Question

Is xy negative?
Statement 1: (x + y)² < (x-y)²
Statement 2: (x - y) is positive

• A. Statement (1) alone is sufficient to answer the question
• B. Statement (2) alone is sufficient to answer the question
• C. Both the statements together are needed to answer the question
• D. Either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. Neither statement (1) nor statement (2) suffices to answer the question.

Answer 1

The Correct Option is A

The correct answer is option (A): Statement (1) alone is sufficient to answer the question

Let's analyze the question and each statement to determine if they are sufficient to answer: "Is xy negative?" This is equivalent to asking whether x and y have opposite signs.

Statement 1

(x + y)^2 < (x - y)^2

Expand both sides:

(x + y)^2 = x^2 + 2xy + y^2
(x - y)^2 = x^2 - 2xy + y^2

So the inequality becomes:

x^2 + 2xy + y^2 < x^2 - 2xy + y^2

Cancel x^2 and y^2 from both sides:

2xy < -2xy

Add 2xy to both sides:

4xy < 0
xy < 0

This directly implies xy is negative. Therefore Statement 1 alone is sufficient.

Statement 2

(x - y) is positive, i.e. x > y.

Knowing only that x > y does not determine the signs of x and y. For example:

• If x = 2, y = 1 then x - y > 0 and xy = 2 > 0.
• If x = -1, y = -2 then x - y = 1 > 0 but xy = 2 > 0.
• If x = 1, y = -1 then x - y = 2 > 0 and xy = -1 < 0.

So Statement 2 alone is insufficient to determine the sign of xy.

Conclusion

Since Statement 1 by itself implies xy < 0, Statement (1) alone is sufficient, and the correct choice is option (A).