Question

On the set of positive rationals, a binary operation $\ast$ is defined by $a\ast b=\frac{2ab}{5}.$ If $2\ast x=3^{-1}$ then $x=$

• A. $\frac{2}{5}$
• B. $\frac{1}{6}$
• C. $\frac{125}{48}$
• D. $\frac{5}{12}$

Answer 1

The correct option is C $\frac{125}{48}$

$a\ast e=a=e\ast a$
where, e= identity element and
$a^{-1}=$inverese of element $a$
$\therefore a\ast e=a\Rightarrow \frac{2ae}{5}=a\Rightarrow e=\frac{5}{2}$
$\because a\ast a^{-1}=e=a^{-1}\ast a$
$3\ast 3^{-1}=e=\frac{2\ast 3\ast 3^{-1}}{5}\Rightarrow 3^{-1}=\frac{5}{2}\times \frac{5}{2}\times \frac{1}{3}=\frac{25}{12}$
Now, $2\ast x=3^{-1}\Rightarrow \frac{2\times 2\times x}{5}=\frac{25}{12}\Rightarrow x=\frac{125}{48}$