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Mathematics>Standard Multiple Choice

Read the following SAT test question and then click on a button to select your answer.

If (x+y)^2 = (x^2) + (y^2), which of the following statements must also be true?

I. x = 0

II.(x-y)^2 = (x^2) + (y^2)

III. xy = 0

(A) None

(B) I only

(C) II only

(D) III only

(E) II and III

答案和解析

答案：E

解析：

The quantity (x + y)^2 can be expressed as (x^2) + (2xy) + (y^2). If (x + y)^2 = (x^2) + (y^2), then 2xy = 0 and xy = 0. Since xy = 0, either x = 0 or y = 0 or both. Therefore, statement III must be true, but statement I, x = 0, is not always true. For statement II, you can write (x-y)^2 = (x^2)-(2xy) + y^2, and since xy = 0, it follows that (x-y)^2 = (x^2) + (y^2). Therefore, both statements II and III must be true.