Step 1: Read the special condition.
Light falls on a glass plate. We are told the reflected ray and the refracted ray make an angle of $90^\circ$ with each other. This is a famous special case in optics.
Step 2: Recall what this case means.
When the reflected and refracted rays are exactly at right angles, the angle of incidence is called Brewster's angle. At this angle the reflected light becomes fully polarised.
Step 3: Write Brewster's law.
Brewster's law says the tangent of this angle equals the refractive index of the glass: $\tan\theta_B = \mu$.
Step 4: Put in the given value.
Here $\mu = \sqrt{3}$, so we need $\tan\theta_B = \sqrt{3}$.
Step 5: Find the angle.
We ask: which angle has a tangent of $\sqrt{3}$? That angle is $60^\circ$, because $\tan 60^\circ = \sqrt{3}$. So $\theta_B = 60^\circ$.
Step 6: State the answer.
Therefore the ray must strike the glass at $60^\circ$ for the reflected and refracted rays to be perpendicular. \[ \boxed{60^\circ} \]