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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS1 Mark
Question
Q+ is the set of all positive rational numbers with the binary operation * defined by $\text{a}*\text{b}=\frac{\text{ab}}2\ \forall\text{ a, b}\in\text{Q}^+$. The inverse of an element $\text{a}\in\text{Q}^+$ is:
  1. $\text{a}$
  2. $\frac{1}{\text{a}}$
  3. $\frac{2}{\text{a}}$
  4. $\frac{4}{\text{a}}$
✓

Answer

  1. $\frac{4}{\text{a}}$

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, $\forall\text{ a}\in\text{Q}^+$

$\frac{\text{ae}}2=\text{a}$ and $\frac{\text{ea}}2=\text{a}$, $\forall\text{ a}\in\text{Q}^+$

$\text{e}=2\in\text{Q}^+, \forall\text{ a}\in\text{Q}^+$

Thus, 2 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 * e = a = e * a

a * b = e and b * a = e

$\frac{\text{ab}}2=2$ and $\frac{\text{ba}}2=2$

$\text{b}=\frac{4}{\text{a}}\in\text{Q}^+$

Thus, $\frac{4}{\text{a}}$ is the inverse of $\text{ a}\in\text{Q}^+$.

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