A uniform rope of mass 0.1kg and length 2.45m hangs from a rigid support. The time taken by the transverse wave formed in the rope to travel through full length of the rope is (g = 9.8m/s²)

Tension at any point on the rope is the weight of the rope below that point

from a point at a distance of 'x' from bottom,

$\mathrm{T}=\mathrm{\mu xg}$

$\mathrm{v}=\sqrt{\frac{\mathrm{\mu xg}}{\mathrm{\mu }}}=\sqrt{\mathrm{xg}}$

$\frac{\mathrm{dx}}{\mathrm{dt}}=\sqrt{\mathrm{xg}}\Rightarrow \frac{\mathrm{dx}}{\sqrt{\mathrm{x}}}=\left(\sqrt{\mathrm{g}}\right)\mathrm{dt}$

on integrating, 

${\left[2\sqrt{\mathrm{x}}\right]}_0^{\mathrm{l}}=\left(\sqrt{\mathrm{g}}\right){\left[\mathrm{t}\right]}_0^T$

$\mathrm{T}=2\sqrt{\frac{l}{\mathrm{g}}}=2\sqrt{\frac{2.45}{9.8}}=\frac{2}{2}=1 \sec$