A particle is taken from point A to point B under the influence of a force field. Now it is taken back from B to A and it is observed that the work done in taking the particle from A to B is not equal to the work done in taking it from B to A. If `W_(nc)` and `W_c` are the work done by non-conservative and conservative forces present in the system, respectively, `DeltaU` is the change in potential energy and `Deltak` is the change in kinetic energy, then

Options

A. (a) `W_(nc)-DeltaU=Deltak`
B. (b) `W_c=-DeltaU`
C. (c) `W_(nc)+W_(c)=Deltak`
D. (d) `W_(nc)-DeltaU=-Deltak`

Explanation

To solve the problem, we need to analyze the situation based on the principles of work, energy, and the characteristics of conservative and non-conservative forces.

### Step-by-Step Solution:

1. **Understanding the Work-Energy Theorem**:
The work-energy theorem states that the total work done on an object is equal to the change in its kinetic energy. Mathematically, this can be expressed as:
\[ W_{total} = \Delta K \]
where \(W_{total}\) is the total work done by all forces acting on the particle, and \(\Delta K\) is the change in kinetic energy.

**Hint**: Recall that the total work done includes contributions from both conservative and non-conservative forces.

2. **Separating Work Done by Forces**:
The total work done can be separated into work done by conservative forces (\(W_c\)) and work done by non-conservative forces (\(W_{nc}\)):
\[ W_{total} = W_c + W_{nc} \]
Therefore, we can rewrite the work-energy theorem as:
\[ W_c + W_{nc} = \Delta K \]

**Hint**: Identify the types of forces acting on the particle and how they contribute to the work done.

3. **Work Done by Conservative Forces**:
For conservative forces, the work done is related to the change in potential energy (\(U\)). The work done by conservative forces can be expressed as:
\[ W_c = -\Delta U \]
where \(\Delta U\) is the change in potential energy.

**Hint**: Remember that the work done by conservative forces is path-independent and depends only on the initial and final positions.

4. **Combining the Equations**:
Substituting the expression for \(W_c\) into the work-energy theorem gives us:
\[ -\Delta U + W_{nc} = \Delta K \]
Rearranging this, we find:
\[ W_{nc} = \Delta K + \Delta U \]

**Hint**: This equation shows how non-conservative work relates to changes in kinetic and potential energy.

5. **Conclusion**:
Based on the analysis, we can conclude that:
- The work done by conservative forces is equal to the negative change in potential energy.
- The total work done includes contributions from both conservative and non-conservative forces, leading to changes in kinetic energy.

Therefore, the correct statements based on the relationships derived are:
- \(W_c = -\Delta U\)
- \(W_{nc} + W_c = \Delta K\)
- \(W_{nc} = \Delta K + \Delta U\)

The correct options are A, B, and C.

### Final Answer:
The correct options are A, B, and C.