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, 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?
Answer & explanation
Answer: (b) The Question can be answered by using either Statement alone.
The expression simplifies to (p − q)², which is positive unless p = q. Each statement on its own rules out p = q, so the question can be answered using either statement alone.
- (p + q)² − 4pq = p² + 2pq + q² − 4pq = (p − q)².
- (p − q)² is positive exactly when p ≠ q; it is 0 when p = q.
- Without the statements, p = q is possible, so the question cannot be answered alone.
- Statement I (p < q) gives p ≠ q, so the expression is positive. Statement II (p > q) also gives p ≠ q.
- Either statement alone is sufficient.
- ✓ Statement I p < q means p ≠ q, so (p − q)² > 0 — the answer is 'yes'.
- ✓ Statement II p > q also means p ≠ q, so (p − q)² > 0 — the answer is 'yes'.
Remember · Simplify before judging sufficiency: (p + q)² − 4pq = (p − q)², so only whether p ≠ q matters.
Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·