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Binary Operations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsBinary Operations1 Mark
MCQ
Let $^*$ be a binary operation on $Q^+$ defined by $\text{a}^*\text{b}=\frac{\text{ab}}{100}\forall\text{ a, b}\in\text{Q}^+$. The inverse of $0.1$ is:
  • ✓
    $10^5$
  • B
    $10^4$
  • C
    $10^6$
  • D
    None of these.
✓

Answer

Correct option: A.
$10^5$
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 $e ^* a = a$, $\forall\text{ a}\in\text{Q}^+$
$\frac{\text{ae}}{100}=\text{a}\text{ and }\frac{\text{ea}}{100}=\text{a},\forall\text{ a}\in\text{Q}^+$
$\text{e}=100,\forall\text{ a}\in\text{Q}^+$
Thus, 100 is the identity element in $Q^+$​​​​​​​ with repect to $^*$.
$0.1 ^* b = e = b ^* 0.1$
$0.1 ^* b = e$ and $b ^* 0.1 = e$
$\frac{(0.1)\text{b}}{100}=100\text{ and }\frac{\text{b}(0.1)}{100}=100$
$\text{b}=\frac{100\times100}{0.1}$
$=10^5\in\text{Q}^+$
Thus, $10^5​​​​​​​$​​​​​​​ is the inverse of $0.1$.

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