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)

• Let f(t)=t+1/t; f'(t)=1-1/t^2, so f decreases on (1/3,1) and increases on (1,2).

• Statement I is not always true: for x=1/2 and y=3/4, f(1/2)=5/2 and f(3/4)=25/12, so x+1/x > y+1/y.

• Let g(t)=sqrt(1+t^2)/t = sqrt(1+1/t^2) for t>0.

• Since x<y and x,y>0, 1/x^2 > 1/y^2, so 1+1/x^2 > 1+1/y^2.

• Therefore sqrt(1+1/x^2) > sqrt(1+1/y^2), i.e. sqrt(1+x^2)/x > sqrt(1+y^2)/y, so II is always correct and option (b) follows.

Why the other options are wrong

(a) I only: Option (a) is wrong because I fails for x=1/2 and y=3/4, where 5/2 > 25/12.
(c) Both I and II: Option (c) is wrong because I is not always correct, as x=1/2 and y=3/4 make x+1/x greater than y+1/y.
(d) Neither I nor II: Option (d) is wrong because II is always correct for 1/3<x<y<2.

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.