निम्नलिखित 5 (पाँच) प्रश्नांशों के लिए निर्देश : इस खंड में प्रत्येक प्रश्नांश में एक प्रश्न है जिसके बाद दो कथन दिए गए हैं। निम्नलिखित अनुदेशों का उपयोग कर प्रत्येक प्रश्नांश का उत्तर दीजिए और तदनुसार अपने प्रत्युत्तर उत्तर-पत्र पर अंकित कीजिए।
प्रश्न : यदि x और y पूर्णांक हैं, तो क्या x सम है?
- कथन I: x²y² सम है।
- कथन II: 1 + x² + y² विषम है।
Read this question in English
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.
(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
उत्तर और व्याख्या · explanation in English
Answer: (c) इस विकल्प का चयन कीजिए, यदि प्रश्न का उत्तर दोनों कथनों का एक साथ उपयोग कर दिया जा सकता है, किन्तु दोनों में से केवल किसी भी एक कथन का उपयोग कर नहीं दिया जा सकता है
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). ·