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Binary Operations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinary Operations1 Mark
Question
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:
  1. $\frac{1}{8}$
  2. $\frac{1}2$
  3. $2$
  4. $4$
✓

Answer

  1. $\frac{1}2$

Solution:

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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