Which pair of the following forces will never give resultant force of `2 N`?

Options

A. `2 N and 2 N`
B. `1 N and 1 N`
C. `1 N and 3 N`
D. `1 N and 4 N`

Explanation

To determine which pair of forces will never give a resultant force of 2 N, we need to analyze the minimum and maximum possible resultant forces that can be obtained from the given pairs of forces.

### Step-by-Step Solution:

1. **Understanding Resultant Forces**:
For two forces \( F_1 \) and \( F_2 \), the resultant force \( R \) can be calculated using:
- Maximum Resultant: \( R_{max} = |F_1| + |F_2| \)
- Minimum Resultant: \( R_{min} = ||F_1| - |F_2|| \)

2. **Analyzing Each Option**:
We will evaluate each pair of forces to see if a resultant of 2 N can be achieved.

- **Option A**: \( F_1 = 0 \, N \) and \( F_2 = 4 \, N \)
- Maximum: \( R_{max} = 0 + 4 = 4 \, N \)
- Minimum: \( R_{min} = |0 - 4| = 4 \, N \)
- Resultant can range from 4 N to 4 N. **(Resultant of 2 N is possible)**

- **Option B**: \( F_1 = 1 \, N \) and \( F_2 = 3 \, N \)
- Maximum: \( R_{max} = 1 + 3 = 4 \, N \)
- Minimum: \( R_{min} = |1 - 3| = 2 \, N \)
- Resultant can range from 2 N to 4 N. **(Resultant of 2 N is possible)**

- **Option C**: \( F_1 = 1 \, N \) and \( F_2 = 1 \, N \)
- Maximum: \( R_{max} = 1 + 1 = 2 \, N \)
- Minimum: \( R_{min} = |1 - 1| = 0 \, N \)
- Resultant can range from 0 N to 2 N. **(Resultant of 2 N is possible)**

- **Option D**: \( F_1 = 1 \, N \) and \( F_2 = 5 \, N \)
- Maximum: \( R_{max} = 1 + 5 = 6 \, N \)
- Minimum: \( R_{min} = |1 - 5| = 4 \, N \)
- Resultant can range from 4 N to 6 N. **(Resultant of 2 N is NOT possible)**

3. **Conclusion**:
The only pair of forces that will never give a resultant force of 2 N is **Option D** (1 N and 5 N).