The refractive index of the material of a prism is $\sqrt{2}$ and its refracting angle is $30^\circ$. One of the refracting surfaces of the prism is made a mirror inwards. A beam of monochromatic light entering the prism from the other face will retrace its path after reflection from the mirrored surface if its angle of incidence on the prism is :-

For the ray to retrace its path after reflection, it must fall normally on the mirrored face, i.e. the angle of incidence at the mirrored face is $0^\circ$.

In a prism $r_1 + r_2 = A$. Here $r_2 = 0^\circ$, so the angle of refraction at the first face is $r_1 = A = 30^\circ$.
Applying Snell's law at the first face: $1 \cdot \sin i = \mu \sin 30^\circ$
$\sin i = \sqrt{2} \times \dfrac{1}{2} = \dfrac{1}{\sqrt{2}}$
$\Rightarrow i = 45^\circ$
So the angle of incidence on the prism must be $45^\circ$.