CBSE Class 11 Mathematics Relations Functions Worksheet Set 03

Case Based MCQs

Ordered Pairs The ordered pair of two elements a and b is denoted by (a, b ) : a is first element (or first component) and b is second element (or second component). Two ordered pairs are equal if their corresponding elements are equal. i.e. (a, b ) = (c , d ) Þ a = c and b = d Cartesian Product of Two Sets For two non-empty sets A and B, the cartesian product A × B is the set of all ordered pairs of elements from sets A and B. In symbolic form, it can be written as A × B = {(a, b ) : a ∈A, b ∈B}

Based on the above topics, answer the following questions.

Question. If (a − 3, b + 7) = (3, 7), then the value of a and b are
(a) 6, 0
(b) 3, 7
(c) 7, 0
(d) 3, −7
Answer : B

Question. If (x + 6, y − 2) = (0, 6), then the value of x and y are
(a) 6, 8
(b) − 6,−8
(c) −6,8
(d) 6,−8
Answer : A

Question. If (x + 2, 4) = (5, 2x + y ), then the value of x and y are
(a) −3 2 ,
(b) 3, 2
(c) −3,−2
(d) 3,−2
Answer : C

Question. Let A and B be two sets such that A × B consists of 6 elements. If three elements of A × B are (1, 4), (2, 6) and (3, 6), then
(a) (A × B) = (B × A)
(b) (A × B) ¹ (B × A)
(c) A × B = {(1,4), (1, 6), (2,4)}
(d) None of the above
Answer : A

Question. If n(A × B ) = 45, then n(A) cannot be
(a) 15
(b) 17
(c) 5
(d) 9
Answer : B

Representation of a Relation A relation can be represented algebraically by roster form or by set-builder form and visually it can be represented by an arrow diagram which are given below
(i) Roster form In this form, we represent the relation by the set of all ordered pairs belongs to R.
(ii) Set-builder form In this form, we represent the relation R from set A to set B as R = {(a, b ) : a ∈ A, b ∈B and the rule which relate the elements of A and B}.
(iii) Arrow diagram To represent a relation by an arrow diagram, we draw arrows from first element to second element of all ordered pairs belonging to relation R.

Based on the above topics, answer the following questions.

Question. Expression of R = {(a,b ): 2a +b =5; a,b ∈ W} as the set of ordered pairs (in roster form) is
(a) R = {(5, 0), (3, 1), (1, 2)}
(b) R = {(0, 5), (1, 3), (1, 2)}
(c) R = {(0, 5), (1, 3), (2, 1)}
(d) None of the above
Answer : C

Question. The relation given in (ii) can be written in set-builder form as
(a) R ={(x, y) :x ∈ P,y ∈ Q and x is the square of y}
(b) R ={(x, y) :x ∈ P,y ∈ Q and y is the square of x}
(c) R ={(x, y) :x ∈ P,y ∈ Q and x =± y}
(d) None of the above
Answer : A

Question. If A = {a, b} and B = {2, 3}, then the number of relations from A to B is
(a) 4
(b) 8
(c) 6
(d) 16
Answer : D

Question. If n (A) = 3 and B = {2, 3, 4, 6, 7, 8}, then the number of relations from A to B is
(a) 2³
(b) 2⁶
(c) 2¹⁸
(d) 2⁹
Answer : C

Assertion-Reasoning MCQs

Directions Each of these questions contains two statements Assertion (A) and Reason (R). Each of the questions has four alternative choices, any one of the which is the correct answer. You have to select one of the codes (a), (b), (c) and
(d) given below.
(a) A is true, R is true; R is a correct explanation of A.
(b) A is true, R is true; R is not a correct explanation of A.
(c) A is true; R is false.
(d) A is false; R is true.

