Q53 / 170 · Manipal (MET) 2017 · Mathematics
In the set $Q^{+}$of all positive rational numbers, the operation * is defined by the formula $a * b=\frac{a b}{6}$. Then, the inverse of 9 with respect
- Correct answerA$4$
- B$3$
- C$\frac{1}{9}$
- D$\frac{1}{3}$
Solution
Let $e$ be the identity.
$\begin{aligned}
& \therefore a * e=a \\
& \Rightarrow \frac{a b}{6}=a \\
& \Rightarrow e=6 \\
& \because a * a^{-1}=e \\
& \therefore a * a^{-1}=6 \\
& \Rightarrow \frac{a a^{-1}}{6}=6 \Rightarrow a^{-1}=\frac{6 \times 6}{a} \\
& \therefore 9^{-1}=\frac{6 \times 6}{9}=4
\end{aligned}$
Hence, the correct answer is option: 'A'.