The correct answer to this question is option 4, which states that neither statement 1 nor statement 2 is correct.

To dissect each statement:

Statement 1 suggests that the Chief Secretary in a State is appointed by the Governor of that State. However, this assertion is incorrect. The Chief Secretary, a senior role in the Indian Administrative Services, is appointed by the Central Government, not the Governor of the respective state.

Statement 2 assumes that the Chief Secretary in a State has a fixed tenure. This statement is also incorrect. In reality, there is no fixed tenure for the post of Chief Secretary in a state. The Chief Secretary serves at the pleasure of the State Government of the day and their tenure can be extended or shortened depending on the will of the government.

Thus, neither of the statements is correct, making option 4 the correct choice for this particular question.