题目信息
In the rectangular coordinate system, line k passes through the point (n,−1). Is the slope of line k greater than zero?
A:Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B:Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C:BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D:EACH statement ALONE is sufficient.
E:Statements (1) and (2) TOGETHER are NOT sufficient.
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正确答案：
C:BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Geometry Simple coordinate geometry
The slope of a line through (n,−1) and (0,0) is
, which is greater than zero if n < 0 and less than zero if n > 0; NOT sufficient.
Given that line k passes through the points (n,−1) and (1,n + 2), then the slope of line k (when it exists) is equal to 
. If n = 0, then the slope of line k is 3, which is positive. However, if n = 2, then the slope of line k is −5, which is negative; NOT sufficient.
Given (1) and (2), it follows that
, which by cross-multiplying is equivalent to (−1)(1 − n) = n(n + 3) when n is not equal to 0 or 1. This is a quadratic equation that can be rewritten as n2 + 2n + 1 = 0, or (n + 1)2 = 0. Therefore, n = −1 and the slope of line k is 
= 1, which is greater than zero.
The correct answer is C;both statements together are sufficient.
The slope of a line through (n,−1) and (0,0) is
, which is greater than zero if n < 0 and less than zero if n > 0; NOT sufficient.
Given that line k passes through the points (n,−1) and (1,n + 2), then the slope of line k (when it exists) is equal to . If n = 0, then the slope of line k is 3, which is positive. However, if n = 2, then the slope of line k is −5, which is negative; NOT sufficient.Given (1) and (2), it follows that
, which by cross-multiplying is equivalent to (−1)(1 − n) = n(n + 3) when n is not equal to 0 or 1. This is a quadratic equation that can be rewritten as n2 + 2n + 1 = 0, or (n + 1)2 = 0. Therefore, n = −1 and the slope of line k is = 1, which is greater than zero.The correct answer is C;both statements together are sufficient.
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