Resultant of which of a following may be equal to zero?

Options

A. `10N ,10N ,10N`
B. `10N ,10N ,25N`
C. `10N ,10N ,35N`
D. None of these

Explanation

To determine which set of vectors can have a resultant equal to zero, we need to analyze the options provided based on the triangle inequality theorem. The triangle inequality states that for any three sides to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

### Step-by-step Solution:

1. **Identify the Options**: We have four options of vectors:
- Option 1: 10N, 10N, 10N
- Option 2: 10N, 10N, 25N
- Option 3: 10N, 10N, 35N
- Option 4: None of these

2. **Apply the Triangle Inequality**:
- For three vectors to have a resultant of zero, they must be able to form a closed triangle. We will check each option to see if they satisfy the triangle inequality.

3. **Check Option 1 (10N, 10N, 10N)**:
- Check: 10 + 10 > 10 (True)
- Check: 10 + 10 > 10 (True)
- Check: 10 + 10 > 10 (True)
- Since all conditions are satisfied, these vectors can form a triangle, and thus, the resultant can be zero.

4. **Check Option 2 (10N, 10N, 25N)**:
- Check: 10 + 10 > 25 (False)
- Since this condition is not satisfied, these vectors cannot form a triangle, and thus, the resultant cannot be zero.

5. **Check Option 3 (10N, 10N, 35N)**:
- Check: 10 + 10 > 35 (False)
- Since this condition is not satisfied, these vectors cannot form a triangle, and thus, the resultant cannot be zero.

6. **Check Option 4 (None of these)**:
- Since we found that Option 1 can have a resultant of zero, this option is incorrect.

7. **Conclusion**: The only option where the resultant can be equal to zero is **Option 1: 10N, 10N, 10N**.

### Final Answer:
The resultant of the vectors in **Option 1 (10N, 10N, 10N)** may be equal to zero.

---