1. Identify which quantities are path functions and which are state functions

   • Heat $q$ is a path function.
   • Work $w$ is a path function.
   • Their sum $q+w$ equals the change in internal energy: $q+w=\Delta U$ Since $\Delta U$ depends only on initial and final states, it is a state function, not a path function.
   • For $H-TS$: $G=H-TS$ where $H$ (enthalpy), $T$ (temperature), and $S$ (entropy) are state variables, so $H-TS$ is also a state function.

2. Evaluate the given items

   • (A) $q+w$ $\rightarrow$ state function
   • (B) $q$ $\rightarrow$ path function
   • (C) $w$ $\rightarrow$ path function
   • (D) $H-TS$ $\rightarrow$ state function

3. Select the correct set

   The path functions are only (B) and (C).

4. Match with options

   • Option A: $(B)$ and $(C)$ ✔
   • Option B: $(A)$ and $(D)$ ✘
   • Option C: $(B), (C)$ and $(D)$ ✘
   • Option D: $(A), (B)$ and $(C)$ ✘

Therefore, the correct answer is Option A.