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Binary Operations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsBinary Operations1 Mark
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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