Relations and Functions Class 12 Mathematics Exam Questions

Please refer to Relations and Functions Class 12 Mathematics Exam Questions provided below. These questions and answers for Class 12 Mathematics have been designed based on the past trend of questions and important topics in your Class 12 Mathematics books. You should go through all Class 12 Mathematics Important Questions provided by our teachers which will help you to get more marks in upcoming exams.

Class 12 Mathematics Exam Questions Relations and Functions

Class 12 Mathematics students should read and understand the important questions and answers provided below for Relations and Functions which will help them to understand all important and difficult topics.

Very Short Answer Type Questions

Question. If R = {(x, y) : x + 2y = 8} is a relation on N, write the range of R.
Answer : Here, R = {(x, y) : x + 2y = 8}, where x, y ∈ N.
For x = 1, 3, 5, …
x + 2y = 8 has no solution in N.
For x = 2, we have 2 + 2y = 8 ⇒ y = 3
For x = 4, we have 4 + 2y = 8 ⇒ y = 2
For x = 6 , we have 6 + 2y = 8 ⇒ y = 1
For x = 8, 10, …
x + 2y = 8 has no solution in N.
∴ Range of R = {y : (x, y) ∈ R} = { 1, 2, 3}

Question. Let R = {(a, a³) : a is a prime number less than 5} be a relation. Find the range of R.
Answer : Given relation is
R = {(a, a³) : a is a prime number less than 5}.
∴ R = {(2, 8), (3, 27)}
So, the range of R is {8, 27}.

Question. Let R be the equivalence relation in the set
A = {0, 1, 2, 3, 4, 5} given by
R = {(a, b) : 2 divides (a – b)}. Write the equivalence class [0].
Answer : Here, R = {(a, b) ∈ A × A : 2 divides (a – b)}
This is the given equivalence relation, where
A = {0, 1, 2, 3, 4, 5}
∴ [0] = {0, 2, 4}.

Question. State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.
Answer : For transitivity of a relation,
If (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R
We have, R = {(1, 2), (2, 1)}
(1, 2) ∈ R and (2, 1) ∈ R but (1, 1) ∉ R
∴ R is not transitive.

Question. 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, state whether f is one-one or not.
Answer : A = {1, 2, 3}, B = {4, 5, 6, 7} and
f = {(1, 4), (2, 5), (3, 6)}
We have, f(1) = 4, f(2) = 5 and f(3) = 6. Distinct elements of A have distinct images in B. Hence, f is a one-one function.

Question. What is the range of the function

Answer :

Question. State whether the function f : N → N given by f(x) = 5x is injective, surjective or both.
Answer : We have, f(x) = 5x
For x₁, x₂ ∈ N.
Let f(x₁) = f(x₂) ⇒ 5x₁ = 5x₂ ⇒ x₁ = x₂
∴ The function is one-one.
Now, f(x) is not onto. Since, for 2 ∈ N (co-domain), there does not exist any x ∈ N (domain) such that
f(x) = 5x = 2
∴ f(x) is injective but not surjective.

Question. Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3}
given by f = {(1, 2), (3, 5), (4, 1)} and
g = {(1, 3), (2, 3), (5, 1)}. Write down gof.
Answer : Here, f = {(1,2), (3, 5), (4, 1)}
⇔ f(1) = 2; f(3) = 5; f(4) = 1                       …(1)
g = {(1, 3), (2, 3), (5, 1)}
⇔ g(1) = 3; g(2) = 3, g(5) = 1                    …(2)
Now, gof : {1, 3, 4} →{1, 3}
Using (1) and (2), we get
(gof) (1) = g(f(1)) = g(2) = 3
(gof) (3) = g(f(3)) = g(5) = 1
(gof) (4) = g(f(4)) = g(1) = 3
∴ gof = {(1, 3), (3, 1), (4, 3)}.

