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

To determine if the product \(xy\) is negative, we need to analyze the given statements one by one and then together to check their sufficiency. 

1. Statement 1: \((x + y)^2 < (x - y)^2\)
   • First, let's expand both squares:
     • \((x + y)^2 = x^2 + 2xy + y^2\)
     • \((x - y)^2 = x^2 - 2xy + y^2\)
   • According to Statement 1:
     • \(x^2 + 2xy + y^2 < x^2 - 2xy + y^2\)
   • By simplifying, we get:
     • \(2xy < -2xy\)
     • Add \(2xy\) to both sides: \(4xy < 0\)
     • So, dividing by 4 gives: \(xy < 0\)
2. Statement 2: \((x - y) > 0\)
   • This statement tells us only that \(x > y\). It does not provide any information about the signs of \(x\) or \(y\) individually or in relation to each other beyond being distinct.
   • Thus, this statement alone is not sufficient to determine if \(xy\) is negative.
3. Conclusion: Based on the analysis:
   • Statement 1 alone is sufficient to determine that \(xy\) is negative.
   • Statement 2 alone is not sufficient.

Therefore, the correct answer is: Statement (1) alone is sufficient to answer the question.