CBSE Class 12 Mathematics Relations and Functions MCQs Set F

Case Based Questions

1. There are two small bookshelves, shelf A and shelf B. Both the shelves have four books each. Shelf A has different books for science students whereas the shelf B has different books for non-science students.

Two books of a shelf are associated to each other by a relation. The difference in their number of pages is at most 10. On the basis of above information, answer the following:

Question. As per the given definition, the relation on shelf A is:
(a) {(Maths, Physics), (Chemistry, Biology)}
(b) {(Maths, Physics), (Chemistry, Biology), (Physics, Biology)}
(c) {(Maths, Maths), (Chemistry, Physics), (Biology, Maths)}
(d) None of the options

Answer: A

Question. With reference to Q1, the relation on shelf A is:
(a) reflexive only
(b) symmetric only
(c) transitive only
(d) None of the options

Answer: D

Question. As per the definition given, the relation on shelf B is:
(a) {(Economics, Accountancy), (Economics, Geography), (Accountancy, History)}
(b) {(Economics, Accountancy), (Economics, History), (Accountancy, History)}
(c) {(History, Geography), (Accountancy, Geography)}
(d) None of the options

Answer: B

Question. With reference to Q3, the relation on the shelf B is:
(a) reflexive only
(b) symmetric only
(c) transitive only
(d) None of the options

Answer: C

Question. Let {(Maths, Maths), (Physics, Physics), (Chemistry, Chemistry), (Biology, Biology)} be a relation defined in a different manner on shelf A. Then the relation is:
(a) reflexive only
(b) identity only
(c) reflexive and identity both
(d) None of the options

Answer: C

2. A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever.

Let I be the set of all citizens of India who were eligible to exercise their voting right in general election held in 2019.
A relation ‘R’ is defined on I as follows:
R = {(V1, V2) : V1, V2 ∈ I and both use their voting right in general election – 2019}

Question. Two neighbours X and Y ∈ I. X exercised his voting right while Y did not cast her vote in general election – 2019. Which of the following is true?
(a) (X, Y) ∈ R
(b) (Y, X) ∈ R
(c) (X, X) ∉ R
(d) (X, Y) ∉ R

Answer: D

Question. Mr. ‘X’ and his wife ‘W’ both exercised their voting right in general election – 2019. Which of the following is true?
(a) Both (X, W) and (W, X) ∈ R
(b) (X,W) ∈ R but (W, X) ∉ R
(c) Both (X, W) and (W, X) ∉ R
(d) (W, X) ∈ R but (X, W) ∉ R

Answer: A

Question. Three friends F1, F2 and F3 exercised their voting right in general election – 2019, then which of the following is true?
(a) (F1, F2) ∈ R, (F2, F3) ∈ R and (F1, F3) ∈ R
(b) (F1,F2 ) ∈ R, (F2, F3) ∈ R and (F1, F3) ∉ R
(c) (F1, F2 ) ∈ R, (F2, F2) ∈ R but (F3, F3) ∉ R
(d) (F1, F2) ∉ R, (F2, F3) ∉ R and (F1, F3) ∉ R

Answer: A

Question. The above defined relation R is
(a) symmetric and transitive but not reflexive
(b) universal relation
(c) equivalence relation
(d) reflexive but not symmetric and transitive

Answer: C

Question. Mr. Shyam exercised his voting right in General Election – 2019, then Mr. Shyam is related to which of the following?
(a) All those eligible voters who cast their votes
(b) Family members of Mr. Shyam
(c) All citizens of India
(d) Eligible voters of India

Answer: A

3. There are two different sections of Class 12: Section A and Section B including girls as well as boys. A college girl Teena forms two sets with these students as her college project. Let A = {a₁, a₂, a₃, a₄, a₅} and B = {b₁, b₂, b₃, b₄}, where ai’s and bi’s are the students of Section A and Section B respectively.

