At what time between 2 o clock and 3 o clock will the hands of a clock point in opposite directions ?

Options

A. `480/11` min past 2
B. `485/11` min past 2
C. `580/11` min past 2
D. `520/11` min past 2

Answer

To find the time between 2 o'clock and 3 o'clock when the hands of a clock point in opposite directions, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Clock Positions**:
- At 2 o'clock, the hour hand is at the 2 position on the clock. The minute hand is at the 12 position.
- The angle between each hour mark on the clock is 30 degrees (360 degrees/12 hours).

2. **Position of the Hour Hand**:
- At 2 o'clock, the hour hand is at 2, which is \(2 \times 30 = 60\) degrees from the 12 o'clock position.

3. **Position of the Minute Hand**:
- Let the number of minutes past 2 o'clock be \(x\).
- The minute hand moves at 6 degrees per minute (360 degrees/60 minutes).
- Therefore, after \(x\) minutes, the minute hand will be at \(6x\) degrees from the 12 o'clock position.

4. **Finding Opposite Directions**:
- For the hands to be in opposite directions, the angle between them must be 180 degrees.
- The angle between the hour hand and the minute hand can be calculated as:
\[ |(60 + 0.5x) - 6x| = 180 \]
- Here, \(0.5x\) is the additional movement of the hour hand due to \(x\) minutes (since the hour hand moves 0.5 degrees per minute).

5. **Setting Up the Equation**:
- This gives us two scenarios:
1. \(60 + 0.5x - 6x = 180\)
2. \(6x - (60 + 0.5x) = 180\)

6. **Solving the First Scenario**:
- Simplifying the first equation:
\[ 60 - 5.5x = 180 \]
\[ -5.5x = 180 - 60 \]
\[ -5.5x = 120 \]
\[ x = \frac{120}{-5.5} \approx -21.82 \text{ (not valid)} \]

7. **Solving the Second Scenario**:
- Simplifying the second equation:
\[ 6x - 60 - 0.5x = 180 \]
\[ 5.5x - 60 = 180 \]
\[ 5.5x = 240 \]
\[ x = \frac{240}{5.5} \approx 43.64 \]

8. **Final Time Calculation**:
- Therefore, the time when the hands are in opposite directions is approximately 43.64 minutes past 2 o'clock.
- This can be rounded to approximately 43 minutes and 38 seconds.

### Conclusion:
The hands of the clock will point in opposite directions at approximately **2:43 and 38 seconds**.