In 12 hours how many times the two hands of clock will be just opposite to each other i.e., they make a straight line having the difference of `180^(@)` between them?

Answer

To solve the problem of how many times the two hands of a clock will be opposite to each other (i.e., making a 180-degree angle) in 12 hours, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Clock Movement**:
- The minute hand completes a full circle (360 degrees) in 60 minutes, moving at a speed of 6 degrees per minute (360 degrees / 60 minutes).
- The hour hand completes a full circle in 12 hours, moving at a speed of 0.5 degrees per minute (360 degrees / 720 minutes).

2. **Finding the Angle Between the Hands**:
- The angle \( \theta \) between the hour hand and the minute hand at any time can be calculated using the formula:
\[ \theta = |30H - 5.5M| \]
where \( H \) is the hour and \( M \) is the minutes past the hour.

3. **Condition for Opposite Position**:
- The hands are opposite to each other when \( \theta = 180 \) degrees. Thus, we set up the equation:
\[ |30H - 5.5M| = 180 \]

4. **Solving the Equation**:
- This equation can yield two scenarios:
1. \( 30H - 5.5M = 180 \)
2. \( 30H - 5.5M = -180 \)

5. **Finding Values of H and M**:
- For each hour \( H \) from 0 to 11 (representing 12 hours), we can solve for \( M \) in both equations.
- For \( H = 0 \) to \( H = 11 \):
- Solve \( 30H - 5.5M = 180 \) and \( 30H - 5.5M = -180 \) to find valid minute values \( M \) that are between 0 and 59.

6. **Counting Valid Occurrences**:
- After solving the equations for each hour, we find that in most hours, the hands will be opposite once. However, between 5 and 6 o'clock and between 6 and 7 o'clock, they will only be opposite once instead of twice due to the overlap of the hour hand's movement.
- Therefore, we find that in 12 hours, the hands will be opposite **11 times**.

### Final Answer:
The two hands of the clock will be opposite to each other **11 times** in 12 hours.