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Binary Operations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinary Operations1 Mark
MCQ
Let $*$ be a binary operation defined on $Q^+$ by the rule $\text{a}^*\text{b}=\frac{\text{ab}}3\forall\text{ a, b}\in \text{Q}^+.$ The inverse of $4 ^* 6$ is:
  • ✓
    $\frac{9}{8}$
  • B
    $\frac{2}3$
  • C
    $\frac{3}2$
  • D
    None of these.
✓

Answer

Correct option: A.
$\frac{9}{8}$
Let e be the identity element in $Q^+$ with respect to $*$ such that
$a ^* e = a = e^ * a, \forall\text{ a}\in\text{Q}^+$
$a^ * e = a$ and $e^ * a = a, \forall\text{ a}\in\text{Q}^+$
Then,
$\frac{\text{ae}}{3}=\text{a}$ and $\frac{\text{ea}}{3}=\text{a},\forall\text{ a}\in\text{Q}^+$
$e = 3, \forall\text{ a}\in\text{Q}^+$
Thus$, 3$ is the identity element in $Q^+$ with respect to $*.$
Let $\text{a}\in\text{Q}^+$ and $\text{b}\in\text{Q}^+$ be the inverse of $a.$ Then,
$a^ * b = e = b^ * a$
$a^ * b = e$ and $b^ * a = e$
$\therefore\ \frac{\text{ab}}3=3$ and $\frac{\text{ba}}3=3$
$\text{b}=\frac{9}{\text{a}}\in\text{Q}^+$
Thus, $\frac{9}{\text{a}}$ is the inverse of $\text{a}\in\text{Q}^+$.
Given: $\text{a}^*\text{b}=\frac{\text{ab}}3$
$4^*6=\frac{4\times6}3=8$
Now,
$\text{a}^{-1}=\frac{9}{\text{a}}$
$(4^*6)^{-1}=8^{-1}$
$=\frac{9}8$

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