Explanation:

Role of Governors in State Universities

• Statement 1 is correct: In most cases, the Governor of the state is the ex-officio chancellor of the universities in that state.
• Statement 2 is correct: While as Governor he functions with the aid and advice of the Council of Ministers, as Chancellor he acts independently of the Council of Ministers and takes his own decisions on all University matters.

Case of Central Universities

• Under the Central Universities Act, 2009, and other statutes, the President of India shall be the Visitor of a central university.
• With their role limited to presiding over convocations, Chancellors in central universities are titular heads, who are appointed by the President in his capacity as Visitor.

Constitutional Provisions Related to Governor

Governor has a dual role vis-s vis State Government

• He is the constitutional head of the state, bound by the advice of his council of ministers (CoM).
• He functions as a vital link between the Union Government and the State Government.

Article 153 says that there shall be a Governor for each State. One person can be appointed as Governor for two or more States. A Governor is appointed by the President and is a nominee of the Central Government.

• Article 163: There is a CoM with the CM at the head to aid and advise the Governor in the exercise of his functions, except some conditions for discretion.
• Article 200: Governor assents, withholds assent, or reserves the bill for the consideration of the President passed by the Legislative Assembly.
• Article 213: Governor may promulgate the Ordinances under certain circumstances.

Hence both statements are correct.