After the time shown and before the clock next strikes the hour, how many times are the hands at right angles?

At 2:00 the minute hand is $60^{\circ }$ behind; by 3:00 it is $270^{\circ }$ ahead (i.e. $90^{\circ }$ behind). It passes $90^{\circ }$ ahead once (at $27\frac{3}{11}$ min past 2); the next right angle is exactly at 3:00, which is not before the hour. So only 1. (2 is the usual "twice an hour" rule.)