After (3) o'clock when will the hour and minute hands first again make a right angle?

Correct answer: after (\frac{360}{11}) minutes

The direct answer is C: \\(\frac{360}{11}\\) minutes after 3 o’clock. At exactly 3:00, the hands are 90° apart. After \\(m\\) minutes, the minute hand moves \\(6m^\circ\\), while the hour hand moves \\(0.5m^\circ\\). Thus the separation decreases at \\(6-0.5=5.5^\circ\\) per minute, so the separation is \\(90-5.5m\\). For the first later right angle, set it equal to 90° after the hands have passed through the other orientation, or equivalently use the relative movement needed to gain 90°: \\(5.5m=180^\circ\\), giving \\(m=180/5.5=360/11\\) minutes. Option C is correct, about 32 minutes 44 seconds after 3, near 3:32:44. Option A, \\(300/11\\), and B, \\(330/11\\), do not produce a 90° separation. Option D, \\(390/11\\), is later than the first occurrence. Common mistake: using only the minute-hand speed and forgetting that the hour hand also moves.