GATE 2024 CH – Question 5
On a given day, how many times will the second-hand and the minute-hand of a clock cross each other during the clock time 12:05:00 hours to 12:55:00 hours?
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Correct answer: (C) 50
Explanation
The second hand turns $360°$ per minute and the minute hand $6°$ per minute, so the second hand gains $354°$ per minute. At 12:05:00 the minute hand is at $30°$ and the second hand at $0°$, so the first crossing needs a gain of $30°$ and each later crossing another $360°$. In 50 minutes the gain is $354 \times 50 = 17700°$. The crossings occur at the gains $30 + 360k \le 17700$, so $k \le 49.08$ and $k = 0, 1, \ldots, 49$, which is 50 crossings.