Is ab positive? 1. (a + b)2 < (a - b)2 2. a = b

In Data Sufficiency, a statement is sufficient when it forces exactly one definite answer to the question posed - a definite "No" counts as sufficient just as much as a definite "Yes" does. A statement is insufficient only when the values it allows can make the answer flip between different outcomes. Evaluate each statement completely on its own first.

Statement 1 alone: (a + b)2 < (a - b)2

1. Expand both squares: a2 + 2ab + b2 < a2 - 2ab + b2

2. Cancel the common a2 and b2 terms from both sides: 2ab < -2ab

3. Combine like terms: 4ab < 0, so ab < 0

4. This is one fixed conclusion in every case allowed by the statement - ab is definitely negative, a firm "No" to "Is ab positive?" - so Statement 1 alone is sufficient.

Statement 2 alone: a = b

1. Substitute b = a into the product: ab = a x a = a2

2. Case a = b = 0: a2 = 0, so ab = 0, which is not positive

3. Case a = b (not equal to) 0: a2 > 0, so ab is positive

4. The same condition allows two different outcomes (ab = 0 and ab > 0), so Statement 2 alone is NOT sufficient.

Cross-check:

• a = 2, b = -3 satisfies (a + b)2 < (a - b)2 (1 < 25) and gives ab = -6 < 0, confirming the negative conclusion from Statement 1.

• a = b = 0 gives ab = 0, while a = b = 4 gives ab = 16 - two different outcomes from Statement 2 alone, confirming it cannot decide the question by itself.

Since Statement 1 alone always fixes one definite answer and Statement 2 alone does not, only Statement 1 is sufficient.