An element e ∈ Q will be the identity element for the operation if

a*e = a = e*a, a ∈ Q

(i) a*b = a − b

If a*e = a,a ≠ 0 ⇒ a −e = a, a ≠ 0 ⇒ e = 0

Also, e*a = a ⇒ e − a = a ⇒ e = 2a

⇒ e = 0 = 2a,a ≠ 0

But the identiry is unique. Hence this operation has no identity.

(ii) a*b = a² + b²

If a*e = a, then a² + e² = a

For a = −2, (−2)² + e² = 4 + e² ≠ −2

Hence, there is no identity element.

(iii) a*b = a + ab

If a*e = a ⇒ a + ae = a ⇒ ae = 0 ⇒ e = 0,a ≠ 0

Also if e*a = a ⇒ e + ea = a ⇒ e = a/(1 - a), a ≠ 1

∴ e = 0 = a/(1 - a), a ≠ 0

But the identity is unique. Hence this operation has no identify.

(iv) a*b = (a − b)²

If a*e = a, then (a − e)² = a. A square is always positive, so for

a = −2,(−2 − e)² ≠ − 2

Hence, there is no identity element.

(v) a*b = ab/4

If a*e = a, then ae /4 = a. 

Hence, e = 4 is the identity element.

∴  a*4 =4 *a = 4a/4 = a

(vi) a*b = ab²

If a*e =a then ae² = a ⇒ e² = 1 ⇒ e = ± 1

But identity is unique. Hence this operation has no identity.

Therefore only part (v) has an identity element.