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we are asked whether xy>0. Two possibilities can make our dream true. Both x and y are positive or both negative.

Statement 1: \(x^3y>0\). If x is negative , y has to be negative. xy>0 in this case. If both x and y are positive still xy has to be positive. SUFFICIENT.

Statement 2: \(x^2\) is always positive but x is not the same as \(x^2\). x could be positive or negative. Thus , this one is NOT Sufficient as we don't know the exact sign of x.

Statement 1: x³y > 0 Since x² is POSITIVE, we can safely divide both sides of the inequality by x² When we do this, we get xy > 0 So, the answer to the target question is YES, xy IS greater than 0 Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x²y > 0 Once again, we can safely divide both sides of the inequality by x² However, this time we get y > 0 So, we now know that y is POSITIVE, but we don't know whether x is positive or negative. As such statement 2 is NOT SUFFICIENT

If you're not convinced, we can always test some values There are several values of x and y that satisfy statement 2. Here are two: Case a: x = 1 and y = 1. In this case, xy = (1)(1) = 1. So, the answer to the target question is YES, xy IS greater than 0 Case b: x = -1 and y = 1. In this case, xy = (-1)(1) = -1. So, the answer to the target question is NO, xy is NOT greater than 0 Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Since even powers are always non-negative, we can ignore the even exponents. Condition 1) \(x^3y > 0\) is equivalent to \(xy > 0\), which is the question itself. Therefore, condition 1) is sufficient. Condition 2) \(x^2y>0\) is equivalent to \(y > 0\). It is not sufficient

Therefore, A is the answer. Answer: A
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