At what time between 10 am and 11 am will the angle between the minute hand…

Concept: On an analog clock measured clockwise from 12, the hour hand's angle is (30H + 0.5M)° and the minute hand's angle is (6M)° for a time of H hours and M minutes. The angle between the hands equals the absolute difference between these two values, and whichever hand's angle value is larger sits closer to the upcoming 12 mark.

Step-by-step:

1. Let the required time be 10 hours and x minutes, with 10 ≤ time < 11.

2. Hour-hand angle = 30(10) + 0.5x = 300 + 0.5x.

3. Minute-hand angle = 6x.

4. Since the hour hand is given to be the one closer to 12, its angle value must be the larger of the two, so (300 + 0.5x) − 6x = 3.

5. Simplify: 300 − 5.5x = 3, so 5.5x = 297, giving x = 54.

6. So the required time is 10:54.

Cross-check: At 10:54, hour-hand angle = 300 + 0.5(54) = 327°, and minute-hand angle = 6(54) = 324°. The difference is exactly 3°, and 327° is the larger value, confirming the hour hand is the one nearer the 12 mark — both conditions in the question are satisfied.

Answer: 10:54.