NCERT solution Class 12 relations and functions

NCERT Solutions

Class 12 Maths

Relations and Functions

NCERT Questions

Ex. 1.1 Q1
Ex. 1.1 Q2
Ex. 1.1 Q3
Ex. 1.1 Q4
Ex. 1.1 Q5
Ex. 1.1 Q6
Ex. 1.1 Q7
Ex. 1.1 Q8
Ex. 1.1 Q9
Ex. 1.1 Q10
Ex. 1.1 Q11
Ex. 1.1 Q12
Ex. 1.1 Q13
Ex. 1.1 Q14
Ex. 1.1 Q15
Ex. 1.1 Q16
Ex. 1.2 Q1
Ex. 1.2 Q2
Ex. 1.2 Q3
Ex. 1.2 Q4
Ex. 1.2 Q5
Ex. 1.2 Q6
Ex. 1.2 Q7
Ex. 1.2 Q8
Ex. 1.2 Q9
Ex. 1.2 Q10
Ex. 1.2 Q11
Ex. 1.2 Q12
Ex. 1.3 Q1
Ex. 1.3 Q2
Ex. 1.3 Q3
Ex. 1.3 Q4
Ex. 1.3 Q5
Ex. 1.3 Q6
Ex. 1.3 Q7
Ex. 1.3 Q8
Ex. 1.3 Q9
Ex. 1.3 Q10
Ex. 1.3 Q11
Ex. 1.3 Q12
Ex. 1.3 Q13
Ex. 1.3 Q14
Ex. 1.4 Q1
Ex. 1.4 Q2
Ex. 1.4 Q3
Ex. 1.4 Q4
Ex. 1.4 Q5
Ex. 1.4 Q6
Ex. 1.4 Q7
Ex. 1.4 Q8
Ex. 1.4 Q9
Ex. 1.4 Q10
Ex. 1.4 Q11
Ex. 1.4 Q12
Ex. 1.4 Q13
Misc. Ex. Q1
Ex. 1.4 Q2
Ex. 1.4 Q3
Misc. Ex. Q4
Misc. Ex. Q5
Misc. Ex. Q6
Misc. Ex. Q7
Misc. Ex. Q8
Misc. Ex. Q9
Misc. Ex. Q10
Misc. Ex. Q11
Misc. Ex. Q12
Misc. Ex. Q13
Misc. Ex. Q14
Misc. Ex. Q15
Misc. Ex. Q16
Misc. Ex. Q17
Misc. Ex. Q18
Misc. Ex. Q19

Ex. 1.4 Q9

Let * be a binary operation on the set Q of rational numbers as follows:

(i) a * b = a − b (ii) a * b = a² + b₂ (iii) a * b = a + ab

(iv) a * b = (a − b)² (v) a ∗ b = ab/4 (vi) a * b = ab²

Find which of the binary operations are commutative and which are associative.

(i) On Q, the operation * is defined as a * b = a − b. It can be observed that:

(1/2) * (1/3) = 1/2 - 1/3 = (3 - 2)/6 = 1/6 and

(1/3) * (1/2) = 1/3 - 1/2 = (2 - 3)/6 = -1/6

So, (1/2) * (1/3) ≠ (1/3) * (1/2), where 1/2, 1/3 ∈ Q

Thus, the operation * is not commutative.

It can also be observed that

(1/2 * 1/3) * 1/4 = (1/2 - 1/3) * 1/4 = (3 - 2)/6 * 1/4 = 1/6 * 1/4 = (2 - 3)/12 = -1/12 and

1/2 *( 1/3 * 1/4) = 1/2 * (1/3 - 1/4) = 1/2 * (4 - 3)/12 = 1/2 * 1/12 = (6 - 1)/12 = 5/12

So, (1/2 * 1/3) * 1/4 ≠ 1/2 * (1/3 * 1/4), where 1/2, 1/3, 1/4 ∈ Q

Thus, the operation * is not commutative.