Question. Let A = {1, 2, 3, 4, 6}. If R is the relation on A defined by {(a, b) : a, b ∈ A, b is exactly divisible by a}.
Assertion (A) The relation R in Roster form is {(6, 3), (6, 2), (4, 2)}.
Reason (R) The domain and range of R is {1, 2, 3, 4, 6}.
Answer : D

Question. Let R be a relation defined by R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}} Then, consider the following
Assertion (A) The domain of R is {0, 1, 2, 3, 4, 5}.
Reason (R) The range of R is {0, 1, 2, 3, 4, 5}.
Answer : C

Question. Assertion (A) The domain of the relation R = {(x + 2, x + 4) : x ∈ N, x < 8} is {3, 4, 5, 6, 7, 8, 9}.
Reason (R) The range of the relation R = {(x + 2, x + 4) : x ∈ N, x < 8} is {1, 2, 3, 4, 5, 6, 7}.
Answer : C

Question. Assertion (A) The range of the function f (x ) = 2 − 3x, x ∈ R, x > 0 is R.
Reason (R) The range of the function f(x ) = x² + 2 is [2, ∝).
Answer : D

Question. Assertion (A) Let A = {1, 2, 3, 5},
B = {4, 6, 9} and R = {(x, y) : |x − y | is odd, x ∈ A, y ∈ B}. Then, domain of R is {1, 2, 3, 5}.
Reason (R) |x |is always positive ∀ x ∈ R.
Answer : B
 

Q.1 Let A = {1,2} and B = {3,4}.Write A×B. How many subsets will A×B have? List them.

Q.2 Let A = {1,2,3,…..,14}. Define a relation R from A to A by R = {(x,y) : 3x-y = 0, where x,y∈A}. Write down its domain, co-domain and range.

Q.3 If f(x) = x2, find f(1.2) - f(1) / 1.2 - 1

Q.4 Find the inverse of the function of x = 3 - 5y / 2y - y

Q.5 If n(A) = 3 and n(B) = 3, then find n(A B). (1 mark)

Q.6 Let R be a relation from Q to Q defined by R = {(a,b):a,b∈Q and a-b∈Z}.Show that (a) (a,a)∈R for all a∈Q (b) (a,b)∈R implies that (b,a)∈R (c) (a,b)∈R and (b,c)∈R implies that (a,c)∈R.

Q.7 Let A = {9, 10, 11, 12, 13} and let f: A → N be defined by f(n) = the highest prime factor of n. Find the range of f.

Q.8 Find the domain of the function f(x) = x² + 2x + 1 / x² - 8x + 12 

Q.9 Let A = {1, 2, 3, 4} and B = {10, 12, 13, 14, 20}. Whether f: A B defined by f(1) = 10, f(2) = 12, f(3) = 13 is a function? 

Q.10 Find the domain and the range of the real function f defined by f (x) = |x – 1|. (2 marks)

Q.11 Examine the relation : R={(2,1),(3,1),(4,1)} and state whether it is a function or not?

Q.12 A function f is defined by f(x) = 3x-4. Write down the value of f(5) and f(-7).

Q.13 Write the domain of the function f(x) = x+1/x²+6x+5

Q.14 Let A = {1, 2, 6, 8} and let R be a relation on A defined by {(a, b): a, b A, b is exactly divisible by a}
a) Write R in roster form.
b) Find the domain of R.
c) Find the range of R.

Q.15 If f and g are two functions such that f(x) = 5x + 2 and g(x) = x2 + 3, then find f + g and f – g. 

Q.16 Write the domain of the function f(x) = x² - 2x + 3 / x² - x - 20

Q.17 A function f is defined by f(x) = 2x – 5. Write down the values of (2 marks)
(i) f(0), (ii) f(7), (iii) f(–3)

Q.18 If f(x) = x² - 1/x². then find the value of : f(x) + f(1/x)

Q.19 Under which condition a relation f from A to B is said to be a function? (1 mark)

Q.20 If A = {a1,a2} and B = {b1,b2,b3}, then write A × B.