Question

On $Q-\left\{1\right\},$ $\ast$ be the binary operation defined as $a\ast b=ab+1.$ Then the identity element for $\ast$ on $Q-\left\{1\right\}$ is

• A. $\frac{a-1}{a}$
• B. $\frac{1}{1+a}$
• C. $\frac{1}{1-a}$
• D. $\frac{a+1}{a}$

Answer 1

The correct option is A $\frac{a-1}{a}$

Let $e$ be the identity element.
$\text{Then}a\ast e=e\ast a=a\forall a\in Q-\left\{1\right\}$
$\Rightarrow ae+1=a$
$\Rightarrow e=\frac{a-1}{a}$