Let * be a binary operation defined on set Q − 1 by the rule a^ *… - Vidyadip

MCQ

Let $*$ be a binary operation defined on set $Q − \{1\}$ by the rule $a^ * b = a + b − ab.$ Then, the identify element for $*$ is:

A
$1$
B
$\frac{\text{a}-1}{\text{a}}$
C
$\frac{\text{a}}{\text{a}-1}$
✓
$0$

✓

Answer

Correct option: D.

$0$

Let e be the identity element in $Q - \{1\}$ with respect to $*$ such that
$a^ * e = a = e^ * a, \forall\text{ a}\in\text{Q}-\{-1\}$
$a^ * e = a$ and $e^ * a = a, \forall\text{ a}\in\text{Q}-\{-1\}$
$a + e - ae = a$ and $e + a - ea = a, \forall\text{ a}\in\text{Q}-\{-1\}$
$e(1 - a) = 0, \forall\text{ a}\in\text{Q}-\{-1\}$
$[\because \text{a}\neq1]$
Thus$, 0$ is the identity element in $Q - \{1\}$ with respect to $*.$

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