एक प्रश्न दिया गया है, जिसके बाद दो कथन I और II दिए गए हैं । उस प्रश्न और उन कथनों पर विचार कीजिए और सही विकल्प को चिह्नित कीजिए ।
प्रश्न :
क्या (p + q)² − 4 pq, जहाँ p, q धनपूर्ण संख्याएँ हैं, धनात्मक है ?
- कथन I: p < q.
- कथन II: p > q.
उपर्युक्त प्रश्न और कथनों के सन्दर्भ में, निम्नलिखित में से कौन-सा एक सही है ?
Read this question in English
A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question:
Is (p + q)² − 4pq, where p, q are natural numbers, positive?
Statement I. p < q.
Statement II. p > q.
Which one of the following is correct in respect of the above Question and the Statements?
(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other statement alone. · (b) The Question can be answered by using either Statement alone. · (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone. · (d) The Question can be answered even without using any of the Statements.
उत्तर और व्याख्या · explanation in English
Answer: (b) इस प्रश्न का उत्तर, दोनों कथनों में से किसी का भी अकेले उपयोग कर दिया जा सकता है ।
The expression simplifies to (p − q)², which is positive unless p = q. Each statement on its own rules out p = q, so the question can be answered using either statement alone.
- (p + q)² − 4pq = p² + 2pq + q² − 4pq = (p − q)².
- (p − q)² is positive exactly when p ≠ q; it is 0 when p = q.
- Without the statements, p = q is possible, so the question cannot be answered alone.
- Statement I (p < q) gives p ≠ q, so the expression is positive. Statement II (p > q) also gives p ≠ q.
- Either statement alone is sufficient.
- ✓ Statement I p < q means p ≠ q, so (p − q)² > 0 — the answer is 'yes'.
- ✓ Statement II p > q also means p ≠ q, so (p − q)² > 0 — the answer is 'yes'.
Remember · Simplify before judging sufficiency: (p + q)² − 4pq = (p − q)², so only whether p ≠ q matters.
Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·