Which pair of the following forces will never give resultant force of `2 N`? (a) 5N & 3N (b) 4N & 10N

Answer

To solve the question, we need to determine which pair of forces will never yield a resultant force of 2 N. We will analyze both options step by step.

### Step 1: Understand the Range of Resultant Forces
The resultant force \( R \) of two forces \( F_1 \) and \( F_2 \) can be calculated using the following rules:
- The maximum resultant force occurs when the two forces are in the same direction:
\[ R_{\text{max}} = F_1 + F_2 \]
- The minimum resultant force occurs when the two forces are in opposite directions:
\[ R_{\text{min}} = |F_1 - F_2| \]
- The resultant force will always lie between \( R_{\text{min}} \) and \( R_{\text{max}} \).

### Step 2: Analyze Option (a): 5 N & 3 N
1. Calculate the maximum resultant:
\[ R_{\text{max}} = 5 \, \text{N} + 3 \, \text{N} = 8 \, \text{N} \]
2. Calculate the minimum resultant:
\[ R_{\text{min}} = |5 \, \text{N} - 3 \, \text{N}| = 2 \, \text{N} \]
3. The range of possible resultant forces for this pair is:
\[ 2 \, \text{N} \leq R \leq 8 \, \text{N} \]
Since 2 N is included in this range, this pair can yield a resultant of 2 N.

### Step 3: Analyze Option (b): 4 N & 10 N
1. Calculate the maximum resultant:
\[ R_{\text{max}} = 4 \, \text{N} + 10 \, \text{N} = 14 \, \text{N} \]
2. Calculate the minimum resultant:
\[ R_{\text{min}} = |4 \, \text{N} - 10 \, \text{N}| = 6 \, \text{N} \]
3. The range of possible resultant forces for this pair is:
\[ 6 \, \text{N} \leq R \leq 14 \, \text{N} \]
Since 2 N is not included in this range, this pair will never yield a resultant of 2 N.

### Conclusion
The pair of forces that will never give a resultant force of 2 N is:
**(b) 4 N & 10 N**