Bunuel wrote:

Is \(|xy| > xy\)?

(1) \(\sqrt{x^2y^2}=xy\)

(2) \(|x|+|y|=x+y\)

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

The first step of VA (Variable Approach) method is modifying the original condition and the question, and rechecking the number of variables and the number of equations.

The question \(|xy| > xy\) is equivalent to \(xy < 0\).

Condition 1)

\(|xy| = xy\) is equivalent ot \(xy \ge 0\).

Since the answer is no, this condition is sufficient by CMT(Common Mistake Type) 1.

Condition 2)

\(|x| + |y| = x + y\)

=> \((|x| + |y|)^2 = (x + y)^2\)

<=> \(|x|^2 + 2|x||y| + |y|^2 = x^2 + 2xy + y^2\)

<=> \(x^2 + 2|x||y| + y^2 = x^2 + 2xy + y^2\)

<=> \(2|x||y| = 2xy\)

<=> \(|xy| = xy\)

<=> \(xy \ge 0\)

Then the answer is no and "no" is also an answer.

By CMT 1, this condition is also sufficient.

Therefore, D is the answer.

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