If vibration of string is to be two times, then tension in the...

Here: Initial vibration $n_1=n$, final vibration $n_2=2 n$ Initial tension $T_1=T$ Vibration of frequency of string $n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$ $\Rightarrow \quad n \propto \sqrt{T}$ Hence, $\frac{n_1}{n_2}=\sqrt{\frac{T_1}{T_2}}$ or $\frac{T_1}{T_2}=$\left(\frac{n_1}{n_2}\right)^2=\left(\frac{n}{2 n}\right)^2=\frac{1}{4}$or$T_2=4 T_1