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)
• (p + q)² − 4pq = p² + 2pq + q² − 4pq = p² − 2pq + q² = (p − q)².
• (p − q)² is positive whenever p ≠ q, and zero when p = q.
• Statement I (p < q) alone gives p ≠ q, so the expression is positive — the question is answered.
• Statement II (p > q) alone also gives p ≠ q — again answered.
• Without a statement p could equal q (value 0), so a statement is needed; each one alone is enough — option (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.