I have come across this question in CAT 2 and I feel the explanation given is wrong.
I feel 1 is insufficient since we cannot tell which quadrant the point lies if x & y are not positive . And the question do not indicate whether X & Y are positive .
Question & MH Answer :
In which quadrant of the coordinate plane does the point (x, y) lie?
(1) |xy| + x|y| + |x|y + xy > 0
(2) -x < -y < |y|
In essence, this question asks whether the signs of both x and y can be determined.
(1) SUFFICIENT: The key to evaluating this statement is to see which values of x and y actually satisfy it ("crack the code"). To do so, consider all possibilities for the signs of x and y.
- x > 0, y > 0: The left side becomes xy + xy + xy + xy = 4xy, which is a positive number; the statement is satisfied.
- x < 0, y > 0: The left side becomes xy - xy + xy - xy = 0, so the statement is not satisfied.
- x > 0, y < 0: The left side becomes xy + xy - xy - xy = 0, so the statement is not satisfied.
- x < 0, y < 0: The left side becomes xy - xy - xy + xy = 0, so the statement is not satisfied.
- Either x or y (or both) is 0: The left side becomes 0 + 0 + 0 + 0 = 0, so the statement is not satisfied.
Therefore, statement (1) can be rephrased simply as "Both x and y are positive." The point (x, y) is thus in the first quadrant.
(2) SUFFICIENT: If -y does not equal |y|, then y must be positive (and -y must be negative). Since -x < -y, we know that -x is also negative, so x is also positive. The point (x, y) is therefore in the first quadrant.
The correct answer is D.