UPSC Prelims 2018 CSAT Paper II · Q23 of 58 Basic Numeracy medium

If X is between -3 and -1, and Y is between -1 and 1, then X^2 - Y^2 is in between which of the following?

  1. (a) -9 and 1
  2. (b) -9 and -1
  3. (c) 0 and 8
  4. (d) 0 and 9 ✓ UPSC's answer

Why the answer is (d)

• X is between -3 and -1, so X^2 is between 1 and 9.

• Y is between -1 and 1, so Y^2 is between 0 and 1.

• The minimum of X^2 - Y^2 occurs at X^2 = 1 and Y^2 = 1, giving 0.

• The maximum of X^2 - Y^2 occurs at X^2 = 9 and Y^2 = 0, giving 9.

• Hence X^2 - Y^2 is between 0 and 9, so option (d) follows.

Why the other options are wrong

(a) -9 and 1
Option (a) is wrong because X^2 - Y^2 cannot be negative, since X^2 is at least 1 and Y^2 is at most 1.
(b) -9 and -1
Option (b) is wrong because the expression is not negative and its upper bound is 9, not -1.
(c) 0 and 8
Option (c) is wrong because the maximum value 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.

Reading the answer is not the same as getting it right under a clock. Practise this question with UPSC's negative marking, and anything you miss goes into an error notebook until you get it right twice.

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