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Binary Operations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsBinary Operations1 Mark
MCQ
On the set $Q^+$ of all positive rational numbers a binary operation $*$ is defined by $\text{a}*\text{b}=\frac{\text{ab}}2\forall\text{ a, b}\in \text{Q}^+$. The inverse of $8$ is:
  • A
    $\frac{1}{8}$
  • ✓
    $\frac{1}2$
  • C
    $2$
  • D
    $4$
✓

Answer

Correct option: B.
$\frac{1}2$
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,
$\frac{\text{ae}}{2}=\text{a}\text{ and }\frac{\text{ea}}{2}=\text{a},\forall\text{ a}\in\text{Q}^+$
$e = 2$, $\forall\text{ a}\in\text{Q}^+$
Thus, 2 is the identity element in $Q^+​​​​​​​$​​​​​​​ with respect to $*$.
Let $\text{b}\in\text{Q}^+$ be the inverse of $8$. Then,
$8 * b = e = b * 8$
$8 * b = e$ and $b * 8 = e$
$\frac{(8)\text{b}}2=2\text{ and }\frac{\text{b}(8)}2=2$ $[\because\ \text{e}=2]$
$b = 12$
Thus, $\frac{1}2$ is the inverse of $8$.

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