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?
Options:
(A) 64
6
11
(B) 66
5
13
(C) 65
5
11
(D) 66
6
13
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.7)**
**1. Kinematics of Clock Hands**:
- **Minute hand angular speed**: $\omega_m = \frac{360^\circ}{60\text{ min}} = 6^\circ/\text{min}$
- **Hour hand angular speed**: $\omega_h = \frac{360^\circ}{720\text{ min}} = 0.5^\circ/\text{min}$
- **Relative angular speed**:
$\omega_{\text{rel}} = \omega_m - \omega_h = 6^\circ - 0.5^\circ = 5.5^\circ/\text{min} = \frac{11}{2}^\circ/\text{min}$
**2. Step-by-Step Derivation**:
- Rohit enters when the hands coincide (shortly after 1 PM, at $1\text{ h } 5\frac{5}{11}\text{ min}$).
- He leaves when the hands coincide again about an hour later (shortly after 2 PM).
- Time elapsed between any two consecutive coincidences of clock hands:
$T = \frac{360^\circ}{\omega_{\text{rel}}} = \frac{360}{11/2} = \frac{720}{11} \text{ minutes}$
- Expressing $\frac{720}{11}$ as a mixed fraction:
$720 = 11 \times 65 + 5 \implies T = 65\frac{5}{11} \text{ minutes}$
**3. Official Verification**:
The verified official IIT answer is **Option (C)** (`65 5/11`).
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