Question

Let $Q^{+}$ be the set of all positive rational numbers.
Let $\ast$ be an operation on $Q^{+}$ defined by $a\ast b=\frac{ab}{2}\forall a,b\in Q^{+}$. Then, the identity element in $Q^{+}$ for the operation $\ast$ is:

• A. 0
• B. 1
• C. 2
• D. $\frac{1}{2}$

Answer 1

Answer: (C)

Let, e be required identity element in $Q^{+}$ for the operation $\ast$.
$\Rightarrow a\ast e=e\ast a=a$ .....(1)
Now, $a\ast e=\frac{ae}{2}$ ......(2)
(By defn. of $a\ast b=\frac{ab}{2}$ (given)) and $e\ast a=\frac{ea}{2}$ .....(3)
$\therefore$ From equation (1) and (2) we have
$\frac{ae}{2}=a$
$\Rightarrow e=2$
Thus, identity element in $Q^{+}$ for $a\ast b=\frac{ab}{2}$ is 2.