2. 外语
  3. In the rectangular coordinate system, line k passes through the point (n,-l). Is the slope of line k greater than zero? (1) Line

In the rectangular coordinate system, line k passes through the point (n,-l). Is the slope of line k greater than zero? (1) Line

admin2022-10-18  52
问题 In the rectangular coordinate system, line k passes through the point (n,-l). Is the slope of line k greater than zero?
(1) Line k passes through the origin.
(2) Line k passes through the point (1 ,n + 2).
选项 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.
答案C
解析 (1)     The slope of a line through (n,-l) and (0,0) is -1/n, which is greater than zero if n < 0 and less than zero if n > 0; NOT sufficient.
(2)     Given that line k passes through the points (n,-l) 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 l. 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/n= 1, which is greater than zero.
The correct answer is C;
both statements together are sufficient.
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