MP Board Class 11th Maths Important Questions Chapter 2 Relations and Functions
Relations and Functions Important Questions
Relations and Functions Objective Type Questions
(A) Choose the correct option :
Question 1.
If A = {2, 4, 5}, B = {7, 8, 9}, then n(A × B) =
(a) 6
(b) 9
(c) 3
(d) 0.
Answer:
(b) 9
Question 2.
If A = { 1, 2, 3, 4, 5} and 5 = {2, 3, 6, 7}, then the number of element in (A × B)∩(B × A) is:
(a) 4
(b) 5
(c) 10
(d) 20.
Answer:
(a) 4
Question 3.
If A and B are two non – empty sets, then:
(a) A × B = {(a, b) : a ∈ B, b ∈ A}
(b) A × B = {(a, b) : a ∈ A, b ∈ B}
(c) {(a, b) : (a, b) ∈ A, (a, b) ∈ B}
(d) None of these.
Answer:
(b) A × B = {(a, b) : a ∈ A, b ∈ B}
Question 4.
If f(x) = log \(\frac { 1+x }{ 1x } \), then find f [ \(\frac { { 2x } }{ 1+{ x }^{ 2 } }\) ] =
(a) [f(x)]^{2}
(b) [f(x)]^{3}
(c) 2 f(x)
(d) 3 f(x).
Answer:
(c) 2 f(x)
Question 5.
Let A = {1,2} and B = {3,4}, then the number of relation from A to B will be:
(a) 2
(b) 4
(c) 8
(d) 16
Answer:
(d) 16
Question 6.
The range of the function f(x) = \(\sqrt { x1 } \) is:
(a) [1, ∞)
(b) [0, ∞)
(c) (0, ∞)
(d) (1, ∞)
Answer:
(b) [0, ∞)
Question 7.
If f(x) = \(\frac { x^{ 2 }1 }{ x^{ 2 }+1 } \), then f ( \(\frac{1}{x}\) ) is:
(a) f(x)
(b) – f(x)
(c) f(x)
(d) \(\frac { 1 }{ f(x) } \)
Answer:
(b) – f(x)
Question 8.
Domain of the function f(x) = \(\frac { 1 }{ \sqrt { 2x3 } } \) is:
(a) R – { \(\frac{3}{2}\) }
(b) ( \(\frac{3}{2}\), ∞)
(c) [ \(\frac{3}{2}\), ∞)
(d) None of these.
Answer:
(b) ( \(\frac{3}{2}\), ∞)
(B) Match the following :
Answer:

 (d)
 (a)
 (c)
 (b)
 (c)
(C) Fill in the blanks:
 If A = {1, 2} and B = {3, 4, 5}, then the number of subsets of A × B is ………………………….
 Range of the function f = {(2, 1), (3, 1), (4, 1), (5, 1)} is …………………………..
 Range of the function f(x) = 11 – 7 sinx is …………………………..
 If f(x) = x^{2} and g(x) = x + 1, ∀ x ∈ R , then (f + g)x is …………………………
 If f(x) = 1 – cosx, then the value of f ( \(\frac { \pi }{ 4 } \) ) …………………………….
 Domain of the function f(x) = \(\frac { 1 }{ \sqrt { (1 – x)(x – 2) } } \) is ……………………………
 If relation R = {(1, 3), (3, 3), (4, 5)} then the value of R^{1} is ……………………………
Answer:
 64
 {1}
 [4, 18]
 x^{2} + x + 1
 1 – \(\frac { 1 }{ \sqrt { 2 } } \)
 (1, 2)
 {(3, 1), (3, 3), (5, 4)}.
(D) Write true/false :
 If A, B, C are three sets, then the value of A × (B∪C) is (A∪B) × (A∪C).
 If A = {x : x^{2} – 5x + 6 = o}, B = {2,4}, C = {4,5}, then A × (B∩C) = {(2, 4), (3, 4)}.
 The relation R = {(2, 1), (3, 2), (4, 3), (5, 4)} is a function.
 If a relation on Z is R = {(x, y) : x, y ∈ Z, x^{2} + y^{2} ≤ 4}, then domain of R is {0, ± 1, ± 2}.
 Domain of the function f(x) = \(\sqrt { a^{ 2 }x^{ 2 } } \), a > 0 is [0, a].
