Which of the following is not a group with respect to the given operation?

Determine whether or not the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $R$,define $*$ by $a * b = ab^2$.

If $*$ is a binary operation defined on $R$ by $a * b = 1 + ab, \forall a, b \in R$. Then the operation $*$ is:

Define a binary operation $^*$ on the set $\{0, 1, 2, 3, 4, 5\}$ as $a \,^* \, b = \begin{cases} a+b, & \text{If } a+b < 6 \\ a+b-6, & \text{If } a+b \geq 6 \end{cases}$. Show that $0$ is the identity for this operation and each element $a \neq 0$ of the set is invertible with $6-a$ being the inverse of $a$.

Show that the operation $*: R \times R \rightarrow R$ defined by $a * b = a + 2b$ is not associative.

The binary operation $*$ on $R - \{-1\}$ defined by $a * b = \frac{a}{b+1}$ is: