The root mean square speed of the molecules of a diatomic gas is v. When the temperature is doubled, the molecules dissociate into two atoms. The new root mean square speed of the atom is

${\mathrm{v}}_{\mathrm{rms}}=\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}$.
According to problem T will becomes T/2 and M will becomes M/2 

$\Rightarrow \mathrm{v}{\text{'}}_{\mathrm{rms}}=\sqrt{\frac{3\mathrm{RT}\text{'}}{\mathrm{M}\text{'}}}=\sqrt{3\mathrm{R}\left(2\mathrm{T}\right)\frac{2}{\mathrm{M}}}=2\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}=2{\mathrm{v}}_{\mathrm{rms}}$

so, the value of v_rms will increase by 2 times i.e. new root mean square velocity will be 2v.