Consider the following statements : 1. Between 3:16 p.m. and 3:17 p.m., both hour hand and minute hand coincide. 2. Between 4:58 p.m. and 4:59 p.m., both minute hand and second hand coincide. Which of the above statements is/are correct?
- (a) 1 only
- (b) 2 only
- (c) Both 1 and 2 ✓ UPSC's answer
- (d) Neither 1 nor 2
Why the answer is (c)
• The minute hand moves at 6° per minute and the hour hand at 0.5° per minute, giving a relative speed of 5.5° per minute. At 3:00 the hour hand sits at 90° while the minute hand is at 0°, so the minute hand must gain 90° to overtake it, which takes 90 ÷ 5.5 = 180/11 ≈ 16.36 minutes, placing the coincidence at roughly 3:16:22, squarely between 3:16 and 3:17 p.m. Statement 1 is therefore correct.
• The second hand moves at 6° per second and the minute hand at 0.1° per second, giving a relative speed of 5.9° per second. At 4:58:00 the minute hand is at 58 × 6° = 348° and the second hand is at 0°, so the second hand must gain 348° to catch the minute hand.
• The time for that catch-up is 348 ÷ 5.9 = 3480/59 ≈ 58.98 seconds, meaning the two hands coincide at approximately 4:58:58.98, which falls between 4:58 and 4:59 p.m. Statement 2 is therefore correct.
• Since both statements are verified as true, the correct choice is (c) Both 1 and 2.
Why the other options are wrong
- (a) 1 only
- Option (a) selects only Statement 1, but Statement 2 is also correct because the second hand and minute hand do coincide at approximately 4:58:58.98, within the stated interval.
- (b) 2 only
- Option (b) selects only Statement 2, but Statement 1 is also correct because the hour and minute hands coincide at approximately 3:16:22, within the stated interval.
- (d) Neither 1 nor 2
- Option (d) claims neither statement is correct, yet both have been shown to hold: the hour-minute coincidence occurs at ~3:16:22 and the minute-second coincidence at ~4:58:58.98.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.