Explanation

(a)
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ v_{rms} = \sqrt{\frac{3RT}{M}}$
$\therefore\ \ \ \ \ \ \ \ \ \ v_{rms}\ \propto \ \sqrt{T}$
$\therefore$ At two different temperatures,
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{v_{rms}}{v_{rms}^{'}} = \sqrt{\frac{T}{T^{'}}}$
Given, $v^{'}_{rms} = 2v_{rms}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{2} = \sqrt{\frac{T}{T^{'}}}\ \ or\ \frac{1}{4} = \frac{T}{T^{'}}$
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T^{'} = 4T$
$\therefore v_{rms}$ gets doubled, when the temperature is increased four times.