If X is between –3 and –1, and Y is between –1 and 1, then X² – Y² is in between which of the following?
- (a) –9 and 1
- (b) –9 and –1
- (c) 0 and 8
- (d) 0 and 9 ✓ UPSC's answer
Why the answer is (d)
• X lies between –3 and –1, so X² lies between 1 and 9.
• Y lies between –1 and 1, so Y² lies between 0 and 1.
• X² – Y² is smallest when X² is smallest and Y² is largest (1 – 1 = 0) and largest when X² is largest and Y² is smallest (9 – 0 = 9).
• Hence X² – Y² lies between 0 and 9, option (d).
Why the other options are wrong
- (a) –9 and 1
- X² – Y² can never be negative here, since X² ≥ 1 ≥ Y².
- (b) –9 and –1
- X² – Y² can never be negative here, since X² ≥ 1 ≥ Y².
- (c) 0 and 8
- X² can approach 9 while Y² approaches 0, so the upper limit is 9, not 8.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2018, held on 3 June 2018. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.