation
Answer: d

Question: let R be the relation in the set N given by R={(a,b):a=b-2,b>6}.Choose the correct answer.
(a) (2,4)€R
(b) (3,8) € R
(c) (6,8)€ R
(d) (8,10) € R
Answer: d

Question: f : [-3π,2π] →[-1,1] defined by f (x) sin = 3x is
(a) into
(b) onto
(c) one one
(d) None of these 
Answer: b

Question: If f : R → R defined by f (x) = 2x-7/4 is an invertible function, then f ^–1 is equal to 
(a) (4x+5)/2
(b) (4x+7)/2
(c) 3x+2/2
(d) 9x+3/5
Answer: b

Question: Given f (x) = log(1+x/1-x) and g(x) = 3x+x³/1+3x² , then fog(x) equals 
(a) – f (x)
(b) 3f(x)
(c) [f (x)]³
(d) None of these
Answer: b

Question: If the binary operation x on the set of integers Z, is defined by a x b = a + 3b², then the value of 8 x 3 is 
(a) 32
(b) 40
(c) 36
(d) 35
Answer: d

Question: Let f: R → R is defined as f(x) = 3x then f is
(a) f is one-one and onto
(b) f is one-one but not onto
(c) f is many-one
(d) f is neither one-one nor onto
Answer: a

Question: 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) symmetric, transitive but not reflexive
(d) neither transitive nor reflexive but symmetric
Answer: d

Question: If N be the set of all natural numbers, consider f : N → N such that f(x) = 2x, ∀ x ∈ N, then f is 
(a) one-one onto
(b) one-one into
(c) many-one onto
(d) None of these
Answer: b

Question: 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 is 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: c

Question: Let f(x) = 2x², g(x) = 3x + 2 and fog (x) = 18 x² + 24x + c, then c = 
(a) 2
(b) 8
(c) 6
(d) 4
Answer: b

Question: Let f(x) = ax-b/cx+d . Then fof (x) = x provided that 
(a) d = – a
(b) d = a
(c) a = b = c = d = 1
(d) a = b = 1
Answer: a

Question: If R is a relation in a set A such that (a, a)∈ R for every a ∈ A, then the relation R is called
(a) symmetric
(b) reflexive
(c) transitive
(d) symmetric or transitive
Answer: b

Question: Let f : R → R be defined by f(x) = 1/x ∀ ∈ R . Then f is
(a) one-one
(b) onto
(c) bijective
(d) f is not defined
Answer: d

Question: If f(x) = |x| and g(x) = |5x – 2|, then 
(a) gof (x) = |5x – 2|
(b) gof (x) = |5| x | – 2|
(c) fog (x) = |5| x | – 2|
(d) fog (x) = |5x + 2|
Answer: b

Question: Let A = {1, 2, 3} and B = {a, b, c}, then the number of bijective functions from A to B are 
(a) 2
(b) 8
(c) 6
(d) 4
Answer: c

Question: If f is an even function and g is an odd function, then the function fog is 
(a) an even function
(b) an odd function
(c) neither even nor odd
(d) a periodic function
Answer: b

Question: A Relation from A to B is an arbitrary subset of:
(a) AxB
(b) BxB
(c) AxA
(d) BxB
Answer: a

Question: If f : R → R be a mapping defined by f(x) = x³ + 5, then f–1(x) is equal to: 
(a) (x + 3)^1/3
(b) (x – 5)^1/3
(c) (5 – x)^1/3
(d) (5 – x)
Answer: b

Question: If f : R → R is given by f (x) =√1-x² , then fof is 
(a) √x
(b) x²
(c) x
(d) 1– x²
Answer: c

Question: Consider the function f in A = R – {2/3} defined as f (x) = 4x+3/6x-4 , then f ^–1 is equal to 
(a) 3+4x/6x-4
(b) 6x-4/3+4x
(c) 3-4x/6x-4
(d) 9+2x/6x-4
Answer: a

Question: A relation R in a set A is called empty relation, if 
(a) no element of A is related to any element of A
(b) every element of A is related to every element of A
(c) some elements of A are related to some elements of A
(d) None of the above
Answer: a

Question: Set A has 3 elements and the set B has 4 elements. Then the number of injective functions that can be defined from set A to set B is
(a) 144
(b) 12
(c) 24
(d) 64
Answer: c

