Let * be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is(i) commutative but not associative (ii) associative but not commutative (iii) neither commutative nor associative (iv) both commutative and associative

Dear Student

Given that * is a binary operation defined on R by a * b = 1 + ab,∀a, b∈R
So, we have a * b = ab + 1 = b * a
So, * is a commutative binary operation.
Now, a * (b * c) = a * (1 + bc) = 1 + a (1 + bc) = 1 + a + abc

(a * b) * c = (1 + ab) * c = 1 + (1 + ab) c = 1 + c + abc
Therefore, a * (b * c) ≠ (a * b) * c
Hence, * is not associative.
Thus, * is commutative but not associative.
Thanks