Given a binary operation $\ast$ in Q defined by $a \ast b=ab^2$:

$\textbf {Step 1: Checking if the operation is Commutative}$:

For an operation $\ast$ to be commutative $a\ast b = b \ast a$.

$ a \ast b = ab^2$, but $b \ast a = ba^2$, which are not equal unless $a=b=1$. Hence the operation $\ast$ is not commutative.

$\textbf {Step 2: Checking if the operation is Associative}$:

For an operation $\ast$ on $A$ is associative $a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in A$

$ a \ast (b \ast c) = a \ast (bc^2) = a(bc^2)^2 = ab^2c^4$ and $(a \ast b) \ast c = (ab^2) \ast c = ab^2c^2$

Since $a\ast ( b \ast c) \neq (a \ast b) \ast c$, the operation $\ast$ is not associative.