Discussion
How many times are the hands of a clock perpendicular in a day ?
Answer: C
The hands are perpendicular 44 times in a day.
The minute hand gains 330° on the hour hand every hour (360° - 30°), so it gains a full circle on it 11 times in 12 hours. In each such lap, the hands are at right angles twice: once when the minute hand is 90° ahead and once when it is 270° ahead (90° behind).
\[ \begin{aligned} \text{in 12 h} &= 11 \times 2 = 22 \\ \text{in 24 h} &= 2 \times 22 = 44 \end{aligned} \]
22 is the count for 12 hours only. 24 and 48 come from assuming two right angles every hour, which ignores that the hands meet only 11 times, not 12, in 12 hours.
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