The correct answer is option 4: "A is false but R is true."

Option 1 states that both A and R are true and R is the correct explanation of A. This is incorrect because A (The British sovereignty continued to exist in free India) is false. After India gained independence from British rule in 1947, British sovereignty ceased to exist.

Option 2 states that both A and R are true but R is not the correct explanation of A. This is incorrect for the same reason as above – `A` is untrue.

Option 3 says that A is true but R is false. As mentioned, A is false, so this choice is incorrect.

Option 4 states that A is false but R is true. This is correct. As stated above, A is false. However, R (The British sovereign appointed the last Governor-General of free India) is true. The last Governor-general of free India, Lord Mountbatten, was indeed appointed by King George VI, the then British monarch.