GATE 2020 EC GA Q.7: solution and answer

Official answer: BSource: Official answer key, GATE 2020 EC (IIT Delhi)

1. Quarter past three means the time is 3:15, so $H=3$ hours and $M=15$ minutes.

2. Minute hand: it turns $360^{\circ }$ in 60 minutes, which is $6^{\circ }$ per minute. At 15 minutes past the hour it is $15\times 6^{\circ }=90^{\circ }$ clockwise from 12, pointing exactly at the 3.

3. Hour hand: it turns $30^{\circ }$ per hour, which is $0.5^{\circ }$ per minute. At 3:15 it is $3\times 30^{\circ }+15\times 0.5^{\circ }=90^{\circ }+7.5^{\circ }=97.5^{\circ }$ clockwise from 12, a quarter of the way from the 3 towards the 4.

4. The angle between the hands is the difference: $97.5^{\circ }-90^{\circ }=7.5^{\circ }$.

5. Check with the clock-angle formula: $$\theta =\left|30H-\frac{11}{2}M\right|=\left|30(3)-\frac{11}{2}(15)\right|=\left|90-82.5\right|=7.5^{\circ }$$ So the answer is (B).