Answer:
 False
 True
 True
 True
 False
(E) Write answer in one word/sentence:
 If f(x) = x^{2} and g(x) = x + 3, x ∈ R, then the value of (fog)(2).
 Range of function f(x) = sin x.
 If mapping f : R → R is defmd by f(x) = x^{2} + 1 the value of f^{1} (26) is:
 If A = {1, 2, 3} and B = {5, 7}, then the value of A×B is:
 Domain of the function f(x) = \(\sqrt { 32x } \) is:
 If function f(x) = \(\frac { { x }^{ 2 } }{ 1{ x }^{ 2 } }\), then the value of f(sin θ) is :
Answer:
 25
 [1, 1]
 {5, 5}
 {(1, 5), (1, 7), (2, 5), (2, 7), (3, 5), (3, 7)}
 (∞, \(\frac { 3 }{ 2 }\))
 tan^{2 }θ
Straight Lines Very Short Answer Type Questions
Question 1.
If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in ( A x B). (NCERT)
Solution:
Given : n(A) = 3, B = {3, 4, 5}, n (B) = 3
∴ n (A x B) = n (A) x n (B) = 3 x 3
⇒ n (A x B) = 9.
Question 2.
If G = {7, 8} and H = {5, 4, 2}, then find G x H. (NCERT)
Solution:
G x H = { 7, 8} x {5, 4, 2}
G x H= { (7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}.
Question 3.
If A x B = { (a, x), (a, y), (b, x), (b, y)} then find A and 5. (NCERT)
Solution:
Given:
A x B = {(a, x), (a, y), (6, x), (6, y)}
A = {a, b} and S = {x, y}.
Question 4.
If A = {1, 2} and B = {3, 4} then find A x B. (NCERT)
Solution:
A x B = { 1, 2} x {3, 4}
A x B = { ( 1, 3), (1, 4), (2, 3), (2, 4)}
Question 5.
The figure shows the relationship between the set P and Q. Write the relation : (NCERT)
 In set builder form
 In roster form
 Find its domain
 Find its range.
Solution:
 Set builder form, R = {(x, y) : y = x – 2, x ∈ P and y ∈ Q}.
 Roster form, R = {(5, 3), (6, 4), (7, 5)}
 Domain = {5, 6, 7}
 Range = {3, 4, 5}.
Question 6.
Let A = (1, 2, 3, 4, 6} and R be the relation on A defined by
{(a, b) : a, b ∈ A, b is exactly divisible by a } :
 Write R in roster form,
 Find the domain of R
 Find the range of R. (NCERT)
Solution:
Given: A = {1, 2, 3, 4, 6}
1. R = {{a, b): a, be A, b is exactly divisible by a }
R= {(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)}.
2. Domain = {1, 2, 3, 4, 6}.
3. Range = {1, 2, 3, 4, 6}.
Question 7.
Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B. (NCERT)
Solution: A = {x, y, z}, B= {1, 2}
n(A) = 3, n(B) = 2
No. of relations from A to B = 2^{mn} = 23^{3/2} = 2^{6}
= 64.
Question 8.
A function f is defined by f (x) = 2x – 5, write down the following values : (NCERT)
 f(0)
 f(7)
 f(3).
Solution:
Given: f(x) = 2x – 5
1. Put x = 0,
f(0) = 2(0) – 5 = 0 – 5 = – 5.
2. Put x = 7,
f(7) = 2 x 7 – 5 = 14 – 5 = 9.
3. Put x = – 3,
f( 3) = 2( 3) – 5 = – 6 – 5 = – 11.
Question 9.
The function ‘t’ which maps temperature in degree celcius into temperature in degree fehrenheit, it is defined as t(c) = \(\frac { 9c }{ 5 }\) +32. Find the following : (NCERT)
 t (0)
 t (28)
 t ( 10)
 Find c, when t (c) = 212.
Solution:
Given :
t(c) = \(\frac { 9c }{ 5 }\) + 32
1. Put c = 0,
t(0) = \(\frac { 9 × 0 }{ 5 }\) + 32
⇒ t(0) = 32
2. Put c = 28,
t(28) = \(\frac { 9 × 28 }{ 5 }\) + 32 = \(\frac { 252}{ 5 }\) + \(\frac { 32}{ 1 }\)
⇒ t(28) = \(\frac { 252 + 160}{ 5 }\) = \(\frac { 412}{ 5 }\)
3. Put c = – 10,
t( 10) = \(\frac { 9 x ( 10)}{ 5 }\) + 32
⇒ t( 10) = – 18 + 32 = 14
4. Put t(c) = 212,
212 = \(\frac { 9c }{ 5 }\) + 32
⇒ \(\frac { 9c }{ 5 }\) = 212 – 32
⇒ \(\frac { 9c }{ 5 }\) = 180
⇒ c = \(\frac { 180 x 5 }{ 9 }\)
Question 10.
