If X is between −3 and −1, and Y is between −1 and 1, then X² − Y² is in between which of the following?
Answer & explanation
Answer: (d) 0 and 9
Squaring removes the signs: X² lies between 1 and 9, and Y² between 0 and 1. So X² − Y² is more than 1 − 1 = 0 and less than 9 − 0 = 9.
- −3 < X < −1 gives 1 < X² < 9.
- −1 < Y < 1 gives 0 ≤ Y² < 1.
- Smallest value of X² − Y²: just above 1 − 1 = 0. Largest: just below 9 − 0 = 9.
- So X² − Y² lies between 0 and 9.
- Check: X = −2.9 and Y = 0 give 8.41, more than 8, so (c) is too narrow.
Remember · For the range of a difference, pair the largest first term with the smallest second term and vice versa; square before combining.
Question and answer: UPSC's official GS Paper II (2018, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·