Statement 1 is incorrect. The Constitution does not contain any specific procedure for the selection and appointment of the Prime Minister. Article 75 says only that the Prime Minister shall be appointed by the president.

Statement 2 is correct.  In other words, the President can act on his discretion (that is, without the advice of the ministers) under the following situations:

1. Appointment of Prime Minister when no party has a clear majority in the Lok Sabha or when the Prime Minister in office dies suddenly and there is no obvious successor.
2. Dismissal of the council of ministers when it cannot prove the confidence of the Lok Sabha.
3. Dissolution of the Lok Sabha if the council of ministers has lost its majority.

Source) Indian Polity by M. Laxmikanth; chapter 19

Statement 1 is correct. In 1980, the Delhi High Court held that the Constitution does not require that a person must prove his majority in the Lok Sabha before he is appointed as the Prime Minister.

Statement 2 is incorrect. The President may first appoint him the Prime Minister and then ask him to prove his majority in the Lok Sabha within a reasonable period. However, there is no mention of this in the Constitution. Article 75 says only that the Prime Minister shall be appointed by the President.  

Source) Indian Polity by M. Laxmikanth; Chapter 19

A no-confidence motion can be moved by any member of the House. It can be moved only in the Lok Sabha and not Rajya Sabha. A mere moving of a no-confidence motion will not dissolve the council of ministers. If the government is not able to prove its majority in the House, then the government of the day has to resign. Hence, statement 1 is incorrect.

Statements 2 and 3 are correct. The resignation or death of an incumbent Prime Minister automatically dissolves the council of ministers and thereby generates a vacuum. The resignation or death of any other minister, on the other hand, merely creates a vacancy which the Prime Minister may or may not like to fill.

Source) Indian Polity by M. Laxmikanth; chapter 19

Statement 1 is correct. Article 74 provides for a council of ministers with the Prime Minister at the head to aid and advise the President in the exercise of his functions. The 42^nd and 44^th Constitutional Amendment Acts have made the advice binding on the President.

Statement 2 is incorrect. According to Article 88, every minister shall have the right to speak and take part in the proceedings of either House, any joint sitting of the Houses and any Committee of Parliament of which he may be named a member. But he shall not be entitled to vote.

Statement 3 is incorrect. In 1971, the Supreme Court held that ‘even after the dissolution of the Lok Sabha, the council of ministers does not cease to hold office. Article 74 is mandatory and, therefore, the president cannot exercise the executive power without the aid and advise of the council of ministers. Any exercise of executive power without the aid and advice will be unconstitutional as being violative of Article 74’.

Source) Indian Polity by M. Laxmikanth; Chapter

As per article 75, the council of ministers shall be collectively responsible to the Lok Sabha. This provision is the foundation of the parliamentary system of government.

Source) Indian Polity by M. Laxmikanth; Chapter

Statement 1 is correct. A member of either House of state legislature belonging to any political party who is disqualified on the ground of defection shall also be disqualified to be appointed as a minister. This provision was added by the 91st Amendment Act of 2003.

Statement 2 is incorrect. The term ‘Inner Cabinet’ or ‘Kitchen Cabinet’ is a informal body consists of the Prime Minister and two to four influential colleagues in whom he has faith and with whom he can discuss every problem. It advises the Prime Minister on important political and administrative issues and assists him in making crucial decisions. It is composed of not only cabinet ministers but also outsiders like friends and family members of the Prime Minister.

Source) Indian Polity by M. Laxmikanth; Chapter 19

Statement 1 is incorrect. The Cabinet Committees are extra-constitutional in emergence. They are not mentioned in the Constitution. However, the Rules of Business provide for their establishment.

The Cabinet Committees are of two types-standing and ad hoc. The former are of a permanent nature while the latter are of a temporary nature. The ad hoc committees are constituted from time to time to deal with special problems. They are disbanded after their task is completed.

Statement 2 is incorrect. The membership of the Cabinet Committees varies from three to eight. They usually include only Cabinet Ministers. However, the non-cabinet Ministers are not debarred from their membership. They not only include the Ministers in charge of subjects covered by them but also include other senior Ministers.

Source) Indian Polity by M. Laxmikanth; Chapter 21

The following four are the more important cabinet committees:

1. The Political Affairs Committee deals with all policy matters pertaining to domestic and foreign affairs.Hence, pair 3 is correctly matched.
2. The Economic Affairs Committee directs and coordinates the governmental activities in the economic sphere. Hence, pair 2 is correctly matched.
3. Appointments Committee decides all higher level appointments in the Central Secretariat, Public Enterprises, Banks and Financial Institutions.Hence, pair 1 is incorrectly matched.
4. Parliamentary Affairs Committee looks after the progress of government business in the Parliament.

Source) Indian Polity by M. Laxmikanth; chapter 21