At what time between 2 and 3 O’clock will the hands of the clock be opposite?

Let the time be 2 : t minutes.

• Angle of hour hand = 30×2 + 0.5t = 60 + 0.5t degrees.

• Angle of minute hand = 6t degrees.

For the hands to be opposite, the difference between the angles must be 180°:

1. |6t − (60 + 0.5t)| = 180

2. Solve the positive case 6t − 60 − 0.5t = 180 ⇒ 5.5t − 60 = 180 ⇒ 5.5t = 240 ⇒ t = 240/5.5 = 480/11 = 43 7/11 minutes.

3. The negative case gives a negative t (not in the interval 0–60), so discard it.

Convert the fractional minute to seconds if desired: 7/11 minute = (7/11)×60 = 420/11 = 38 2/11 seconds (≈ 38.18 s).

Final answer: 2 hr 43 7/11 min (approximately 2:43:38.18).