= Solution
Within the stated small-volume-change approximation, treat $V,\kappa_T,\alpha$ in the differential constitutive law as constants. Then
$$
\delta W=V_1\kappa_T P\,dP-V_1\alpha P\,dT.
$$
Path (a) first changes <temperature> at <pressure> $P_1$, then changes <pressure> at fixed <temperature> $T_2$. The two contributions are $-V_1\alpha P_1(T_2-T_1)$ and $V_1\kappa_T(P_2^2-P_1^2)/2$. Therefore
$$
\boxed{W_a=\frac12V_1\kappa_T(P_2^2-P_1^2)-V_1\alpha P_1(T_2-T_1).}
$$
Work is path dependent even though the endpoints and approximate total <volume> change are the same.
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