At which one of the following times, do the hour hand and the minute hand of the clock make an angle of 180° with each other ?

(a) At 7:00 hours
(b) Between 7:00 hours and 7:05 hours
(c) At 7:05 hours
(d) Between 7:05 hours and 7:10 hours ✓ UPSC's answer

Why the answer is (d)

• At 7:00, the hour hand is at 210° and the minute hand is at 0°, creating an initial angle of 150°.

• The minute hand moves at 6° per minute, while the hour hand moves at 0.5° per minute, resulting in a relative speed of 5.5° per minute.

• To reach 180°, the minute hand must gain 30° on the hour hand (180° - 150°).

• The time required is calculated as 30° / 5.5° per minute, which equals 5 5/11 minutes.

• This places the exact time at 7:05 and 5/11 minutes, which falls between 7:05 and 7:10.

• Therefore, option (d) is the correct interval.

Why the other options are wrong

(a) At 7:00 hours: At 7:00, the angle between the hands is 150°, not 180°.
(b) Between 7:00 hours and 7:05 hours: Between 7:00 and 7:05, the angle increases from 150° to approximately 177.27°, never reaching 180°.
(c) At 7:05 hours: At exactly 7:05, the angle is approximately 177.27°, which is less than 180°.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2021, held on 10 October 2021. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.