Reasoning : Logical Reasoning & Analytical Ability

Q 77 / 325

UPSC CSE Prelims 2023

Is p greater than q?

Statement-1: p×q is greater than zero.

Statement-2: p2 is greater than q2.

Which one of the following is correct in respect of the above Question and the Statements?

EXPLANATION

Correct Option: 4

The question seeks to determine if p > q.

Consider Statement-1: p \times q > 0. This condition implies that p and q must have the same sign (both positive or both negative).

  • If p=2 and q=1, then \(p \times q = 2 > 0\), and p > q.
  • If p=-2 and q=-1, then \(p \times q = 2 > 0\), and p < q.

Since both p > q and p < q are possible outcomes, Statement-1 alone is insufficient to answer the question.

Consider Statement-2: p^{2} > q^{2}. This condition implies that |p| > |q|.

  • If p=2 and q=1, then \(p^{2} = 4 > q^{2} = 1\), and p > q.
  • If p=-2 and q=1, then \(p^{2} = 4 > q^{2} = 1\), and p < q.

Since both p > q and p < q are possible outcomes, Statement-2 alone is insufficient to answer the question.

Consider both Statement-1 and Statement-2 together: From Statement-1, p and q have the same sign. From Statement-2, |p| > |q|.

  • If p and q are both positive: Given |p| > |q|, it follows that p > q. (e.g., p=2, q=1).
  • If p and q are both negative: Given |p| > |q|, it implies that p is numerically larger than q but negative. Consequently, p < q. (e.g., p=-2, q=-1, where |-2| > |-1| but -2 < -1).

As the combined statements allow for scenarios where p > q and scenarios where p < q, the question "Is p greater than q?" cannot be definitively answered even by using both statements together.

Incorrect Options:

Options 1, 2, and 3 are incorrect because, as demonstrated, neither statement alone nor both statements together provide a conclusive answer to whether p > q.