Teena decides to explore these sets for various types of relations and functions using the information given, answer the following:

Question. Teena wishes to know the number of reflexive relations defined on set A. How many such relations are possible?
(a) 0
(b) 25
(c) 220
(d) None of the options

Answer: C

Question. Let R : A → A, R = {(x, y) : x and y are students of same sex}. Then relation R is:
(a) reflexive only
(b) reflexive and symmetric but not transitive
(c) reflexive and transitive but not symmetric
(d) an equivalence relation

Answer: D

Question. Teena and her friend Reena interested to know the number of symmetric relations defined on both the sets A and B, separately. She decides to find the symmetric relation on set A, while Reena decides to find the symmetric relation on set B. What is difference between their result?
(a) 1024
(b) 210 (15)
(c) 210(31)
(d) None of the options

Answer: C

Question. Let R : A → B, R = {(a₁, b₁), (a₁, b₂), (a₂, b₁), (a₃, b₃), (a₄, b₂), (a₅, b₂)}, then R is:
(a) neither one-one nor onto
(b) one-one but, not onto
(c) only onto, but not one-one
(d) not a function

Answer: D

Question. From the given sets n(A × A), n(B × B) and n(A × B) are respectively:
(a) 25, 16 and 20
(b) 20, 10 and 15
(c) 22, 14 and 18
(d) 24, 15 and 19

Answer: A

4. 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.
Then, A = {S, D} and 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

Answer: A

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

Answer: A

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

Answer: D

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

Answer: D

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

Answer: B

5. Raji visited the exhibition along with her family. The exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a parabola as given by y = x².

Question. Let f : R → R be defined by f (x) = x² is
(a) neither surjective nor injective
(b) surjective
(c) injective
(d) bijective

Answer: A

Question. Let f : N → N be defined by f (x) = x² is
(a) surjective but not injective
(b) surjective
(c) injective
(d) bijective

Answer: C

Question. Let f : {1, 2, 3,….} → {1, 4, 9,….} be defined by f (x) = x² is
(a) bijective
(b) surjective but not injective
(c) injective but not surjective
(d) neither surjective nor injective

Answer: A

Question. Let : N → R be defined by f (x) = x² . Range of the function among the following is
(a) {1, 4, 9, 16,…}
(b) {1, 4, 8, 9, 10,…}
(c) {1, 4, 9, 15, 16,…}
(d) {1, 4, 8, 16,…}

Answer: A

Question. The function f : Z→Z defined by f (x) = x² is
(a) neither injective nor surjective
(b) injective
(c) surjective
(d) bijective

Answer: A

6. An organization conducted bike race under two different categories–boys and girls. Totally there were 250 participants. Among all of them finally three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for the final race. Let B = {b₁, b₂, b3} and G = {g₁, g₂} where B represents the set of boys selected and G the set of girls who were selected for the final race.

Ravi decides to explore these sets for various types of relations and functions.

Question. Ravi wishes to form all the relations possible from B to G. How many such relations are possible?
(a) 2⁶
(b) 2⁵
(c) 0
(d) 2³

Answer: A

Question. Let R : B → B be defined by R = {(x, y): x and y are students of same sex}, Then this relation R is
(a) equivalence
(b) reflexive only
(c) reflexive and symmetric but not transitive
(d) reflexive and transitive but not symmetric

Answer: A

Question. Ravi wants to know among those relations, how many functions can be formed from B to G?
(a) 2²
(b) 21²
(c) 3²
(d) 2³

Answer: D

Question. Let R : B → G be defined by R = {(b₁, g₁), (b₂, g₂), (b₃, g₁)}, then R is
(a) injective
(b) surjective
(c) neither surjective nor injective
(d) surjective and injective

Answer: B

Question. Ravi wants to find the number of injective functions from B to G. How many injective functions are possible?
(a) 0
(b) 2!
(c) 3!
(d) 0!

Answer: A

7. Students of Grade 12, 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.

Question. Let relation R be defined by R = {(L1, L2): L1 || L2 where L1, L2 ∈ L} then R is
(a) equivalence relation
(b) only reflexive relation
(c) not reflexive relation
(d) symmetric but not transitive relation

Answer: A

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

Answer: A

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

Answer: A

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

Answer: A

Question. Let R = {(L 1 , L 2 ) : L 1 is parallel to L 2 and L 1 : 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

Answer: A