GATE 2025 PH – Question 7
Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time (in minutes) did Rohit spend at the restaurant?
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Correct answer: (C) 65 5 11
Explanation
Hour hand and minute hand coincide again after the minute hand has gained a full circle on the hour hand.
**Angular speeds.**
- Minute hand: $360^\circ$ per 60 min $=6^\circ$/min.
- Hour hand: $360^\circ$ per 720 min $=0.5^\circ$/min.
**Relative speed:** $6-0.5=5.5=\dfrac{11}{2}$ degrees per minute.
**Time between successive coincidences:** the minute hand must gain $360^\circ$:
$$t=\frac{360}{11/2}=\frac{720}{11}=65\frac5{11}\text{ minutes}.$$
Rohit enters at one coincidence (a little after 1 PM) and leaves at the next, so he spends **$65\frac5{11}$ minutes** (option C).