How many times do the hands of a clock point in exactly opposite directions between 5 a.m. and 11 a.m.?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 5
The hands are opposite once in every hour except between 5 and 6, because the opposite position for that hour comes exactly at 6:00. Measured from 12, the minute hand gains 5.5° a minute on the hour hand.
At 5:00 the minute hand is 150° behind the hour hand. To be opposite it must gain \(150^\circ+180^\circ=330^\circ\), which takes \(\frac{330}{5.5}=60\) minutes
So the first time after 5 a.m. is 6:00 (hour hand on 6, minute hand on 12)
Then once in each of the hours 7–8, 8–9, 9–10 and 10–11 (at about 7:05, 8:11, 9:16 and 10:22)
Between 11 and 12 the next one comes at \(27\frac{3}{11}\) minutes past 11, after 11 a.m.
Number of times \(=1+4=5\)