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, y और z पूर्णांक हैं, जिनमें से प्रत्येक 1 से बड़ा है, तो क्या x एक अभाज्य संख्या है?
- कथन I: xy² = 116
- कथन II: xz = 261
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, y and z are integers, each greater than 1, then is x a prime number?
Statement I. xy² = 116
Statement II. xz = 261
(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: (a) इस विकल्प का चयन कीजिए, यदि प्रश्न का उत्तर, इन कथनों में से केवल एक का उपयोग कर दिया जा सकता है, किन्तु दूसरे कथन का उपयोग कर नहीं दिया जा सकता है
Factorise both numbers: 116 = 2² × 29 and 261 = 3² × 29. The first leaves only y = 2 and x = 29, a prime; the second allows x = 3, 9, 29 or 87, so it cannot decide.
- 116 = 2² × 29. With y > 1, the only square factor is y² = 4, so y = 2 and x = 29.
- 29 is prime, so Statement I alone answers the question (yes).
- 261 = 3² × 29. With x, z > 1, x can be 3, 9, 29 or 87.
- 3 and 29 are prime but 9 and 87 are not, so Statement II alone cannot answer.
- One statement alone works and the other does not.
- ✓ Statement I Sufficient alone: 116 = 4 × 29 forces y = 2 and x = 29, which is prime.
- ✗ Statement II Not sufficient alone: 261 = 9 × 29 allows x = 3 or 29 (prime) and x = 9 or 87 (not prime).
Remember · Prime factorisation first; then ask which factor splits are allowed by the given conditions (here, every variable > 1).
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). ·