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 :
मान लीजिए, P, Q, R, S भिन्न शून्येतर अंक हैं । यदि PP × PQ = RRSS, जहाँ P ≤ 3 और Q ≤ 4, तो Q किसके बराबर है ?
- कथन I: R = 1.
- कथन II: S = 2.
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:
Let P, Q, R, S be distinct non-zero digits. If PP × PQ = RRSS, where P ≤ 3 and Q ≤ 4, then what is Q equal to?
Statement I. R = 1.
Statement II. S = 2.
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: (d) इस प्रश्न का उत्तर, इन कथनों में से किसी का उपयोग किए बिना भी दिया जा सकता है ।
Dividing both sides by 11 gives P × PQ = 100R + S, which must be at least 100. With P ≤ 3 and Q ≤ 4, only 33 × 34 = 1122 works, so Q = 4 without using either statement.
- PP = 11 × P and RRSS = 1100R + 11S = 11 × (100R + S). So P × PQ = 100R + S.
- R is non-zero, so P × PQ must be at least 100. P = 1 gives at most 1 × 14 = 14; P = 2 gives at most 2 × 24 = 48. Too small.
- P = 3: 3 × (30 + Q) = 90 + 3Q reaches 100 only for Q = 4, giving 102, so R = 1 and S = 2.
- Check: 33 × 34 = 1122, and the digits 3, 4, 1, 2 are distinct.
- Q = 4 follows from the question alone; neither statement is needed.
- ✗ Statement I Not needed: R = 1 already follows from the question itself.
- ✗ Statement II Not needed: S = 2 also follows from the question itself.
Remember · PP and RRSS are multiples of 11 — divide it out first. Digit limits often settle the question on their own.
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). ·