Step 1: Understanding the Question.
This question asks for the exact date on which India stopped having a Governor General as its constitutional head and moved to a different form of headship.
Step 2: Key Fact.
On 15th August 1947, India became independent, but it did not immediately become a republic. It stayed a dominion within the British Commonwealth, with a Governor General as the formal head of state, representing the British Crown.
Step 3: Detailed Explanation.
Lord Mountbatten was the first Governor General of independent India, followed by C. Rajagopalachari, the only Indian to hold that post.
This dominion arrangement continued for about two and a half years while the Constituent Assembly finished drafting the Constitution of India.
On 26th January 1950, the new Constitution came into force, India became a sovereign republic, and the post of Governor General was abolished. Dr. Rajendra Prasad took over as the first President of India on that day.
15th August 1947 only marks independence, not the end of the Governor General's post, so option (A) is a common trap.
9th August 1948 and 2nd October 1950 do not correspond to any change in the head of state and are simply distractor dates.
Step 4: Final Answer.
The Governor General's rule ended on 26th January 1950, so option (C) is correct.