The magnitude of pairs of displacement vectors are give. Which pairs of displacement vectors cannot be added to give a resultant vector of magnitude 13 cm :-
`(i) 4 cm, 12 cm" "(ii) 4 cm, 8 cm`
`(iii) 6 cm, 8cm" "(iv) 1 cm, 15 cm`

Options

A. (ii, iv)
B. (i,ii)
C. (i, iii)
D. (ii, iii)

Answer

To determine which pairs of displacement vectors cannot be added to give a resultant vector of magnitude 13 cm, we will analyze each pair of vectors based on the properties of vector addition.

### Step-by-Step Solution:

1. **Understanding Vector Addition**:
- The resultant vector \( R \) from two vectors \( A \) and \( B \) can vary based on the angle between them. The maximum resultant occurs when the vectors are in the same direction (0 degrees), and the minimum resultant occurs when they are in opposite directions (180 degrees).
- The range of possible resultant magnitudes is given by:
\[ R_{\text{max}} = A + B \]
\[ R_{\text{min}} = |A - B| \]

2. **Analyzing Each Pair**:
- **Pair (i): 4 cm, 12 cm**
- \( R_{\text{max}} = 4 + 12 = 16 \, \text{cm} \)
- \( R_{\text{min}} = |4 - 12| = 8 \, \text{cm} \)
- The range is from 8 cm to 16 cm. Since 13 cm is within this range, this pair can give a resultant of 13 cm.

- **Pair (ii): 4 cm, 8 cm**
- \( R_{\text{max}} = 4 + 8 = 12 \, \text{cm} \)
- \( R_{\text{min}} = |4 - 8| = 4 \, \text{cm} \)
- The range is from 4 cm to 12 cm. Since 13 cm is outside this range, this pair cannot give a resultant of 13 cm.

- **Pair (iii): 6 cm, 8 cm**
- \( R_{\text{max}} = 6 + 8 = 14 \, \text{cm} \)
- \( R_{\text{min}} = |6 - 8| = 2 \, \text{cm} \)
- The range is from 2 cm to 14 cm. Since 13 cm is within this range, this pair can give a resultant of 13 cm.

- **Pair (iv): 1 cm, 15 cm**
- \( R_{\text{max}} = 1 + 15 = 16 \, \text{cm} \)
- \( R_{\text{min}} = |1 - 15| = 14 \, \text{cm} \)
- The range is from 14 cm to 16 cm. Since 13 cm is outside this range, this pair cannot give a resultant of 13 cm.

3. **Conclusion**:
- The pairs of displacement vectors that cannot be added to give a resultant vector of magnitude 13 cm are:
- (ii) 4 cm, 8 cm
- (iv) 1 cm, 15 cm

### Final Answer:
The pairs that cannot produce a resultant of 13 cm are (ii) 4 cm, 8 cm and (iv) 1 cm, 15 cm.