Question

Is \(x>0\)? Statement (I): \(xy+4=0\) Statement (II): \(y<0\)

• A. Statement I alone is sufficient, but Statement II alone is not sufficient.
• B. Statement II alone is sufficient, but Statement I alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Even both statements together are not sufficient.

Hint:

A negative product means the factors have opposite signs. Once the sign of one factor is known, the sign of the other becomes immediately known.

Answer 1

The Correct Option is C

Concept: To determine the sign of a variable, we often examine the sign of a product. A negative product means one factor is positive and the other is negative.

Step 1: Analyze Statement (I). Given: \[ xy+4=0 \] \[ xy=-4. \] Thus the product is negative. One variable is positive and the other is negative. We cannot determine which one is positive. Hence Statement (I) alone is insufficient.

Step 2: Analyze Statement (II). Given: \[ y<0. \] Nothing is known about \(x\). Therefore Statement (II) alone is insufficient.

Step 3: Combine both statements. From Statement (I): \[ xy=-4. \] From Statement (II): \[ y<0. \] Since the product is negative and \(y\) is negative, \[ x \text{ must be positive}. \] Hence \[ x>0. \] The answer is definitely “Yes.” Therefore both statements together are sufficient.