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UPSC Prelims 2025 CSAT, question 74

Topic: Data sufficiency

Q74

A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.

Question :

Is (p+q)2−4pq(p+q)^2-4pq, where p,qp,q are natural numbers, positive ?

Statement-I : p<qp<q.

Statement-II : p>qp>q.

Which one of the following is correct in respect of the above Question and the Statements ?

  1. (a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other statement alone.
  2. (b) The Question can be answered by using either Statement alone.Correct answer
  3. (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
  4. (d) The Question can be answered even without using any of the Statements.

Solution.

Answer: (b) The Question can be answered by using either Statement alone.

  1. Simplify the algebraic expression: (p+q)2−4pq=p2+2pq+q2−4pq=(p−q)2(p + q)^2 - 4pq = p^2 + 2pq + q^2 - 4pq = (p - q)^2.
  2. Statement I alone is sufficient: if p<qp < q, then p−q≠0p - q \ne 0, so (p−q)2>0(p - q)^2 > 0 (strictly positive).
  3. Statement II alone is sufficient: if p>qp > q, then p−q≠0p - q \ne 0, so (p−q)2>0(p - q)^2 > 0 (strictly positive); thus the Question can be answered using either Statement alone.

Trap: Without the statements, p=qp = q is possible and gives (p−q)2=0(p - q)^2 = 0 (which is not positive), so the Question cannot be answered without using the statements.