Find the domain and range of the function f(x) = x.
Solution:
f(x) = – x, f(x) < 0
Domain of f = R.
Range of f = {y : y ∈ R, y ≤ 0} = ( ∞, 0].
Straight Lines Short Answer Type Questions
Question 1.
Find the domain and range of the function f(x) = \(\sqrt { 9 – { x }^{ 2 } }\).
Solution:
Given : f(x) = \(\sqrt { 9 – { x }^{ 2 } }\)
Value of f(x) is real, if f(x) ≥ 0
9 – x^{2} ≥ 0
⇒ – (x^{2} – 9) ≥ 0
⇒ x^{2} – 9 ≤ 0
⇒ (x + 3)(x – 3) ≤ 0
∴ Domain = [ 3, 3].
Question 2.
If f(x) = x^{2}, then find \(\frac { f(1.1) – f(1)}{ (1.1 – 1)}\)
Solution:
f(x) = x^{2}
f(1.1) = (1.1)^{2} = 1.21
f(1) = 1^{2} = 1
Question 3.
Find domain of function f(x) = \(\frac { { x }^{ 2 } + 2x + 1 }{ { x }^{ 2 } – 8x + 12 }\)
Solution:
Given function is :
f(x) will be defined if,
x^{2} – 8x + 12 ≠ 0
⇒ x^{2} – 6x – 2x + 12 ≠ 0
⇒ x(x – 6) – 2(x – 6) ≠ 0
⇒ (x – 2)(x – 6) ≠ 0
x ≠ 2 and x ≠ 6
Domain of function = R – {2, 6}.
Question 4.
Let f : g → R → R be defined by f(x) = x + 1 and g(x) = 2x – 3 respectively, then find :
 f + g
 \(\frac { f }{ g }\)
Solution:
Given: f(x) = x + 1, g (x) = 2x – 3
1. (f + g)x = f(x) + g(x)
= x + 1 + 2x – 3
∴ (f + g)x = 3x – 2
2. \(\frac { f }{ g }\)(x) = \(\frac { f(x)}{ g(x) }\) = \(\frac { x +1 }{ 2x – 3}\)
Question 5.
If f(x) = x^{2} – \(\frac { 1 }{ { x }^{ 2 } }\), then prove that:
f(x) + f\(\frac { 1 }{ x }\) = 0
Solution:
Question 6.
If f(x) = \(\frac { { x }^{ 2 } }{ 1{ x }^{ 2 } }\), then prove that:
f(sinθ) = tan^{2 }θ.
Solution:
Question 7.
If f(x) = x^{3} + 3x + tanx, then prove that f(x) is an odd function.
Solution:
Given : f(x) = x^{3} + 3x + tanx
f( x) = ( x)^{3} + 3( x) + tan( x)
= – x^{3} – 3x – tanx
= – (x^{3} + 3x + tanx)
= – f(x).
Hence, f(x) is an odd function.
Question 8.
If f(x) = x^{2} + 2xsinx + 3, then prove that f(x) is an even function.
Solution:
Given: f(x) = x^{2} + 2x sin x + 3
f( x) = ( x)^{2} + 2( x) sin( x) + 3
= x^{2} + 2xsinx + 3, [∵ sin( x) = – sinx]
= f(x).
Hence, f(x) is an even function
Question 9.
If f(x) = x^{2}, g(x) = x + 2, ∀ x ∈R, then find gof and fog. Is gof = fog.
Solution:
Given : f(x) = x^{2}, g(x) = x + 2
fog(x) = f[g(x)]
= f(x + 2) = (x+2)^{2}.
gof(x) = g[f(x)]
= g(x^{2}) = x^{2} + 2
Hence, fog(x) ≠ gof(x).
Question 10.
If f(x) = e^{2x} and g(x) = log \(\sqrt {x}\), x> 0, then find the value of fog(x) and gof(x).
Solution: