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UPSC Prelims 2025 CSAT, question 76

Topic: Number system

Q76

Let p+q=10p+q=10, where p,qp,q are integers.

Value−I=Maximumvalueofp×qwhenp,qarepositiveintegers.Value-I=Maximum value of p \times q when p,q are positive integers.

Value−II=Maximumvalueofp×qwhenp≥−6,q≥−4.Value-II=Maximum value of p \times q when p \ge -6,q \ge -4.

Which one of the following is correct ?

  1. (a) Value−I<Value−IIValue-I<Value-II
  2. (b) Value−II<Value−IValue-II<Value-I
  3. (c) Value−I=Value−IIValue-I=Value-IICorrect answer
  4. (d) Cannot be determined due to insufficient data

Solution.

Answer: (c) Value−I=Value−IIValue-I=Value-II

  1. For integers p+q=10p + q = 10, the product is pq=p(10−p)=25−(p−5)2p q = p(10 - p) = 25 - (p - 5)^2, which attains its global maximum of 2525 at p=5,q=5p = 5, q = 5.
  2. Since (p,q)=(5,5)(p, q) = (5, 5) consists of positive integers (giving Value-I=25\text{Value-I} = 25) and also satisfies p≥−6,q≥−4p \ge -6, q \ge -4 (giving Value-II=25\text{Value-II} = 25), we have Value-I=Value-II\text{Value-I} = \text{Value-II}.