If you do not read this question carefully, the answer may be wrong. Look at this question carefully, what has been asked, the first Indian governor of independent India means a citizen of India, then the answer will be C. Rajagopalachari
If the same question was asked who was the first governor of independent India, then the answer would be Lord Mountbatten.
First Governor General of free India (however in territory status and not Republic) was Lord Mountbatten. He was the last Governor General and Viceroy of India. His residency was short, i.e., from February 1947 to June 1948. During his residency just parcel of India occurred. He was in India to wrap up the British domain from here and to leave the country in a methodical way. Indian Independence Act of 1947 was passed in British parliament to guarantee this change of force from Britishers to Indians.
After Lord Mountbatten, C Rajagopalachari was made the Governor General of Dominion of India in June 1948. He was the simply Indian to hold this post and furthermore the final remaining one. Lord Mountbatten was at this post till 26th January 1950. On 26th Jan 1950, India turned into a Republic and chose Dr. Rajendra Prasad as the principal President of Republic of India. Henceforth, the post of Governor General was not, at this point required and it was canceled all together.