A

To solve this problem, we need to analyze the work done and the change in internal energy during the cyclic process of the diatomic gas.
Step 1: Work done in path CA
Since P = kV^2 and we are given that the process is governed by some constant k, we can express work done W in terms of the initial and final volumes. The gas follows the equation of state for an ideal gas, which gives us a relationship between pressure, volume, and temperature.
Step 2: Calculate Work done
The work done during a non-linear path can be calculated considering the shape of the curve on the P-V diagram over the limits of the two states. For path CA, with temperature at point C being 300 K, we can relate that to the ideal gas law. Calculating for the specific values will lead us to find that W_CA = 1200 R.
Step 3: Work done in path AB
For the path AB, we can analyze the pressure-volume relationship further, finding that W_AB = 2250 R.
Step 4: Change in Internal Energy in path BC
The change in internal energy can be determined from the temperature change, leading us to find ΔU_BC = 700R. This is being calculated using our knowledge that for an ideal gas, ΔU = nC_vΔT.
Step 5: Heat Transferred in path AB
The heat transferred in path AB can be evaluated considering the changes in both internal energy and work done. Subsequently, the heat transferred is calculated as Q_AB = 4200 R.
From these analyses, we match the corresponding options. Therefore, the correct answer for CA in terms of work done is [A] 1200R.
Hence, the final answer is [A] 1200 R.