Q98 / 180 · KCET 2019 · Mathematics
On the set of positive rationals, a binary operation * is defined by \( a^{*} b=\frac{2 a b}{5} \). If \( 2^{*} x=3^{-1} \)
then \( x= \)
then \( x= \)
- A\( \frac{5}{12} \)
- Correct answerB\( \frac{125}{48} \)
- C\( \frac{1}{6} \)
- D\( \frac{2}{5} \)
Solution
(B)
\[ \begin{array}{l} a^{*} e=a \\ a^{*} e=a \Rightarrow \frac{2 a e}{5}=a \Rightarrow e=\frac{5}{2} \\ a^{*} a^{-1}=e \Rightarrow \frac{2 a a^{-1}}{5}=\frac{5}{2} \Rightarrow a^{-1}=\frac{25}{4 a} \\ 2^{*} x=3^{-1} \Rightarrow \frac{2(2 x)}{5}=\frac{25}{4(3)} \\ \Rightarrow x=\frac{125}{48} \end{array} \]
\[ \begin{array}{l} a^{*} e=a \\ a^{*} e=a \Rightarrow \frac{2 a e}{5}=a \Rightarrow e=\frac{5}{2} \\ a^{*} a^{-1}=e \Rightarrow \frac{2 a a^{-1}}{5}=\frac{5}{2} \Rightarrow a^{-1}=\frac{25}{4 a} \\ 2^{*} x=3^{-1} \Rightarrow \frac{2(2 x)}{5}=\frac{25}{4(3)} \\ \Rightarrow x=\frac{125}{48} \end{array} \]
Hence, the correct answer is option: 'B'.