## 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 \(\frac { 1+x }{ 1-x } \), 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) = \(\sqrt { x-1 } \) 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) = \(\frac { 1 }{ \sqrt { 2x-3 } } \) 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) = \(\sqrt { 3-2x } \) 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: