The integers r and s are distinct, r ≠ 0, and s ≠ 0. If r^2s^2 = -rs,

The integers r and s are distinct, \(r ≠ 0\), and \(s ≠ 0\). If \(r^2s^2 = -rs\), which of the following must be true?

I. r = -1
II. s = 1
III. r - s = 0

A. None
B. I alone
C. II alone
D. III alone
E. I, II and III

\(r^2s^2=-rs\) --> as neither of unknowns equal to 0 we can safely reduce this equation by \(rs\) (as \(rs\neq{0}\)) --> \(rs=-1\).

I. r = -1 --> must not be true as \(r\) could be 1 and \(s\) could be -1;

II. s = 1 --> the same here: must not be true as \(r\) could be 1 and \(s\) could be -1;

III. r - s = 0 --> and again the same example works: must not be true as \(r\) could be 1 and \(s\) could be -1 --> \(r-s=1-(-1)=2\neq{0}\) (in fact this statement is never true).

Answer: A (None).

The question asks which of the following MUST be true, or which of the following is ALWAYS true no matter what set of numbers you choose. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

As for "COULD BE TRUE" questions:
The questions asking which of the following COULD be true are different: if you can prove that a statement is true for one particular set of numbers, it will mean that this statement could be true and hence is a correct answer.

Hope it helps.