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The correct answer is option 4 - 2 only. Let`s dissect each statement to understand why this is the correct answer.
Statement 1 states that the Union Executive consists of the President and the Council of Ministers with the Prime Minister as the head. This statement is incorrect because the Union Executive consists of the President, Vice-President, and the Council of Ministers with the Prime Minister as the head. So option 1 is incorrect.
Statement 2 states that the President may resign his office by writing under his hand-addressed to the Vice-President. This statement is correct. The President has the right to resign by writing a letter to the Vice-President. So option 2 is correct.
Statement 3 states that the executive power of the Union is vested in the Prime Minister. This statement is incorrect. The executive power of the Union is vested in the President and it is exercised by the Council of Ministers with the Prime Minister as the head. So option 3 is incorrect.
Based on the analysis of each statement, we can conclude that only statement 2 is correct. Therefore, the correct answer is option 4 - 2 only.