Two beams, A and B, of plane polarized light with mutually perpendicular planes of polarization are seen through a polaroid. From the position when the beam A has maximum intensity (and beam B has zero intensity), a rotation of Polaroid through 30° makes the two beams appear equally bright. If the initial intensities of the two beams are IA and IB respectively, then IA/IB equals:

Let the initial intensity of beam A be IA and that of beam B be IB.
When the polaroid is in a position where beam A has maximum intensity, its intensity is IA and beam B has zero intensity.
When the polaroid is rotated by 30°, the intensity of A becomes IA cos²30° and the intensity of B becomes IB cos²(60°) = IB cos²(90° - 30°) = IB sin²30°.
Since the two beams appear equally bright after rotation, we have:
IA cos²30° = IB sin²30°
IA (√3/2)² = IB (1/2)²
IA (3/4) = IB (1/4)
IA/IB = (1/4)/(3/4) = 1/3
Therefore, IA/IB = 1/3