Each of the questions given below consists of a statement and/or a question…

Concept:

In a data-sufficiency question, a statement (or a pair of statements) is sufficient only if it forces exactly one answer to the question asked in every case consistent with it. To judge this, test each statement alone against all cases it allows, then test the statements together, and stop as soon as the answer becomes unique in one direction.

Application:

1. Statement I alone (xy < 0): this only fixes that x and y have opposite signs, not their magnitudes. With x = 1, y = -1, xy = -1 < 0 and x + y = 0; with x = 3, y = -1, xy = -3 < 0 and x + y = 2. Two different sums are consistent with Statement I, so it alone is not sufficient.

2. Statement II alone (x2 = y2): this only fixes that x and y have equal magnitude, allowing x = y or x = -y, with sign unspecified. With x = y = 4, x + y = 8; with x = 1, y = -1, x + y = 0. Two different sums are consistent with Statement II, so it alone is not sufficient.

3. Statements I and II together: Statement I forces x and y to have opposite signs, so x cannot equal y (equal values with opposite signs would both have to be 0, but xy = 0 contradicts xy < 0). Statement II restricts x to either y or -y. Since the x = y branch is ruled out by Statement I, only x = -y remains, and that gives x + y = 0 in every case consistent with both statements together.

Cross-check:

• x = 2, y = -2 satisfies both statements (xy = -4 < 0 and 4 = 4) and gives x + y = 0.

• No pair satisfying both xy < 0 and x2 = y2 can avoid x = -y, so no counter-example with a different sum exists once both statements are used together.

Since neither statement alone pins down a single value of x + y, but the two statements together always force x + y = 0, the data in both Statements I and II together are necessary to answer the question.