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rajatvmittal
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Statement 1: |xy| + x|y| + |x|y + xy > 0rajatvmittal wrote: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|
Plug in one point from each quadrant: (1,1), (-1,1), (-1,-1), and (1, -1).
Case 1: (x,y) = (1,1)
Here, |xy| + x|y| + |x|y + xy = |1*1| + 1|1| + |1|1 + 1*1 = 4.
Case 2: (x,y) = (-1,1)
Here, |xy| + x|y| + |x|y + xy = |-1*1| + -1|1| + |-1|1 + -1*1 = 0.
Case 3: (x,y) = (1,-1)
Here, |xy| + x|y| + |x|y + xy = |1*-1| + 1|-1| + |1|-1 + 1*-1 = 0.
Case 4: (x,y) = (-1,-1)
Here, |xy| + x|y| + |x|y + xy = |-1*-1| + -1|-1| + |-1|-1 + -1*-1 = 0.
Only Case I satisfies the constraint that |xy| + x|y| + |x|y + xy > 0, implying that (x,y) must be in quadrant I.
SUFFICIENT.
Statement 2: -x < -y < |y|
Multiplying the entire expression by -1, we get:
x > y > -|y|.
y > -|y| requires that y>0.
Since x>y, we get:
x>y>0.
Thus, (x,y) lies in quadrant I.
SUFFICIENT.
The correct answer is D.
