एक प्रश्न दिया गया है जिसके बाद दो कथन I और II दिए गए हैं। प्रश्न और कथनों पर विचार कीजिए।
- प्रश्न: m और n के मान क्या हैं, जहाँ m और n धनपूर्ण संख्याएँ हैं?
- कथन–I: m + n > mn और m > n.
- कथन–II: m और n का गुणनफल 24 है।
उपर्युक्त प्रश्न और कथनों के संबंध में, निम्नलिखित में से कौन-सा सही है?
Read this question in English
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question. What are the values of m and n, where m and n are natural numbers?
Statement–I. m + n > mn and m > n.
Statement–II. The product of m and n is 24.
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 cannot be answered even by using both the Statements together
उत्तर और व्याख्या · explanation in English
Answer: (c) प्रश्न का उत्तर, दोनों कथनों का एक-साथ प्रयोग कर दिया जा सकता है, किन्तु अकेले किसी भी एक कथन का प्रयोग कर नहीं दिया जा सकता है
m + n > mn with m > n forces n = 1, but leaves m open; mn = 24 alone allows several pairs. Together, n = 1 and m = 24.
- m + n > mn is the same as mn − m − n + 1 < 1, i.e. (m − 1)(n − 1) < 1.
- For natural numbers this means (m − 1)(n − 1) = 0; since m > n ≥ 1, n = 1. Statement I fixes n = 1 but any m ≥ 2 works, so it is not sufficient.
- Statement II alone: mn = 24 allows (24, 1), (12, 2), (8, 3), (6, 4) and their reversals, so it is not sufficient.
- Together: n = 1 and mn = 24 give m = 24. Check: 24 + 1 = 25 > 24.
Remember · Rewrite m + n > mn as (m − 1)(n − 1) < 1; with natural numbers, one of them must be 1.
Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·