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GATE 2022 AR — Planning – Question 10

General Aptitude · Quantitative Aptitude: Data Interpretation, Arithmetic and Number Computation · 2 marks · MTA

In a 12-hour clock that runs correctly, how many times do the second, minute, and hour hands of the clock coincide, in a 12-hour duration from 3 PM in a day to 3 AM the next day?

  1. 11
  2. 12
  3. 144
  4. 2

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Correct answer: MTA

Explanation

The final key awards **marks to all**; retain that scoring. For an ideal continuously moving clock, the actual triple-coincidence count in the stated interval is one.

Let t be minutes after a 12 o'clock triple coincidence. Minute and hour hands meet when (6−0.5)t=360k, so t=720k/11. Second and minute hands meet when (360−6)t=360j, so t=60j/59.

Equating gives 708k=11j. Since 708 and 11 are coprime, k is a multiple of 11; all three hands coincide every 720 minutes, namely every twelve hours.

From 3 PM to 3 AM, only midnight is such a time. One is absent from the options, explaining why no particular listed answer is required.

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