The correct answer is option 2, Article 75. Article 75 of the Indian Constitution empowers the President to appoint the Prime Minister of India. This article states that the President shall appoint the Prime Minister, who is usually the leader of the political party or coalition of parties that has a majority in the Lok Sabha, the lower house of parliament.
It is important to understand the roles and powers of the President and the Prime Minister in the Indian political system. The President is the head of state and holds a ceremonial position, while the Prime Minister is the head of government and holds the executive powers. The President acts on the advice of the Prime Minister and the Council of Ministers.
Options 1, 3, and 4 are not correct. Article 74 deals with the Council of Ministers and their aid and advice to the President. Article 76 deals with the Attorney General of India, and Article 77 deals with the conduct of government business and the duties of the Prime Minister.
Therefore, the correct article that empowers the President to appoint the Prime Minister of India is Article 75.