a
(a) At angular frequency \(\Omega\), the current in \(RC\) circuit is given by

\({i_{rms}} = \frac{{{V_{rms}}}}{{\sqrt {{R^2} + {{\left( {\frac{1}{{\omega C}}} \right)}^2}} }}\)                                                  ......\((i)\)

Also \(\frac{{{i_{rms}}}}{2} = \frac{{{V_{rms}}}}{{\sqrt {{R^2} + {{\left( {\frac{1}{{\frac{\omega }{3}C}}} \right)}^2}} }} = \frac{{{V_{rms}}}}{{\sqrt {{R^2} + \frac{9}{{{\omega ^2}{C^2}}}} }}\)                 ......\((ii)\)

From equation \((i)\) and \((ii)\) we get

\(3{R^2} = \frac{5}{{{\omega ^2}{C^2}}} \Rightarrow \frac{{\frac{1}{{\omega C}}}}{R} = \sqrt {\frac{3}{5}} \)==> \(\frac{{{X_C}}}{R} = \sqrt {\frac{3}{5}} \)