Question. If f : R → R defined as

invertible function, write f ^–1(x).
Answer :

Question. If f : R → R is defined by f(x) = (3 – x³)^1/3, then find fof(x).
Answer : f : R → R and f(x) = (3 – x³)^1/3
∴ fof(x) = f(f(x)) = f [3 – x3)^1/3]
= [3 – {(3 – x³)^1/3}3]^1/3
= [3 – (3 – x³)]^1/3 = (3 – 3 + x³)^1/3 = x

Question. If f : R → R is defined by f (x) = 3x + 2, find f (f (x)).
Answer : We have, f(x) = 3x + 2
∴ f (f (x)) = f (3x + 2) = 3(3x + 2) + 2 = 9x + 8

Question. If the function f : R → R, defined by f (x) = 3x – 4, is invertible, find f ^–1
Answer : We have, f(x) = 3x – 4
Let f(x) = y ⇒ x = f ^–1(y)

Question. If f : R → R defined by

an invertible function, find f^–1.
Answer :

Question. If f(x) = x + 7 and g(x) = x – 7, x ∈ R, find (fog) (7).
Answer : f(x) = x + 7 and g(x) = x – 7
So, fog (x) = f(g(x)) = f (x – 7) = x – 7 + 7 = x
⇒ fog (x) = x
∴ fog (7) = 7

Question. If f : R → R is defined by
f(x) = x² – 3x + 2, find f(f(x)).
Answer : f(f(x)) = f (x² – 3x + 2)
= (x² – 3x + 2)² – 3 (x² – 3x + 2) + 2
= x⁴ + 9x² + 4 – 6x³ – 12x + 4x² – 3x² + 9x – 6 + 2
= x⁴ – 6x³ + 10x² – 3x.

CASE STUDY

Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1,2,3,4,5,6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1,2,3,4,5,6}

Question. Let 𝑅∶ 𝐵→𝐵 be defined by R = {(𝑥,𝑦): 𝑦 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑥 } is
a. Reflexive and transitive but not symmetric
b. Reflexive and symmetric and not transitive
c. Not reflexive but symmetric and transitive
d. Equivalence

Question. Raji wants to know the number of functions from A to B. How many number of functions are possible?
a. 6²
b. 2⁶
c. 6!
d. 2¹²

Question. Let R be a relation on B defined by R = {(1,2), (2,2), (1,3), (3,4), (3,1), (4,3), (5,5)}. Then R is
a. Symmetric
b. Reflexive
c. Transitive
d. None of these three

Question. Let 𝑅:𝐵→𝐵 be defined by R={(1,1),(1,2), (2,2), (3,3), (4,4), (5,5),(6,6)}, then R is
a. Symmetric
b. Reflexive and Transitive
c. Transitive and symmetric
d. Equivalence

CASE STUDY:

Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line 𝑦=𝑥−4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.

Question. Let relation R be defined by R = {(𝐿₁,𝐿₂): 𝐿₁║𝐿₂ where L₁,L₂ € L} then R is______ relation
a. Equivalence
b. Only reflexive
c. Not reflexive
d. Symmetric but not transitive

Question. Let R = { (𝐿1,𝐿2)∶ 𝐿1┴𝐿2 where L1,L2 € L } which of the following is true?
a. R is Symmetric but neither reflexive nor transitive
b. R is Reflexive and transitive but not symmetric
c. R is Reflexive but neither symmetric nor transitive
d. R is an Equivalence relation

Question. The function f: R→R defined by 𝑓(𝑥)=𝑥−4 is___________
a. Bijective
b. Surjective but not injective
c. Injective but not Surjective
d. Neither Surjective nor Injective

Question. Let 𝑓: 𝑅→𝑅 be defined by 𝑓(𝑥)=𝑥−4. Then the range of 𝑓(𝑥) is ________
a. R
b. Z
c. W
d. Q

Question. Let R = {(L₁ , L₂ ) : L₁ is parallel to L₂ and L₁ : y = x – 4} then which of the following can be taken as L 2 ?
a. 2x-2y+5=0
b. 2x+y=5
c. 2x + 2y + 7 =0
d. x+y=7