A clock set right at 12 noon gains 6 minutes every hour. It now shows the time in the picture. At this moment, what angle do the hands of a CORRECT clock make?

The dial shows 5:30, i.e. 330 min after noon. It runs 66 min per true hour, so the true time is $330\times \frac{60}{66}=300$ min after noon: 5:00. At 5:00 the hands are $150^{\circ }$ apart. (15° is the angle on the fast clock itself.)