UPSC CSE Prelims 2025

Correct Option (2)

To determine if the expression is positive, first simplify it:

The given expression is (p + q)² - 4pq.

Using the algebraic identity (a + b)² = a² + 2ab + b², we expand (p + q)² as p² + 2pq + q².

Substituting this into the expression:

p² + 2pq + q² - 4pq

= p² - 2pq + q²

This simplifies to (p - q)², based on the identity (a - b)² = a² - 2ab + b².

The question now becomes: Is (p - q)² positive?

For any real numbers p and q, the square of their difference, (p - q)², is always non-negative (i.e., (p - q)² ≥ 0).

• (p - q)² = 0 if and only if p - q = 0, which implies p = q.
• (p - q)² > 0 if and only if p - q ≠ 0, which implies p ≠ q.

Since p and q are natural numbers, the expression (p - q)² is positive if and only if p ≠ q.

Now, evaluate the given statements:

• Statement I: p < q.

  This condition explicitly states that p is not equal to q (p ≠ q). Therefore, (p - q)² will be positive. Statement I alone is sufficient to answer the question.

• Statement II: p > q.

  This condition also explicitly states that p is not equal to q (p ≠ q). Therefore, (p - q)² will be positive. Statement II alone is also sufficient to answer the question.

Since both Statement I and Statement II individually provide sufficient information to determine that the expression is positive, the question can be answered by using either statement alone.

Incorrect Options:

• Option 1: The Question can be answered by using one of the Statements alone, but cannot be answered using the other statement alone.

  This option is incorrect because both Statement I and Statement II are individually sufficient to answer the question, as demonstrated above.

• Option 3: The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

  This option is incorrect because the question can be answered using either statement alone. Combining both statements is not necessary.

• Option 4: The Question can be answered even without using any of the Statements.

  This option is incorrect. Without any information about the relationship between p and q, we cannot definitively determine if (p - q)² is strictly positive. If p = q, the expression would be 0, not positive. Thus, the statements are necessary to ascertain p ≠ q.