Directions for the following 5 (five) items: Each item in this section contains a question followed by two statements. Answer each item using the following instructions and mark your response on the Answer Sheet accordingly.
Question: For two distinct real numbers x and y, which of them is bigger?
- Statement I: x² < y < 1
- Statement II: y < √x < 1
Answer & explanation
Answer: (d) Select this option if the question cannot be answered even using any of the statements
Both statements together only trap y between x² and √x, and for 0 < x < 1 the number x itself lies inside that gap. So y can sit on either side of x and the question stays open.
- Statement I: x = 0.9, y = 0.85 fits (0.81 < 0.85 < 1) and gives x > y; x = 0.1, y = 0.5 also fits and gives y > x. Not sufficient.
- Statement II: √x < 1 means 0 ≤ x < 1. x = 0.25, y = 0.4 fits (0.4 < 0.5) with y > x; x = 0.25, y = 0.1 fits with y < x. Not sufficient.
- Together: x² < y < √x with 0 < x < 1. Take x = 0.25: y must lie between 0.0625 and 0.5.
- y = 0.4 gives y > x, y = 0.1 gives y < x. Still not decided.
- Check: for 0 < x < 1, x² < x < √x, so x always lies inside the allowed range of y.
- ✗ Statement I Not sufficient alone: x = 0.9, y = 0.85 and x = 0.1, y = 0.5 both satisfy it but order x and y differently.
- ✗ Statement II Not sufficient alone: with x = 0.25, both y = 0.4 and y = 0.1 are below √x = 0.5, yet one is above x and one below.
Remember · For numbers between 0 and 1, x² < x < √x. A range that straddles x never decides which is bigger.
Question and answer: UPSC's provisional GS Paper II (2026, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·