Step 1: Understand the question.
We need the first Governor-General of independent India.
Step 2: The setting.
After 15 August 1947 India was a Dominion until it became a Republic on 26 January 1950. During this time it had a Governor-General.
Step 3: Identify the person.
Lord Louis Mountbatten, the last Viceroy, became the first Governor-General of independent India and served until June 1948.
Step 4: Rule out others.
C. Rajagopalachari was the first Indian Governor-General, after Mountbatten. Warren Hastings was Governor-General of Bengal. Rajendra Prasad became the first President.
Step 5: Conclude.
The first Governor-General of independent India was Lord Mountbatten.
Answer: Lord Louis Mountbatten