After the time shown, when do the hands first make a right angle?

At 2:00 the hour hand is 60° ahead. The minute hand gains 5.5° a minute; it must catch up 60° and get 90° ahead, i.e. gain 150°: 150 ÷ 5.5 = 27 3/11 min past 2. (5 5/11 uses 90 − 60; 16 4/11 uses only 90.)