A particle of mass $m$ goes counterclockwise around a square path ABCDA with vertices at $A(-1,-1), B(1,-1), C(1,1)$ and $D(-1,1)$ in a force, $\vec{F}=\alpha x \sin (k x) \hat{x}$. Here, $\alpha$ and $k$ are positive constants. Consider the work done along the following paths:
(i) $A \rightarrow B$
(ii) $B \rightarrow C$
(iii) $A \rightarrow B \rightarrow C \rightarrow D$
(iv) $A \rightarrow B \rightarrow C \rightarrow D \rightarrow A$

The work done is zero for

• A

  paths (i) and (ii) only.
• B

  paths (i), (iii) and (iv) only.
• Correct answer
  C

  paths (ii), (iii) and (iv) only.
• D

  paths (ii) and (iv) only.

Hence, the correct answer is option: 'C'.