Two point objects of masses m and 4m are at rest at an infinite separation. They move towards each other under mutual gravitational attraction. If G is the universal gravitational constant, then at a separation r

For (A) : By COME total mechanical energy of the two objects remains same. Initially , both potential and kinetic energy is zero.
For (B) : By using reduced mass concept
$\frac{1}{2}{\mathrm{\mu v}}_{\text{rel }}^2-\frac{\mathrm{G}\left(4\mathrm{m}\right)\left(\mathrm{m}\right)}{\mathrm{r}}=0\text{ where }\mathrm{\mu }=\frac{\left(\mathrm{m}\right)\left(4\mathrm{m}\right)}{\mathrm{m}+4\mathrm{m}}=\frac{4}{5}\mathrm{m}\Rightarrow {\mathrm{v}}_{\text{rel }}=\sqrt{\frac{10\mathrm{Gm}}{\mathrm{r}}}$
$\text{ For }\left(\mathrm{C}\right)\text{ total }\mathrm{KE}=-\mathrm{PE}=-\left(-\frac{\mathrm{G}\left(\mathrm{m}\right)\left(4\mathrm{m}\right)}{\mathrm{r}}\right)=\frac{4{\mathrm{Gm}}^2}{\mathrm{r}}$