Option 1 states that the Chairman and the Members of the Union Public Service Commission (UPSC) are appointed by the President, which is correct. The UPSC is a constitutional body responsible for conducting examinations and appointments to civil services and positions in the government of India. The Chairman and Members of the UPSC are indeed appointed by the President.

Option 2 states that the Chairman and the Members of the UPSC are eligible for further employment under the Government. This statement is incorrect. According to the UPSC (Exemption from Consultation) Regulations, 1958, once appointed as Chairman or Member of the UPSC, they are disqualified from future employment under the Government of India or the Government of a state.

Therefore, only statement 1 is correct, and statement 2 is incorrect.

The correct answer is "option 1 - 1 only".