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Option 1 states that the resignation of the Prime Minister of India means the resignation of the whole Union Cabinet. This statement is correct. When the Prime Minister resigns, it leads to the dissolution of the entire Union Cabinet.
Option 2 states that the President of India can dismiss the Prime Minister on the grounds of bad performance even when he enjoys the support of the majority in the Parliament. This statement is incorrect. The President does not have the power to dismiss the Prime Minister on his own. The Prime Minister can only be removed if he or she loses the majority support in the Parliament or if the Prime Minister voluntarily resigns.
Option 3 states that it is on the advice of the Prime Minister that the President of India summons and prorogues the Parliament and dissolves the Lok Sabha. This statement is correct. The Prime Minister advises the President on matters related to the functioning of the Parliament, such as summoning and proroguing sessions and dissolving the Lok Sabha.
In summary, the correct statements are 1 and 3, so the correct answer is option 3.