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.