How many times the hour hand and the minute hand coincide in a clock between 10:00 a.m. and 2:00 p.m. (same day)?

Correct Option (1)

The hour hand and minute hand of a clock coincide at specific intervals. Over a 12-hour period, they coincide 11 times. This is because the coincidence that would typically occur between 11 and 12, and between 12 and 1, both happen precisely at 12 o'clock, counting as a single event for this two-hour span.

Considering the period from 10:00 a.m. to 2:00 p.m.:

• Between 10:00 a.m. and 11:00 a.m.: The hands coincide once (approximately at 10:55 a.m.).
• Between 11:00 a.m. and 1:00 p.m.: This two-hour interval includes the critical point of 12:00 p.m. The hands coincide exactly at 12:00 p.m. This single coincidence accounts for the period from 11:00 a.m. to 12:00 p.m. and also from 12:00 p.m. to 1:00 p.m. Thus, for the entire 11:00 a.m. to 1:00 p.m. duration, they coincide only once.
• Between 1:00 p.m. and 2:00 p.m.: The hands coincide once (approximately at 1:05 p.m.).

Summing these instances, the total number of times the hour hand and minute hand coincide between 10:00 a.m. and 2:00 p.m. is 1 + 1 + 1 = 3 times.

Incorrect Options:

Options 2 (4 times), 3 (5 times), and 4 (6 times) are incorrect. These options do not align with the established pattern of clock hand coincidences, particularly the unique behavior around the 12 o'clock mark where a single coincidence covers two distinct hourly intervals (11:00-12:00 and 12:00-1:00).