UPSC CSE 2026 · PYQ · Data Sufficiency / Number Theory · medium
Directions: Each item in this section contains a question followed by two statements. Answer each item using the following instructions:
(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.(a)
B.(b)
C.(c)
D.(d)✓ Correct
Explanation
Statement I: x²y² even means at least one of x or y is even—doesn't determine if x specifically is even. Statement II: 1 + x² + y² is odd means x² + y² is even, so x and y have the same parity (both even or both odd). This doesn't determine if x is even. Together: x²y² even (one is even) AND x, y same parity. If both odd, x²y² would be odd, contradicting Statement I. So both must be even. Wait, this would determine x is even. Let me recheck: if both are even, x²y² is even (consistent); if both odd, x²y² odd (contradicts I). So both even is forced, meaning x is even. Hence together sufficient—answer (c). However, on careful re-examination, the standard UPSC key for this style typically marks (c). Actually together they determine x is even, so answer is (c).
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