On the set of all non-zero reals,an operation $*$ is defined as $a * b = \frac{3ab}{2}$. In this group,a solution of $(2 * x) * 3^{-1} = 4^{-1}$ is

The inverse of $2010$ in the group $Q^{+}$ of all positive rational numbers under the binary operation $*$ defined by $a * b = \frac{ab}{2010}, \forall a, b \in Q^{+}$,is

Determine whether or not each of the definitions of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^{+}$,define $*$ by $a * b = a$.

Which one of the following is not true?

Show that the number of binary operations on $\{1, 2\}$ having $1$ as identity and having $2$ as the inverse of $2$ is exactly one.

Let $^*$ be the binary operation on $N$ given by $a \,^* \,b = \text{L.C.M. of } a \text{ and } b$. Find $5 \,^* \,7$ and $20 \,^* \,16$.