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 ✓ UPSC's answer
Why the answer is (d)
• Let p and q be odd integers, and r and s be even integers.
• Statement 1: (p - r) is (odd - even) = odd. (qs) is (odd * even) = even. The product of an odd and an even number is even, so Statement 1 is correct.
• Statement 2: (q - s) is (odd - even) = odd. q² is (odd * odd) = odd. s is even. The product (odd * odd * even) is even, so Statement 2 is correct.
• Statement 3: (q + r) is (odd + even) = odd. (q + r)² is (odd * odd) = odd. (p + s) is (odd + even) = odd. The product of two odd numbers is odd, so Statement 3 is correct.
• Since all three statements are correct, option (d) is the answer.
Why the other options are wrong
- (a) 1 and 2 only
- This option incorrectly excludes Statement 3, which is correct because the product of two odd terms is odd.
- (b) 2 and 3 only
- This option incorrectly excludes Statement 1, which is correct because the product includes an even factor (qs).
- (c) 1 and 3 only
- This option incorrectly excludes Statement 2, which is correct because the product includes an even factor (s).
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.