(ii) On Q, the operation * is defined as a * b = a² + b²

For a, b ∈ Q, we have

a * b = a² + b² = b² + a² = b * a

So, a * b = b * a

Thus, the operation * is commutative.

It can be observed that

(1 * 2) * 3 = (1² + 2²) * 3 = (1 + 4) * 3 = 5 * 3 = 5² + 3² = 34 and

1 * (2 * 3) = 1 * (2² + 3²) = 1 * (4 + 9) = 1 * 13 = 1² + 13² = 170

So, (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ Q

Thus, the operation * is not associative.

(iii) On Q, the operation * is defined as a * b = a + ab.

It can be observed that

1 * 2 = 1 + 1 × 2 = 1 + 2 = 3

2 * 1 = 2 + 2 × 1 = 2 + 2 = 4

So, 1 * 2 ≠ 2 * 1, where 1, 2 ∈ Q

Thus, the operation * is not commutative.

It can also be observed that

(1 * 2) * 3 = (1+ 1 × 2) * 3 = (1 + 2) * 3 = 3 * 3 = 3 + 3 × 3 = 3 + 9 = 12 and

1 * (2 * 3) = 1 * (2 + 2 × 3) = 1 * (2 + 6) = 1 * 8 = 1 + 1 × 8 =1 + 8 = 9

So, (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ Q

Thus, the operation * is not associative.

(iv) On Q, the operation * is defined by a * b = (a − b)²

For a, b ∈ Q, we have

a * b = (a − b)²

b * a = (b − a)² = [− (a − b)]² = (a − b)²

So, a * b = b * a

Thus, the operation * is commutative.

It can be observed that

(1 * 2) * 3 = (1 – 2)² * 3 = (– 1)² * 3 = 1 * 3 = (1 – 3)² = (– 2)² = 4

and

1 * (2 * 3) = 1 * (2 – 3)² = 1 * (– 1)² = 1 * 1 = (1 – 1)² = 0

So, (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ Q

Thus, the operation * is not associative.

(v) On Q, the operation * is defined as a*b = ab/4

For a, b ∈ Q, we have

a*b = ab/4 = ba/4 = b*a

So, a * b = b * a

Thus, the operation * is commutative.

For a, b, c ∈ Q, we have

(a ∗ b) ∗ c = (ab/4) ∗ c = {(ab/4) . c}/4 = abc/16 and

a ∗ (b ∗ c) = a ∗ ( bc/4) = {a. (bc/4)}/4 = abc/16

So, (a * b) * c = a * (b * c), where a, b, c ∈ Q

Thus, the operation * is associative.

(vi) On Q, the operation * is defined as a * b = ab2

It can be observed that

(1/2) * (1/3) = (1/2) * (1/3)² = (1/2).(1/9) = 1/18 and

(1/3) * (1/2) = (1/3) * (1/2)² = (1/3).(1/4) = 1/12

So, (1/2) * (1/3) ≠ (1/3) * (1/2), where 1/2 and 1/3 ∈ Q

Thus, the operation * is not commutative.

It can also be observed that

(1/2 * 1/3) * 1/4 = [1/2(1/3)²] ∗ 1/4 = 1/18 ∗ 1/4 = 1/18 . (1/4)² = 1/(18 × 16) = 1/288 and

1/2 * (1/3 * 1/4) = (1/2)[1/3(1/4)²] = 1/2 ∗ 1/48 = 1/2 . (1/48)² = 1/(2 × 2304) = 1/4608

So, (1/2 * 1/3) * 1/4 ≠ 1/2 * (1/3 * 1/4) , where 1/2, 1/3, 1/4 ∈ Q

Thus, the operation * is not associative.

Hence, the operations defined in (ii), (iv), (v) are commutative and the operation defined in (v)

is associative.

Relations and Functions Class 12 Maths NCERT Chapter 1 NCERT Solutions

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