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The correct answer is option 4 - "2 only". Let`s analyze each statement:
1. The Union Executive consists of the President and the Council of Ministers with the Prime Minister as the head.
Statement 1 is incorrect because the Union Executive consists of the President, the Vice-President, and the Council of Ministers with the Prime Minister as the head.
2. The President may, by writing under his hand addressed to the Vice-President, resign his office.
Statement 2 is correct. According to Article 56 of the Indian Constitution, the President has the power to resign by writing under his hand, addressed to the Vice-President.
3. Executive power of the Union is vested in the Prime Minister.
Statement 3 is incorrect. The executive power of the Union is vested in the President of India, not the Prime Minister. The President is the head of the Union Executive, while the Prime Minister is the head of the Council of Ministers.
Therefore, only statement 2 is correct.