A prism of apex angle A=60^° is made of glass of refractive index √2 and is surrounded by air. A narrow monochromatic ray passes through the…

For $n=\sqrt2$ the critical angle is $\theta_c=45^\circ$. At minimum deviation $r_1=r_2=A/2=30^\circ$ and $\sin i=\sqrt2\sin30^\circ=1/\sqrt2$, so $i=45^\circ$ and $\delta_{min}=2i-A=30^\circ$: (A) is correct. In (B) the angle of refraction at the first face is $30^\circ$, not $45^\circ$ (the angle of incidence is $45^\circ$), so (B) is incorrect. (C) Grazing incidence gives $\sin r_1=1/\sqrt2$, $r_1=45^\circ$ (the largest possible), hence $r_2=60^\circ-45^\circ=15^\circ<\theta_c$; the ray therefore emerges (at about $21.5^\circ$), so (C) is incorrect. (D) Since $r_1\le\theta_c$, we have $r_2=A-r_1\ge A-45^\circ$; for $A>90^\circ$, $r_2>45^\circ=\theta_c$ for every ray, so all are totally internally reflected. (D) is correct.