Question: Is m \gt n if m, n are real numbers?<br>Statement-I:<br>m=(1-p)(p^{2}+p+1) and n=(p+1)(p^{2}-p+1)<br>Statement-II:<br>m=pn

Consider the following for the next ten (10) items that follow :<br>Mark option (a) if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.<br>Mark option (b) if the question can be answered by using either statement alone.<br>Mark option (c) if the question can be answered by using both the statements together, but cannot be answered using either statement alone.<br>Mark option (d) if the question cannot be answered even by using both the statements together.

  1. A. The question can be answered by using one of the statements alone, but cannot be answered using the other statement alone
  2. B. The question can be answered by using either statement alone
  3. C. The question can be answered by using both the statements together, but cannot be answered using either statement alone
  4. D. The question cannot be answered even by using both the statements together ✓

Correct Answer: D. The question cannot be answered even by using both the statements together

Explanation

From Statement-I: m = 1-p^3 and n = 1+p^3. The difference m-n = -2p^3. The sign of this difference depends on the sign of p, which is unknown. So Statement-I is not sufficient. From Statement-II: m=pn gives no information about their comparative values. Combining both gives 1-p^3 = p(1+p^3) \implies p^4+p^3+p-1=0. Since p can take multiple values satisfying this (both positive and negative), we still cannot determine if m \gt n.

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