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## Tamilnadu Samacheer Kalvi 10th Maths Solutions Chapter 1 Relations and Functions Ex 1.6

Multiple Choice Questions

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: n(A × B) = 6

n(A) = 2

n(A × B) = n(A) × n(B)

6 = 2 × n(B)

n(B) = \(\frac { 6 }{ 2 } \) = 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: n(A × B) = 2 × 4 = 8

(A × C) = 2 × 2 = 4

n(B × C) = 4 × 2 = 8

n(C × D) = 2 × 4 = 8

n(A × C) = 2 × 2 = 4

n(A × D) = 2 × 4 = 8

n(B × D) = 4 × 4 = 16

∴ (A × C) ⊂ (B × D)

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}) 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:

Prime number less than 13 = {2, 3, 5, 7, 11}

Range (R) = {(x, x^{2})}

Range = {4, 9, 25, 49, 121} (square of x)

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}

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:

Different elements of A has different images in B.

∴ It is one-to-one function.

Question 10.

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

(1) \(\frac{3}{2 x^{2}}\)

(2) \(\frac{2}{3 x^{2}}\)

(3) \(\frac{2}{9 x^{2}}\)

(4) \(\frac{1}{6 x^{2}}\)

Answer:

(3) \(\frac{2}{9 x^{2}}\)

Hint:

Question 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 f o g 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: f = {(0, 1)(2, 0)(3, -4) (4, 2) (5, 7)}

g = {(0,2)(l,0)(2,4)(-4,2)(7,0)}

fog = f[g(x)]

f [g(0)] = f(2) = 0

f [g(1)] = f(0) = 1

f [g(2)] = f(4) = 2

f[g(-4)] = f(2) = 0

f[g(7)] = f(0) = 1

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:

\(\sqrt{1+x^{2} y^{2}} \leq \sqrt{\left(1+x^{2}\right)} \sqrt{\left(1+y^{2}\right)}\)

⇒ f(xy) ≤ f(x) . f(y)

Question 14.

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

(1) (-1,2)

(2) (2,-1)

(3) (-1,-2)

(4) (1,2)

Answer:

(2) (2, -1)

Hint: g (x) = αx + β

g(1) = α(1) + β

1 = α + β ….(1)

g (2) = α (2) + β

3 = 2α + β ….(2)

Solve the two equations we get

α = 2, β = -1

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]

= x^{3} + 3x^{2} + 3x + 1 – x^{3} + 3x^{2} – 3x + 1 = 6x^{2} + 2

It is a quadratic function.