(i) On Q, the operation * is defined as a*b = a − b.

It can be observed that for 2,3,4 ∈ Q, we have 2*3 = 2 − 3 = −1 and 3*2 = 3 − 2 = 1 

⇒ 2*3 ≠ 3*2

Thus, the operation is not commutative.

It can also be observed that

(2*3)*4 = (−1)*4 = −1 − 4 = − 5

and 2*(3*4) = 2*(−1) = 2 − (−1) = 3 

2*(3*4) ≠ 2* (3*4)

Thus, the operation * is not associative.

(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

Therefore, a*b = b*a

Thus, the operation * is commutative.

It can be observed that

(1* 2)* 3 = (1² + 2²) * 3 (1 + 4) * 4 = 5 * 4 = 5² + 4² = 41

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

⇒ (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

⇒ 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 = 3*3 = 3 + 3 × 3 = 3 + 9 = 12

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

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

Therefore, a*b = b*a 

Thus, the operation * is commutative. It can be observed that 

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

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

⇒ (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 

Therefore, 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

Therefore, (a*b)*c = a*(b*c).

Thus, the operation * is associative.

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

It can be observed that for 23 ∈ Q

2 * 3 = 2 x 3² = 18 and 3*2 = 3 x 2² = 12

Hence, 2*3 ≠ 3*2

Thus, the operation is not commutative.

It can also be observed that for 1,2,3 ∈ Q 

(1*2)*3 = (1. 2²)*3 = 4 *3 = 4.3² = 36 

and 1*(2* 3) = 1*(2. 3²) = 1 *18 = 1.18² = 324 

⇒ (1*2)*3 ≠ 1*(2*3)

Thus, the operation * is not associative. 

Hence, the operations defined in parts (ii), (iv), (v) are commutative and the operation defined in part (v) is associative.