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)

2

Answer 2

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.