Oh yes, no need of brute force.

The question is:
Is |xy| > (x^2)(y^2)?

Thought No. 1: Both LHS and RHS are positive quantities. Hmm.
Thought No. 2: Oh! We're squaring x and y on the RHS. That would mean that the RHS has really large quantities.
Thought No. 3: When will this inequality hold true then?
Thought No. 4: Um, when we square a number between -1 and 1, that is if |n| < 1, then n^2 < |n|. For example if you square (1/2) or (-1/2), it becomes 1/4.
Thought No. 5: So, |xy| > x^2y^2, when -1 < x < 1 AND -1 < y < 1. If any one of x or y exceeds 1, the RHS will become really large and the inequality will not hold true.
Thought No. 6(and now we look at the statements): If only 0 < x < 1, but y > 1 then the RHS may become larger than LHS. If 0 < y < 1, but x > 1, then the RHS may become larger than LHS. Both the Statements together tell us that x and y are small fraction quantities. The given inequality will hold true when they are combined.

[spoiler](C)[/spoiler] is the answer.

By the way, what would your answer have been if the Statements were:
Statement(1): 1 < x < 3/2
Statement(2): 4/3 < y < 7/4
?

A quick way to show insufficiency for each of the two statements individually.

Dabral