Two solid spheres of radii r and 2r, made of the same material, are kept in contact. The mutual gravitational force of attraction between them is proportional to

If $\rho$ is the density of the material of each sphere, then the mass of the sphere of radius r is $M_1=\frac{4\mathrm{\pi }}{3}{r}^3\rho$ and the mass of the sphere of radius 2r is $M_2=\frac{4\mathrm{\pi }}{3}{\left(2r\right)}^3\rho .$

Distance between their centres is d = r + 2r = 3r. 

Now F = $\frac{G{M}_1{M}_2}{d^2}=\frac{G\times \left({\displaystyle \frac{4\mathrm{\pi }}{3}}\right)r^3\rho \times \left({\displaystyle \frac{4\mathrm{\pi }}{3}}\right){\left(2r\right)}^3\rho }{9r^2}$

which gives $F\propto r^4$, which is choice ( 4).