Let p + q = 10, where p, q are integers. Value-I = Maximum value of p × q when p, q are positive integers. Value-II = Maximum value of p × q when p ≥ -6, q ≥ -4. Which one of the following is correct ?
- (a) Value-I < Value-II
- (b) Value-II < Value-I
- (c) Value-I = Value-II ✓ UPSC's answer
- (d) Cannot be determined due to insufficient data
Why the answer is (c)
• For Value-I, p and q are positive integers summing to 10. The product p × q is maximized when the numbers are closest to each other, i.e., p = 5 and q = 5, giving a maximum value of 25.
• For Value-II, p and q are integers with p ≥ -6 and q ≥ -4, and p + q = 10. This implies q = 10 - p.
• Substituting the constraint q ≥ -4 into the equation gives 10 - p ≥ -4, which simplifies to p ≤ 14.
• The constraint p ≥ -6 remains, so the possible integer values for p range from -6 to 14.
• The product p × q becomes p(10 - p) = 10p - p². This is a downward-opening parabola with its vertex at p = 5.
• Since p = 5 is within the valid range [-6, 14], the maximum value occurs at p = 5, yielding q = 5 and a product of 25. Thus, Value-I equals Value-II.
Why the other options are wrong
- (a) Value-I < Value-II
- Value-I is 25 and Value-II is 25, so Value-I is not less than Value-II.
- (b) Value-II < Value-I
- Value-II is 25 and Value-I is 25, so Value-II is not less than Value-I.
- (d) Cannot be determined due to insufficient data
- The constraints provide sufficient data to determine the exact maximum values for both cases.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.