For 1/3 < x < y < 2, which of the following statements is/are always correct?

Quantitative Aptitude ·Previously asked in UPSC Civil Services Examination 2026

View the full solved paper: GS Paper II

Question

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.

  1. A. I only
  2. B. II only (Correct answer)
  3. C. Both I and II
  4. D. Neither I nor II

Correct Answer

Option B — II only

Detailed Solution & Explanation

The correct answer is Option B (II only).

Key Points

  • Statement I is not always true. The function f(t) = t + 1/t decreases on (0, 1) and increases on (1, ∞), so for x < y straddling 1 (e.g. x = 0.5, y = 1) we get f(x) > f(y), violating I.
  • Statement II is always true. Write √(1+t²)/t = √(1/t² + 1), which decreases as t increases. Since x < y, √(1+x²)/x > √(1+y²)/y, i.e. √(1+y²)/y < √(1+x²)/x.

Additional Information

  • Only the second inequality holds throughout the interval, so Option B.