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 pair of forces cannot be added to give a resultant force of 4 N, we will analyze each option based on the properties of vector addition.

### Step-by-Step Solution:

1. **Understanding Resultant Force**:
The resultant force from two forces can be calculated using the triangle law of vector addition. The resultant \( R \) must lie between the minimum and maximum values formed by the two forces, which is given by:
\[ |F_1 - F_2| \leq R \leq F_1 + F_2 \]
where \( F_1 \) and \( F_2 \) are the magnitudes of the two forces.

2. **Analyzing Option 1 (2 N and 8 N)**:
- Minimum force: \( |2 - 8| = 6 \) N
- Maximum force: \( 2 + 8 = 10 \) N
- Therefore, the range of possible resultant forces is \( 6 \leq R \leq 10 \).
- Since 4 N is not within this range, **Option 1 cannot give a resultant of 4 N**.

3. **Analyzing Option 2 (2 N and 2 N)**:
- Minimum force: \( |2 - 2| = 0 \) N
- Maximum force: \( 2 + 2 = 4 \) N
- The range of possible resultant forces is \( 0 \leq R \leq 4 \).
- Since 4 N is within this range, **Option 2 can give a resultant of 4 N**.

4. **Analyzing Option 3 (2 N and 6 N)**:
- Minimum force: \( |2 - 6| = 4 \) N
- Maximum force: \( 2 + 6 = 8 \) N
- The range of possible resultant forces is \( 4 \leq R \leq 8 \).
- Since 4 N is within this range, **Option 3 can give a resultant of 4 N**.

5. **Analyzing Option 4 (2 N and 4 N)**:
- Minimum force: \( |2 - 4| = 2 \) N
- Maximum force: \( 2 + 4 = 6 \) N
- The range of possible resultant forces is \( 2 \leq R \leq 6 \).
- Since 4 N is within this range, **Option 4 can give a resultant of 4 N**.

### Conclusion:
The only pair of forces that cannot be added to give a resultant force of 4 N is **Option 1 (2 N and 8 N)**.