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UPSC Prelims 2025CSATGeneral Mental AbilityData Sufficiency - Algebra

Q44. 44. 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.✓ Correct
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.

Detailed Solution

✓ Correct Answer: Option B

First, simplify the expression: (p + q)² - 4pq = p² + 2pq + q² - 4pq = p² - 2pq + q² = (p - q)² The question asks: Is (p - q)² positive? A square of any real number is always non-negative (≥ 0). It is positive if and only if the base is not zero.

So, (p - q)² is positive if p - q ≠ 0, which means p ≠ q. Statement I: p < q. If p < q, then p ≠ q. Therefore, (p - q)² will be positive. This statement alone is sufficient. Statement II: p > q.

If p > q, then p ≠ q. Therefore, (p - q)² will be positive. This statement alone is sufficient. Since either Statement I alone or Statement II alone is sufficient to answer the question, the correct option is (b).

This question tests basic algebraic identities and properties of numbers, a common element in CSAT.

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