at what temperature the r.m.s speed of N₂ will be tripled and K.E will be 4 times .

Dear student,
$v_{rms}= \sqrt{\frac{3RT}{M}} and K.E. = \frac{3}{2}RT$
For calculating temperature at which rms speed is tripled , we assume that T₁ is the initial temp at which rms speed is v₁ and T₂ is the temp. at which rms speed is tripled .
$$\begin{gathered} \frac{v_1}{3v_1}= \sqrt{\frac{3R{T}_1 }{M }}\times \sqrt{\frac{M}{3R{T}_2}} \\ \frac{1}{3}= \sqrt{\frac{T_1}{T_2}} \\ {T}_2 = 9T_1 \end{gathered}$$
So, the temp. will be 9 times the initial temp. to triple the rms speed.
Now, to increase the K.E. 4 times of initial ; similarly
$$\begin{gathered} \frac{K{E}_1}{4 K{E}_1}= \frac{3 R{T}_1}{2}\times \frac{2}{3 R{T}_2} \\ \frac{1}{4}=\frac{T_1}{T_2} \\ {T}_2 = 4T_1 \end{gathered}$$
Thus to increase K.E. 4 times temperature should also be the 4 times of initial temperature.

Regards