Question : 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 ?

(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 cannot be answered even by using both the Statements together. ✓ UPSC's answer

Why the answer is (d)

• Statement 1 states that p × q > 0, which implies that p and q have the same sign (both positive or both negative), but it does not determine their relative magnitudes.

• Statement 2 states that p² > q², which implies that |p| > |q|, meaning the absolute value of p is greater than the absolute value of q, but it does not specify the signs of p and q.

• Using Statement 1 alone, if p = 2 and q = 1, p > q is true; if p = -1 and q = -2, p > q is false. Thus, Statement 1 alone is insufficient.

• Using Statement 2 alone, if p = 2 and q = 1, p > q is true; if p = -2 and q = 1, p > q is false. Thus, Statement 2 alone is insufficient.

• Combining both statements, we know p and q have the same sign and |p| > |q|. If both are positive (e.g., p = 2, q = 1), then p > q. If both are negative (e.g., p = -2, q = -1), then p < q.

• Since the relationship between p and q depends on whether they are positive or negative, the question cannot be answered definitively even with both statements together.

Why the other options are wrong

(a) The Question can be answered by using one of the Statements alone, but cannot be answered…: Neither statement alone is sufficient to determine if p is greater than q.
(b) The Question can be answered by using either Statement alone.: Neither statement alone is sufficient to determine if p is greater than q.
(c) The Question can be answered by using both the Statements together, but cannot be…: Even with both statements, the sign of the numbers remains ambiguous, making the answer indeterminate.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.