Q. 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)?

Explanation : 

Hour Hand & Minute Hand

1) First, let’s understand how the hands move:
Hour hand makes a complete 360° rotation in 12 hours, so it moves at 360°/12 = 30° per hour or 0.5° per minute
Minute hand makes a complete 360° rotation in 1 hour, so it moves at 360°/60 = 6° per minute

2) For hands to coincide, they must point at the same angle. Let’s use the formula:
Let ‘t’ be minutes after start time
Minute hand angle = 6t degrees
Hour hand angle = (initial hour angle + 0.5t) degrees
When they coincide: 6t = 30h + 0.5t
Where h is hours passed from start of clock (0 to 12)

3) From 10:00 AM to 2:00 PM is 4 hours
At 10:00, hour hand is at 300°
Coincidences happen when: 6t = 300 + 0.5t
5.5t = 300
t = 54.545… minutes after 10:00

4) Similarly, we can find all coincidences:
First: Around 10:54 AM
Second: Around 12:00 noon
Third: Around 1:05 PM

Therefore, in the given 4-hour period from 10:00 AM to 2:00 PM, the hour and minute hands coincide 3 times.

The answer is a) 3 times.