## Tamilnadu State Board Class 10 Maths Solutions Chapter 1 Relations and Functions Ex 1.6

Question 1.

If n(A × B) = 6 and A = {1, 3} then n(B) is

(1) 1

(2) 2

(3) 3

(4) 6

Answer:

(3) 3

Hint:

If n(A × B) = 6

A = {1, 1}, n(A) = 2

n(B) = 3

Question 2.

A = {a, b, p}, B = {2, 3}, C = {p, q, r, s} then

n[(A ∪ C) × B] is

(1) 8

(2) 20

(3) 12

(4) 16

Answer:

(3) 12

Hint:

A = {a, b,p}, B = {2,3}, C = {p, q, r, s}

n (A ∪ C) × B

A ∪C = {a, b,p, q, r,s}

(A ∪C) × B = {{a, 2), (a, 3), (b, 2), (b, 3), (p, 2), (p, 3), (q, 2), (q, 3), (r, 2), (r, 3), (s, 2), (s, 3)

n [(A ∪ C) × B] = 12

Question 3.

If A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5,6,7,8} then state which of the following statement is true.

(1) (A × C) ⊂ (B × D)

(2) (B × D) ⊂ (A × C)

(3) (A × B) ⊂ (A × D)

(4) (D × A) ⊂ (B × A)

Answer:

(1) (A × C) ⊂ (B × D)]

Hint:

A = {1, 2}, B = {1, 2, 3, 4},

C = {5, 6}, D ={5,6, 7, 8}

A × C ={(1,5), (1,6), (2, 5), (2, 6)}

B × D = {(1,5),(1,6),(1,7),(1,8),(2,5),(2,6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8)}

∴ (A × C) ⊂ B × D it is true

Question 4.

If there are 1024 relations from a set A = {1, 2, 3, 4, 5} to a set B, then the number of elements in B is

(1) 3

(2) 2

(3) 4

(4) 6

Answer:

(2) 2

Hint:

n(A) = 5

n(B) = x

n(A × B) = 1024 = 2^{10}

2^{5x} = 2^{10}

⇒ 5x = 10

⇒ x =2

Question 5.

The range of the relation R = {(x, x^{2})|x is a prime number less than 13} is

(1) {2,3,5,7}

(2) {2,3,5,7,11}

(3) {4,9,25,49,121}

(4) {1,4,9,25,49,121}

Answer:

(3) {4, 9, 25, 49,121}]

Hint:

R = {(x, x^{2})/X is a prime number < 13}

The squares of 2, 3, 5, 7, 11 are

{4, 9, 25,49,121}

Question 6.

If the ordered pairs (a + 2,4) and (5,2a+b)are equal then (a,b) is

(1) (2,-2)

(2) (5,1)

(3) (2,3)

(4) (3,-2)

Answer:

(4) (3,-2)

Hint:

(a + 2, 4), (5, 2a + b)

a + 2 = 5

a = 3

2a + b = 4

6 + b = 4

b = -2

Question 7.

Let n(A) = m and n(B) = n then the total number of non-empty relations that can be defined from A to B is

(1) m^{n}

(2) n^{m}

(3) 2^{mn}-1

(4) 2^{mn}

Answer:

(4) 2^{mn}

Hint:

n(A) = m,n(B) = n

n(A × B) = 2^{mn}

Question 8.

If {(a,8),(6,b)}represents an identity function, then the value of a and b are respectively

(1) (8,6)

(2) (8,8)

(3) (6,8)

(4) (6,6)

Answer:

(1) (8,6)

Hint:

{{a, 8), (6,b)}

a = 8

b = 6

Question 9.

Let A = {1,2,3,4} and B = {4,8,9,10}. A function f : A → B given by f = {(1,4),(2,8),(3,9),(4,10)} is a

(1) Many-one function

(2) Identity function

(3) One-to-one function

(4) Into function

Answer:

(3) One-to one function

Hint:

A = {1, 2, 3, 4), B = {4, 8, 9,10}

Question 10.

If f(x) = 2x^{2} and g (x) = \(\frac { 1 }{ 3x } \), Then fog is

Answer:

Hint:

Queston 11.

If f: A → B is a bijective function and if n(B) = 7, then n(A) is equal to

(1) 7

(2) 49

(3) 1

(4) 14

Answer:

(1) 7

Hint:

In a bijective function, n(A) = n(B)

⇒ n(A) = 7

Question 12.

Let f and g be two functions given by f = {(0,1), (2,0), (3, -4), (4,2), (5,7)} g = {(0,2), (1, 0), (2, 4), (-4, 2), (7,0)} then the range of fog is

(1) {0,2,3,4,5}

(2) {-4,1,0,2,7}

(3) {1,2,3,4,5}

(4) {0,1,2}

Answer:

(4) {0,1,2}

Hint:

gof = g(f(x))

fog = f(g(x))

= {(0,2),(1,0),(2,4),(-4,2),(7,0)}

Range of fog = {0,1,2}

Question 13.

Let f(x) = \(\sqrt{1+x^{2}}\) then

(1) f(xy) = f(x),f(y)

(2) f(xy) > f(x),f(y)

(3) f(xy) < f(x).f(y)

(4) None of these

Answer:

(3) f(xy) < f(x).f(y)

Hint:

Question 14.

If g = {(1, 1),(2, 3),(3, 5),(4, 7)} is a function given by g(x) = ∝x + β then the values of ∝ and β are

(1) (-1,2)

(2) (2,-1)

(3) (-1,-2)

(4) (1,2)

Answer:

(2) (2,-1)

Hint:

g(x) = αx + β

α = 2

β = -1

g(x) = 2x – 1

g(1) = 2(1) – 1 = 1

g(2) = 2(2) – 1 = 3

g(3) = 2(3) – 1 = 5

g(4) = 2(4) – 1 = 7

Question 15.

f(x) = (x + 1)^{3} – (x – 1)^{3} represents a function which is

(1) linear

(2) cubic

(3) reciprocal

(4) quadratic

Answer:

(4) quadratic

Hint:

f(x) = (x + 1)^{3} – (x – 1)^{3}

= x^{3} + 3x^{2} + 3x + 1 -[x^{3} – 3x^{2} + 3x – 1]

It is a quadratic function.