For 1/3 < x < y < 2, which of the following statements is/are always correct? I. x + 1/x < y + 1/y II. √(1 + y²)/y < √(1 + x²)/x Select the answer using the code given below.
- (a) I only
- (b) II only ✓ UPSC's answer
- (c) Both I and II
- (d) Neither I nor II
Why the answer is (b)
• Statement I is not always correct. t + 1/t falls for t < 1 and rises for t > 1. With x = 0·5 and y = 0·9: x + 1/x = 2·5 but y + 1/y ≈ 2·01, so the inequality fails.
• Statement II is always correct. √(1 + t²)/t = √(1/t² + 1), which keeps falling as t increases. Since y > x > 0, the value at y is smaller than the value at x.
• Hence II only: (b).
Why the other options are wrong
- (a) I only
- Statement I fails for x = 0·5, y = 0·9.
- (c) Both I and II
- Statement I is not always true.
- (d) Neither I nor II
- Statement II is always true because √(1/t² + 1) decreases as t increases.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.