At what time between 9PM and 10PM minute hand and hour hand will be opposite to each other?
The hour hand and the minute hands will be opposite to each other when the angle between them=Ā $$180^{\circ\ }$$.
At 9PM, angle between the two hands= 270$$^{\circ\ }$$
The minute hand covers 360$$^{\circ\ }$$ in 60 minutes. So, in 1 minute, it coversĀ $$\ \frac{\ 360^{\circ\ }}{60}$$= 6$$^{\circ\ }$$
Similarly for the hour hand, it coversĀ 360$$^{\circ\ }$$ in 12 hours or 720 minutes.Ā
.'. The hour hand coversĀ $$\ \frac{\ 360^{\circ\ }}{720}$$= 0.5$$^{\circ\ }$$.
Using the concept of relative speed, since both the hands move in the same directions, the effective speed= (6-0.5)$$^{\circ\ }$$ =5.5 orĀ $$\ \frac{\ 11}{2}$$ degreesĀ per minute.
Effective distance to cover=Ā $$270^{\circ}-180^{\circ\ }=\ 90^{\circ\ }$$
Therefore time taken=Ā $$\ \frac{\ 90}{\ \frac{\ 11}{2}}=\ \ \frac{\ 180}{11}=\ 16\ \frac{\ 4}{11}$$ minutes after 9 PM.Ā
Option B is our answer.
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