A beam of unpolarised light is incident from air onto a transparent...

Since the reflected light is completely plane-polarised, the light must be incident at Brewster's angle ($i_p$).

At Brewster's angle, the reflected and refracted rays are mutually perpendicular, which gives the relation:
$i_p + r = 90^{\circ}$

Given the angle of refraction $r = 30^{\circ}$:
$i_p + 30^{\circ} = 90^{\circ}$
$i_p = 60^{\circ}$

According to Brewster's law, the refractive index $\mu$ of the medium is given by:
$\mu = \tan(i_p)$
$\mu = \tan(60^{\circ}) = \sqrt{3}$

Answer: $\sqrt{3}$