Q. 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?
(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.
(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.
Correct Answer : b)
UPSC Prelims 2025 CSAT
Explanation :
1. Expand the expression:
(p + q)² – 4pq
= p² + 2pq + q² – 4pq
= p² – 2pq + q²
= (p – q)².
2. We ask: Is (p – q)² positive?
– (p – q)² is zero exactly when p = q.
– It is positive whenever p ≠ q.
3. Examine each statement:
Statement I: p < q
→ Then p ≠ q, so (p – q)² > 0.
→ The expression is positive.
Statement II: p > q
→ Again p ≠ q, so (p – q)² > 0.
→ The expression is positive.
Each statement by itself guarantees p ≠ q, hence the expression is positive. Therefore, either statement alone is sufficient to answer the question.
Final choice: (b) The Question can be answered by using either Statement alone.
