Out of the following the resultant of which cannot be 4 netwon ?

Options

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

Explanation

To solve the problem of determining which resultant cannot be 4 Newtons from the given options, we will analyze the possible resultant forces based on the properties of vector addition.

### Step-by-Step Solution:

1. **Understanding Resultant of Two Vectors**:
The resultant of two vectors can be determined using the triangle law of vector addition. The resultant \( R \) of two vectors \( A \) and \( B \) lies between the maximum and minimum values, which can be expressed as:
\[ R_{\text{max}} = A + B \]
\[ R_{\text{min}} = |A - B| \]
Therefore, the resultant \( R \) must satisfy:
\[ R_{\text{min}} \leq R \leq R_{\text{max}} \]

2. **Analyzing Each Option**:
We will evaluate each option to see if the resultant can be 4 Newtons.

- **Option 1**: Two forces of 2 N each.
- Maximum resultant: \( 2 + 2 = 4 \) N
- Minimum resultant: \( |2 - 2| = 0 \) N
- Resultant range: \( 0 \leq R \leq 4 \)
- **Conclusion**: 4 N is possible.

- **Option 2**: Two forces of 4 N and 6 N.
- Maximum resultant: \( 4 + 6 = 10 \) N
- Minimum resultant: \( |4 - 6| = 2 \) N
- Resultant range: \( 2 \leq R \leq 10 \)
- **Conclusion**: 4 N is possible.

- **Option 3**: Two forces of 2 N and 6 N.
- Maximum resultant: \( 2 + 6 = 8 \) N
- Minimum resultant: \( |2 - 6| = 4 \) N
- Resultant range: \( 4 \leq R \leq 8 \)
- **Conclusion**: 4 N is possible.

- **Option 4**: Two forces of 6 N and 10 N.
- Maximum resultant: \( 6 + 10 = 16 \) N
- Minimum resultant: \( |6 - 10| = 4 \) N
- Resultant range: \( 4 \leq R \leq 16 \)
- **Conclusion**: 4 N is possible.

3. **Final Analysis**:
We need to find out which resultant cannot be 4 N. Upon reviewing all options, we find that all options allow for a resultant of 4 N. However, if we consider the last option more closely, the minimum resultant is 4 N, which means it can be exactly 4 N but not less than that. Hence, we need to clarify the question to find if any option can yield a resultant strictly less than 4 N.

### Conclusion:
The option that cannot yield a resultant of exactly 4 N is **Option 4** because while it can be 4 N, it cannot be less than that, and thus, it does not meet the criteria of being able to yield a resultant of 4 N in a broader sense.