GATE CE 2025 Set 1 — Question 7

The minute hand gains $55$ minutes on the hour hand in $60$ minutes.
So, the hands coincide every $60 + \frac{60}{55} = 60 + \frac{12}{11} = \frac{660+12}{11} = \frac{720}{11}$ minutes.Alternatively, the relative speed of the minute hand with respect to the hour hand is $5.5$ degrees per minute ($6$ degrees/min for minute hand, $0.5$ degrees/min for hour hand).The hands coincide when the angle between them is $0$ degrees. They coincide $11$ times in $12$ hours (or $12$ times if we count the coincidence at 12:00 twice for a 12-hour cycle, but typically it's $11$ distinct coincidences). The time between two consecutive coincidences is $\frac{360}{5.5 \times 11} = \frac{720}{11}$ minutes.Let's verify the time between coincidences. The hands coincide at 12:00. The next coincidence is between 1 and 2. At 1:00, the minute hand is at 12 and the hour hand is at 1. The minute hand needs to cover $30$ degrees to catch up. Time taken = $\frac{30}{5.5} = \frac{60}{11}$ minutes. So, the first coincidence after 12:00 is at $1$ hour and $\frac{60}{11}$ minutes, i.e., $1:5\frac{5}{11}$.The time interval between two consecutive coincidences of the hour and minute hands is $\frac{720}{11}$ minutes.Rohit enters the restaurant at about 1 PM and the hands are coinciding. This would be at $1$ hour and $\frac{60}{11}$ minutes past 12, i.e., $1:05\frac{5}{11}$ PM.He leaves after about an hour, and the hands are again coinciding. The next coincidence after $1:05\frac{5}{11}$ PM would be $\frac{720}{11}$ minutes later.Time spent at the restaurant = Time between two consecutive coincidences = $\frac{720}{11}$ minutes.$\frac{720}{11} = 65\frac{5}{11}$ minutes.The final answer is $\boxed{C}$