The correct option is C commutative but not associative
Checking commutativity
Given : ∗ be binary operation defined on R by
a ∗ b=1 + ab, ∀ a,b ∈ R
⇒ b ∗ a=1 + ba, ∀ a,b ∈ R
Since, for any a,b ∈ R
ab = ba
(∵ Multiplication is commutative for real numbers)
⇒ 1+ab=1+ba
⇒ (a∗b)=(b∗a) ∀ a,b ∈ R
Therefore, ∗ is commutative binary operation
Checking associativity
a∗(b∗c)=1+a(b∗c)
=1+a(1+bc)
=1+a+abc ...(1)
and
(a∗b)∗c=1+(a∗b)∗c
=1+(1+ab)c
=1+c+abc ...(2)
Clearly, from equation (1) and (2)
1+a+abc ≠ 1+c+abc
∴ a∗(b∗c) ≠ (a∗b)∗c
Therefore, ∗ is not associative binary operation.
Hence, ∗ is commutative but not associative.
∴ Option (A) is correct.