Question

Is \(x\) negative? Statements: (I) \(x^2>0\) (II) \(x^3\geq 0\)

• A. Statement (I) alone is sufficient to answer the question, but (II) alone is not sufficient.
• B. Statement (II) alone is sufficient to answer the question, but (I) alone is not sufficient.
• C. Both the statements (I) and (II) are sufficient to answer the question, but neither statement alone is sufficient.
• D. Both the statements (I) and (II) together are not sufficient to answer the question and additional data are required.

Hint:

Even powers remove sign information, but odd powers preserve sign. This is very useful in data sufficiency.

Answer 1

The Correct Option is B

Concept: To determine whether \(x\) is negative, we check whether the given conditions uniquely fix the sign of \(x\). Important properties:
• If \(x^2>0\), then \(x\neq0\), but \(x\) may be positive or negative.
• If \(x^3\geq0\), then \(x\geq0\).

Step 1: Checking Statement (I).
Given: \[ x^2>0 \] This means: \[ x\neq0 \] Possible values: \[ x=2 \quad (\text{positive}) \] or \[ x=-2 \quad (\text{negative}) \] Both satisfy the condition. So, Statement (I) alone is not sufficient.

Step 2: Checking Statement (II).
Given: \[ x^3\geq0 \] Since cube preserves the sign: \[ x\geq0 \] Thus, \(x\) cannot be negative. So the answer to “Is \(x\) negative?” is definitely No. Hence, Statement (II) alone is sufficient.