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.
Question: If x and y are integers, then is x even?
- Statement I: x²y² is even.
- Statement II: 1 + x² + y² is odd.
Answer & explanation
Answer: (c) Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone
Statement I says at least one of x and y is even; Statement II says x and y have the same parity. Only together do they force both to be even, which answers the question.
- Statement I: x²y² = (xy)² is even, so xy is even, so at least one of x, y is even. x could be odd (x = 1, y = 2). Not sufficient.
- Statement II: 1 + x² + y² is odd, so x² + y² is even, so x and y are both even or both odd. Not sufficient.
- Together: they share parity and at least one is even, so both are even. x is even — answered.
- Check: x = 1, y = 1 satisfies II but not I; x = 2, y = 2 satisfies both.
- ✗ Statement I Not sufficient alone: x = 1, y = 2 gives x²y² = 4 (even) with x odd, while x = 2, y = 1 gives x even.
- ✗ Statement II Not sufficient alone: it only says x and y have the same parity; x = y = 1 and x = y = 2 both fit.
Remember · Parity items: turn each statement into a plain rule (at least one even / same parity), then combine the rules.
Question and answer: UPSC's provisional GS Paper II (2026, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·