UPSC Prelims 2026 · CSAT
Question 21: Basic Numeracy
For 1/3 < x < y < 2, which of the following statements is/are always correct?
- Ix + 1/x < y + 1/y
- II√(1 + y^2)/y < √(1 + x^2)/x
Select the answer using the code given below.
AnswerUPSC’s answer is (b).
Let f(t) = t + 1/t. It falls for t below 1 and rises above 1, so it is not monotonic on (1/3, 2). Take x = 1/2 and y = 1: f(x) = 2.5 and f(y) = 2, so I fails. For II, √(1 + t²)/t = √(1 + 1/t²), which keeps falling as t grows. Since x < y, the value at y is smaller than at x, so II always holds.
- I. Counterexample x = 1/2, y = 1: 2.5 is not less than 2.
- II. √(1 + 1/t²) decreases as t increases, so the larger value gives the smaller result.
How you could have got this
- If you were sure statement I is not correct, (a) and (c) go: each of them includes I.
What you needed to know: Counterexample x = 1/2, y = 1: 2.5 is not less than 2.
- Then, if you were sure statement II is correct, (d) goes: it leaves II out.
What you needed to know: √(1 + 1/t²) decreases as t increases, so the larger value gives the smaller result.
Only (b) is left. This one needs every statement.
This follows UPSC’s provisional key; the final key could change it.
Which options go is worked out from UPSC’s key and the options, not by a model.
Why each option is right or wrong
- (a) Wrong: I fails at x = 1/2, y = 1.
- (b) Right: II holds for every x < y, while I does not.
- (c) Wrong: I fails, so both cannot be always correct.
- (d) Wrong: II always holds.
Only the provisional key (27 May 2026) was found; the final key replaces it when found.
Answer: UPSC’s key. Explanation: written by UPSC Grade, not by UPSC.
Source: UPSC, Civil Services (Preliminary) Examination 2026, General Studies Paper II (CSAT), Series A. The paper on upsc.gov.in.