Question

At what time between 9PM and 10PM minute hand and hour hand will be opposite to each other?

• A. 9:15\(\frac{4}{11}\)
• B. 9:16\(\frac{4}{11}\)
• C. 9:12\(\frac{4}{11}\)
• D. None of these

Answer 1

The Correct Option is B

To determine the time between 9 PM and 10 PM when the minute and hour hands of a clock are opposite each other, let's follow the logic of clock angles. Each hour, the hour hand moves \(30^\circ\) (since \(360^\circ / 12 = 30^\circ\)). Each minute, the hour hand moves \(0.5^\circ\) (since \(30^\circ/60 = 0.5^\circ\)). The minute hand moves \(6^\circ\) per minute (since \(360^\circ/60 = 6^\circ\)).
At 9 PM, the hour hand is at \(270^\circ\) (as \(9 \times 30^\circ = 270^\circ\)).
Let \(m\) be the number of minutes past 9 PM where the hands are \(180^\circ\) apart:
1. Position of the hour hand = \(270^\circ + 0.5m\).
2. Position of the minute hand = \(6m\).
3. Since they are opposite, their angles satisfy:
\[\begin{align}(6m - (270 + 0.5m)) &= 180\\6m - 270 - 0.5m &= 180\\5.5m &= 450\\m &= \frac{450}{5.5} = \frac{900}{11} = 81\frac{9}{11}\end{align}\]
Thus, the time is \(9:16\frac{4}{11}\). Therefore, the correct choice is 9:16\(\frac{4}{11}\).