In 12 hours how many times a minute-hand and hour-hand of a clock makes `90^(@)` between them or becomes perpendicular to each other?

Options

A. A. 2
B. B. 4
C. C. 3
D. D. 1

Answer

To solve the problem of how many times the minute hand and hour hand of a clock become perpendicular (90 degrees apart) in 12 hours, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Speeds of the Hands:**
- The minute hand moves at a speed of 6 degrees per minute.
- The hour hand moves at a speed of 0.5 degrees per minute.

2. **Calculate the Relative Speed:**
- The relative speed between the minute hand and the hour hand is calculated as:
\[ \text{Relative Speed} = \text{Speed of Minute Hand} - \text{Speed of Hour Hand} = 6 - 0.5 = 5.5 \text{ degrees per minute} \]

3. **Convert Time from Hours to Minutes:**
- We need to consider the total time in minutes for 12 hours:
\[ \text{Total Time} = 12 \text{ hours} \times 60 \text{ minutes/hour} = 720 \text{ minutes} \]

4. **Calculate the Total Distance Covered:**
- The total distance covered by the relative speed in 720 minutes is:
\[ \text{Total Distance} = \text{Relative Speed} \times \text{Total Time} = 5.5 \text{ degrees/minute} \times 720 \text{ minutes} = 3960 \text{ degrees} \]

5. **Determine the Number of Circles Made:**
- Since one complete circle is 360 degrees, we can find the number of complete circles made by dividing the total distance by 360 degrees:
\[ \text{Number of Circles} = \frac{3960 \text{ degrees}}{360 \text{ degrees/circle}} = 11 \text{ circles} \]

6. **Calculate the Number of Times the Hands are Perpendicular:**
- In one complete circle, the minute hand and hour hand are perpendicular to each other 2 times. Therefore, in 11 circles, they will be perpendicular:
\[ \text{Total Perpendicular Instances} = 11 \text{ circles} \times 2 = 22 \text{ times} \]

### Final Answer:
The minute hand and hour hand of a clock become perpendicular to each other **22 times** in 12 hours.

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