Which of the following pairs of forces cannot be added to give a resultant force of `4N`?

Options

A. `2N` and `8N`
B. `2N` and `2N`
C. `2N` and `6N`
D. `2N` and `4N`

Explanation

To determine which pairs of forces cannot be added to give a resultant force of 4 N, we need to analyze the maximum and minimum resultant forces that can be achieved with each pair. The resultant force depends on the angle between the two forces.

### Step-by-Step Solution:

1. **Identify the Forces**: Let's consider the first pair of forces: 2 N and 8 N.

2. **Calculate Maximum Resultant**: The maximum resultant occurs when the forces are in the same direction (0 degrees apart).
\[ R_{\text{max}} = F_1 + F_2 = 2 \, \text{N} + 8 \, \text{N} = 10 \, \text{N} \]

3. **Calculate Minimum Resultant**: The minimum resultant occurs when the forces are in opposite directions (180 degrees apart).
\[ R_{\text{min}} = |F_1 - F_2| = |8 \, \text{N} - 2 \, \text{N}| = 6 \, \text{N} \]

4. **Determine the Range of Resultants**: The resultant force can vary between the minimum and maximum values:
\[ R_{\text{range}} = [6 \, \text{N}, 10 \, \text{N}] \]

5. **Check if 4 N is in the Range**: The resultant force of 4 N does not fall within the range of 6 N to 10 N. Therefore, it is impossible to achieve a resultant of 4 N with the forces of 2 N and 8 N.

6. **Conclusion**: Since the resultant of 4 N is not within the possible range of resultant forces for the pair of 2 N and 8 N, this pair cannot be added to give a resultant force of 4 N.

### Final Answer:
The pair of forces (2 N and 8 N) cannot be added to give a resultant force of 4 N.