Passage

Directions for the following 5 (five) items : Each item in this section contains a question followed by two statements. Answer each item using the following instructions and mark your response on the Answer Sheet accordingly. (a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement (b) Select this option if the question can be answered using either statement alone (c) Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone (d) Select this option if the question cannot be answered even using any of the statements

Question : If x and y are integers, then is x even? Statement I : x²y² is even. Statement II : 1 + x² + y² is odd.

(a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement
(b) Select this option if the question can be answered using either statement alone
(c) Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone ✓ UPSC's answer
(d) Select this option if the question cannot be answered even using any of the statements

Why the answer is (c)

• Statement I: x²y² even means xy is even, so at least one of x, y is even. x could be odd (x = 1, y = 2) or even. Not sufficient.

• Statement II: 1 + x² + y² odd means x² + y² is even, so x and y have the same parity (both odd or both even). Not sufficient.

• Together: same parity, and they cannot both be odd because then xy would be odd. So both are even, and x is even.

• Both statements are needed together: (c).

Why the other options are wrong

(a) Select this option if the question can be answered using one of these statements alone…: Neither statement alone decides the parity of x.
(b) Select this option if the question can be answered using either statement alone: Neither statement alone is sufficient.
(d) Select this option if the question cannot be answered even using any of the statements: Together the statements do force x to be even.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.