Question

On the set Q⁺ of all positive rational numbers a binary operation * is defined by \[a * b = \frac{ab}{2} \text{ for all, a, b }\in Q^+\]. The inverse of 8 is _________ .

Options

• `1/8`

• `1/2`

• 2

• 4

Answer 1

`1/2`

Let e be the identity element in Q⁺ with respect to * such that

\[a * e = a = e * a, \forall a \in Q^+ \]
\[a * e = a \text{ and }e * a = a, \forall a \in Q^+ \]
\[\text{ Then}, \]
\[\frac{ae}{2} = a \text{ and }\frac{ea}{2} = a, \forall a \in Q^+ \]
\[e = 2, \forall a \in Q^+\]

Thus, 2 is the identity element in Q⁺ with respect to *

\[\text{ Let }b \in Q^+ \text{ be the inverse of 8 . Then },\]
\[8 * b = e = b * 8\]
\[8 * b = e \text { and }b * 8 = e\]
\[\frac{8b}{2} = 2 \text{ and }\frac{b\left( 8 \right)}{2}=2\left[ \because e = 2 \right]\]
\[b = \frac{1}{2}\]
\[\text{Thus },\frac{1}{2} \text{ is the inverse of 8 } . \]