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 - 4 pq, where p, q are natural numbers, positive ? Statement I : p < q. Statement II : p > q. Which one of the following is correct in respect of the above Question and the Statements ?
- (a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other statement alone.
- (b) The Question can be answered by using either Statement alone. ✓ UPSC's answer
- (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
- (d) The Question can be answered even without using any of the Statements.
Why the answer is (b)
• The expression (p + q)^2 - 4pq simplifies algebraically to p^2 + 2pq + q^2 - 4pq, which equals p^2 - 2pq + q^2.
• This further simplifies to (p - q)^2, which is a perfect square.
• Since p and q are natural numbers, (p - q)^2 is always non-negative (greater than or equal to zero).
• The question asks if the value is positive. A perfect square is positive if and only if the base (p - q) is not zero.
• Statement I (p < q) implies p ≠ q, so (p - q)^2 > 0, making the answer 'Yes'.
• Statement II (p > q) implies p ≠ q, so (p - q)^2 > 0, making the answer 'Yes'.
• Since each statement alone is sufficient to determine that the expression is positive, the correct option is (b).
Why the other options are wrong
- (a) The Question can be answered by using one of the Statements alone, but cannot be answered…
- Both Statement I and Statement II individually imply that p is not equal to q, making each statement sufficient on its own.
- (c) The Question can be answered by using both the Statements together, but cannot be…
- The statements are not mutually exclusive in a way that requires combining them; each independently ensures p ≠ q, so they are sufficient alone.
- (d) The Question can be answered even without using any of the Statements.
- Without any statement, p could equal q (e.g., p=1, q=1), resulting in (1-1)^2 = 0, which is not positive, so the question cannot be answered without the statements.
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.