A uniform thin rope of length 12 m and mass 6 kg hangs vertically from a rigid support and a block of mass 2 kg is attached to its free end. A transverse shortwave train of wavelength 4 cm is produced at the lower end of the rope. What is the wavelength of the wave train (in cm) when it reaches the top of the rope?  (Answer the nearest integer)

As wave train moves up over string tension and hence wave speed also charges. Also, frequency(f) wave train remains constant as it depends only on the source.
 
Now, Using  $v=\sqrt{\frac{T}{\mu }}$ 
Where , V= wave speed, T= tension in string and $\mu$ = mass per unit length of string.
Tension  $T_1$  at bottom= 2 kg,
Tension $T_2$  at top=(2+6)kg=8kg
If  ${\lambda }_1$ and  ${\lambda }_2$  wavelengths at bottom and top, then
 $$\begin{gathered} \frac{v_1}{v_2}=\frac{f{\lambda }_1}{f{\lambda }_2}=\frac{\sqrt{\frac{T_1}{\mu }}}{\sqrt{\frac{T_2}{\mu }}} \\ \Rightarrow {\lambda }_2={\lambda }_1\sqrt{\frac{T_2}{T_1}} \\ =4\times \sqrt{\frac{8g}{2g}}=8cm \end{gathered}$$