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Question
Let * be a binary operation defined on Q+ by the rule
\[a * b = \frac{ab}{3} \text{ for all a, b } \in Q^+\] The inverse of 4 * 6 is ___________ .
Options
`9/8`
`2/3`
`3/2`
none of these
Solution
\[\frac{9}{8}\]
Let e be the identity element in Q+ with respect to * such that
\[a * e = a = e * a, \forall a \in Q^+ \]
\[ a * e = a \text{ and }e * a = a, \forall a \in Q^+ \]
\[\text{ Then }, \]
\[\frac{ae}{3} = a \text{ and }\frac{ea}{3} = a, \forall a \in Q^+ \]
\[ \Rightarrow e = 3 , \forall a \in Q^+\]
Thus, 3 is the identity element in Q+ with respect to *.
\[\text{ Let }a \in Q^+ \text{ and }b \in Q^+ \text{ be the inverse of a . Then },\]
\[a * b = e = b * a\]
\[a * b = e \text{ and }b * a = e\]
\[ \therefore \frac{ab}{3} = 3 \text{ and }\frac{ba}{3}=3\]
\[b = \frac{9}{a} \in Q^+ \]
\[\text{ Thus },\frac{9}{a} \text{ is the inverse of a } \in Q^+ . \]
\[\text{ Given } : a * b = \frac{ab}{3}\]
\[4 * 6 = \frac{4 \times 6}{3} = 8\]
\[Now,\]
\[ a^{- 1} = \frac{9}{a}\]
\[ \left( 4 * 6 \right)^{- 1} = 8^{- 1} \]
\[ = \frac{9}{8}\]
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