How many times are the hour hand and the minute hand of a clock of a right angles during their motion from 1 : 00 pm to 10 : 00 pm ?

Answer: Option E

Explanation in Detail:

To determine how many times the hour hand and minute hand of a clock form a right angle between 1:00 pm and 10:00 pm, we analyze their angular positions over this period.

1. Calculating Angular Movements:
- Minute Hand: The minute hand moves 360 degrees in 60 minutes, so in 9 hours (from 1:00 pm to 10:00 pm), it covers:
     $$360\text{ degrees/hour}\times 9\text{ hours}=3240\text{ degrees}$$
- Hour Hand: The hour hand moves 30 degrees in 60 minutes (or 0.5 degrees per minute), covering:
    $$30\text{ degrees/hour}\times 9\text{ hours}=270\text{ degrees}$$
2. Relative Angular Distance:
- The difference in their angular positions over 9 hours is:
    $$3240\text{ degrees (minute hand)}-270\text{ degrees (hour hand)}=2970\text{ degrees}$$
3. Calculating Right Angles:
- A right angle is formed every 180 degrees.
- Therefore, the number of times they form a right angle is:
     $$\frac{2970\text{ degrees}}{180\text{ degrees}}=16.5$$
Rounding down, they form a right angle 16 times.
4. Identifying Times of Right Angles:
- The first right angle occurs at 1:21.8181... pm.
- Subsequent right angles occur approximately every 32.7878... minutes until the last right angle at 9:32.7272... pm.
5. List of Times:
- The hour and minute hands form a right angle at the following times:
- 1:21.8181 pm
- 1:54.5454 pm
- 2:27.2727 pm
- 3:00 pm
- 3:32.7272 pm
- 4:05.4545 pm
- 4:38.1818 pm
- 5:10.9090 pm
- 5:43.6363 pm
- 6:16.3636 pm
- 6:49.0909 pm
- 7:21.8181 pm
- 7:54.5454 pm
- 8:27.2727 pm
- 9:00 pm
- 9:32.7272 pm
Therefore, the hour hand and minute hand of the clock form a right angle 16 times between 1:00 pm and 10:00 pm, as calculated based on their angular movements and the criteria for forming right angles.