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.