Question: A relation R in a set A is said to be an equivalence relation, if R is 
(a) symmetric only
(b) reflexive only
(c) transitive only
(d) All of these
Answer: d

Question: Let A = {x : -1 ≤ x ≤ 1} and f : A → A is a function defined by f(x) = x |x| then f is
(a) a bijection
(b) injection but not surjection
(c) surjection but not injection
(d) neither injection nor surjection
Answer: a

Question: Set A has 3 elements and set B has 4 elements. Then the number of injective functions that can be defined from set A to set B is
(a) 144
(b) 12
(c) 24
(d) 64
Answer: c

Question: If f (x) = sin x + cos x, g (x) = x² – 1, then g (f (x)) is invertible in the domain 
(a) [0,π/2]
(b) [-π/4 , π/4]
(c) [-π/2 , π/2]
(d) [0, π]
Answer: b

Question: The relation “less than” in the set of natural numbers is : 
(a) only symmetric
(b) only transitive
(c) only reflexive
(d) equivalence relation
Answer: b

Question: Let g(x) = x² – 4x – 5, then 
(a) g is one-one on R
(b) g is not one-one on R
(c) g is bijective on R
(d) None of these
Answer: b

Question: If R = {(x, y) : x is father of y}, then R is
(a) reflexive but not symmetric
(b) symmetric and transitive
(c) neither reflexive nor symmetric nor transitive
(d) Symmetric but not reflexive
Answer: c

Question: Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defined by y = 2×4, is
(a) one-one onto
(b) one-one into
(c) many-one onto
(d) many-one into
Answer: c

Question: Let f : R → R be defined as f (x) = 2x³. then 
(a) f is one-one onto
(b) f is one-one but not onto
(c) f is onto but not one-one
(d) f is neither one-one nor onto
Answer: a

Question: Let R be a relation on the set L of lines defined by l1 R l2 if l1 is perpendicular to l2, then relation R is 
(a) reflexive and symmetric
(b) symmetric and transitive
(c) equivalence relation
(d) symmetric
Answer: d

Question: 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: d

Question: Let A = {1, 2, 3} and R = {(1, 2), (2, 3)} be a relation in A. 
Then, the minimum number of ordered pairs may be added, so that R becomes an equivalence relation, is
(a) 7
(b) 5
(c) 1
(d) 4
Answer: a

Question: The maximum number of equivalence relations on the set A = {1, 2, 3} are
(a) 1
(b) 2
(c) 3
(d) 5
Answer: d

Question: Let P = {(x, y) l x² + y² = 1, x, y ∈ R}. Then, P is
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) Anti-symmetric
Answer: b

Question: Let R be a relation on set of lines as L₁ R L₂ if L₁ is perpendicular to L₂. Then
(a) R is Reflexive
(b) R is transitive
(c) R is symmetric
(d) R is an equivalence relation
Answer: c

Read Assertion and reason carefully and write correct option for each question

(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.

Question:
Assertion (A)The relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),(3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.
Reason (R)R is not symmetric, as (1, 2) ∈R but (2, 1) ∉R. Similarly, R is not transitive, as (1, 2) ∈R and (2, 3) ∈R
but (1, 3) ∉R.
Answer: a

Question:
Assertion (A) Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. R is an equivalence relation
Reason (R) Since R is reflexive, symmetric but R is not transitive.
Answer: c

Question:
Assertion (A) The Modulus Function f :R→R, given by f (x) = | x | is not one one and onto function
Reason (R) The Modulus Function f :R→R, given by f (x) = | x | is bijective function
Answer: c

Question:
Assertion (A) Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function From A to B. Then f is one-one.
Reason (R) Since the function f : N→N, given by f (x) = 2x, is not onto
Answer: b

Question:
Assertion (A)The relation R in the set Z of integers given by R = {(a, b) : 2 divides a – b} is reflexive and symmetric
Reason (R)R is reflexive, as 2 divides (a – a) for all a ∈Z.
Answer: b

Question:
Assertion (A)A function f : X →Y is said to be one-one and onto (or bijective)
Reason (R) if f is both one-one and onto.
Answer: a

Question:
Assertion (A) The relation R in R defined as R = {(a, b) :a≤b} is not equivalence relation.
Reason (R) Since R is not reflexive but it is symmetric and transitive.
Answer: c

Question:
Assertion (A) Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Then f is one-one.
Reason (R) f is bijective function
Answer: c