Question

Is x positive? Statement (I): \( xy = 6 \)
Statement (II): \( x(y^2) = 12 \)

• A. Statement (I) alone is sufficient.
• B. Statement (II) alone is sufficient.
• C. Both statements (I) and (II) are sufficient.
• D. Neither statement is sufficient.

Hint:

In a system of equations, if one equation contains a higher power of a variable than the other, division is a powerful tool to eliminate variables.

Answer 1

The Correct Option is C

Concept: This problem requires determining the sign of a variable \( x \) given a system of non-linear equations.

Step 1: Analyze individual statements. Statement (I) \( xy = 6 \): We cannot determine the sign of \( x \) because \( y \) could be positive (making \( x \) positive) or negative (making \( x \) negative). Insufficient. Statement (II) \( xy^2 = 12 \): We know \( y^2 \) is always non-negative. However, \( x \) could still be positive or negative depending on other conditions. Insufficient.

Step 2: Combine the statements. We have a system: \[ (1) \quad xy = 6 \] \[ (2) \quad xy^2 = 12 \] To isolate \( y \), divide equation (2) by equation (1): \[ \frac{xy^2}{xy} = \frac{12}{6} \] \[ y = 2 \] Now, substitute \( y = 2 \) back into equation (1): \[ x(2) = 6 \implies x = 3 \] Since \( x = 3 \) is definitively positive, the combined information is sufficient. { Both statements (I) and (II) are sufficient.}