मान लीजिए p, q, r और s भिन्न धन पूर्णांक हैं। मान लीजिए p, q विषम हैं और r, s सम हैं। निम्नलिखित कथनों पर विचार कीजिए :
- 1.(p − r)²(qs) सम है।
- 2.(q − s)q²s सम है।
- 3.(q + r)²(p + s) विषम है।
उपर्युक्त कथनों में से कौन-से सही हैं?
Read this question in English
Let p, q, r and s be distinct positive integers. Let p, q be odd and r, s be even. Consider the following statements:
1. (p − r)²(qs) is even.
2. (q − s)q²s is even.
3. (q + r)²(p + s) is odd.
Which of the statements given above are correct?
(a) 1 and 2 only · (b) 2 and 3 only · (c) 1 and 3 only · (d) 1, 2 and 3
उत्तर और व्याख्या · explanation in English
Answer: (d) 1, 2 और 3
With p, q odd and r, s even, statement 1 contains the even factor qs and statement 2 the even factor s, so both are even. In statement 3, q + r and p + s are both odd, so the product is odd. All three are correct.
- Parity rules: odd ± even = odd; any product with an even factor is even; odd × odd = odd.
- Statement 1: (p − r)² is odd, but qs = odd × even is even, so the product is even. Correct.
- Statement 2: s is even, so (q − s)q²s is even. Correct.
- Statement 3: q + r is odd, so (q + r)² is odd; p + s is odd; odd × odd = odd. Correct.
- Check with p = 1, q = 3, r = 2, s = 4: (−1)² × 12 = 12 (even); (−1) × 9 × 4 = −36 (even); 5² × 5 = 125 (odd).
- ✓ 1. qs is odd × even = even, so the whole product is even whatever (p − r)² is.
- ✓ 2. The factor s is even, so the product is even.
- ✓ 3. q + r and p + s are each odd + even = odd, and a product of odd numbers is odd.
Remember · For parity questions, plug in small numbers of the right parity and test each statement; it is faster than algebra.
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). ·