Both clocks were set right at 12 noon. The one on the right gains 2 minutes every hour. After how many hours will its minute hand first point exactly opposite to the minute hand of the correct clock?

The minute hands are opposite when the fast clock is 30 minutes ahead (or 30 minutes plus a whole number of hours). At 2 min per hour that first happens after $\frac{30}{2}=15$ h. (30 h waits for a 60-min gap, when the minute hands coincide again; 90 h uses the hour hands.)