**Question:**

Consider a binary operation * on $\mathbf{N}$ defined as $a^{*} b=a^{3}+b^{3}$. Choose the correct answer.

(A) Is * both associative and commutative?

(B) Is * commutative but not associative?

(C) Is * associative but not commutative?

(D) Is * neither commutative nor associative?

**Solution:**

On $\mathbf{N}$, the operation ${ }^{*}$ is defined as $a^{*} b=a^{3}+b^{3}$.

For, $a, b, \in \mathbf{N}$, we have:

$a^{*} b=a^{3}+b^{3}=b^{3}+a^{3}=b^{*} a$ [Addition is commutative in $\mathbf{N}$ ]

Therefore, the operation * is commutative.

It can be observed that:

$(1 * 2) * 3=\left(1^{3}+2^{3}\right) * 3=9 * 3=9^{3}+3^{3}=729+27=756$

$1 *(2 * 3)=1 *\left(2^{3}+3^{3}\right)=1 *(8+27)=1 \times 35=1^{3}+35^{3}=1+(35)^{3}=1+42875=42876$

$\therefore\left(1^{*} 2\right)^{*} 3 \neq 1^{*}\left(2^{*} 3\right) ;$ where $1,2,3 \in \mathbf{N}$

Therefore, the operation * is not associative.

Hence, the operation * is commutative, but not associative. Thus, the correct answer is B.