a) In a frame of reference `S_1`( though the net force is zero, the net acceleration is not zero.
b) In a frame of reference `S_2` though the net force is not zero, the net acceleration is zero.
c) In a frame of reference`S_3` the net acceleration is zero whenever the net force is zero.

Options

A. `S_(1)` and `S_3` are inertial and `S_2` is non-inertial
B. `S_(1)` and `S_2` are non-inertial and `S_3` is inertial
C. `S_(1), S_(2), S_(3)` are non-inertial
D. `S_(1), S_(2), S_(3)` are inertial

Answer

To solve the problem, we need to analyze the statements regarding the frames of reference \( S_1 \), \( S_2 \), and \( S_3 \) based on the definitions of inertial and non-inertial frames.

### Step-by-Step Solution:

1. **Understanding Inertial and Non-Inertial Frames**:
- Inertial frames are those where Newton's laws of motion hold true. This means that if the net force acting on an object is zero, then the acceleration of that object must also be zero.
- Non-inertial frames are those where Newton's laws do not hold in their standard form. This can occur when there is acceleration present in the frame itself or when fictitious forces are acting.

2. **Analyzing Frame \( S_1 \)**:
- The statement says that in frame \( S_1 \), the net force is zero, but the net acceleration is not zero.
- According to the definition of inertial frames, if the net force is zero, the acceleration must also be zero. Since this condition is violated (net acceleration is not zero), frame \( S_1 \) is a **non-inertial frame**.

3. **Analyzing Frame \( S_2 \)**:
- The statement for frame \( S_2 \) indicates that the net force is not zero, but the net acceleration is zero.
- If the net force is not zero, then according to Newton's second law (\( F = ma \)), the acceleration must also be non-zero. Since this condition is also violated (net acceleration is zero), frame \( S_2 \) is also a **non-inertial frame**.

4. **Analyzing Frame \( S_3 \)**:
- The statement for frame \( S_3 \) mentions that the net acceleration is zero whenever the net force is zero.
- This is consistent with the definition of an inertial frame. If the net force is zero, then the acceleration is indeed zero, which means that frame \( S_3 \) is an **inertial frame**.

5. **Conclusion**:
- Based on the analysis:
- \( S_1 \) is non-inertial.
- \( S_2 \) is non-inertial.
- \( S_3 \) is inertial.
- Therefore, the correct option is that \( S_1 \) and \( S_2 \) are non-inertial frames, while \( S_3 \) is an inertial frame.

### Final Answer:
The correct option is **Option 2**: \( S_1 \) and \( S_2 \) are non-inertial, and \( S_3 \) is inertial.