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23 May 2019, 03:28
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FRESH GMAT CLUB TESTS QUESTION
In the xy-plane line k passes through the origin. Is the slope of line k is greater than 1 ?
(1) Line k does not pass through any point (a, b) where a and b are positive and a > b.
(2) Line k passes through the point (c, d +1) where c and d are consecutive integers and c > d.
M36-42
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15 Jul 2019, 06:05
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Bunuel
What is the slope of a line? It is change in y for every unit change in x. Slope = 1 when y changes by 1 with 1 unit change in x. Slope is greater than 1 when y changes by more than 1 for every 1 unit change in x. Slope is less than 1 when y changes by less than 1 for every one unit change in x.
Read about the slope of a line here: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2010/1 ... he-graphs/
Review the diagram given in the post showing various slopes.
Line k passes through origin. A line with slope > 1 will lie between the green line and y axis. In the first quadrant, its y co-ordinate > x co-ordinate. In the third quadrant, its x co-ordinate > y co-ordinate. So to have slope > 1, k must be a line passing through 1st and 3rd quadrant lying between the green line (slope = 1) and Y axis.
(1) Line k does not pass through any point (a, b) where a and b are positive and a > b.
If a and b are positive, we are talking about the first quadrant. k does not pass through any point such that a > b so k does not have slope between 0 and 1. k could have slope of 1 or greater than 1 or negative slope.
Not sufficient.
(2) Line k passes through the point (c, d +1) where c and d are consecutive integers and c > d.
c = d+1 (c and d are consecutive such that c > d)
Line k passes through (c, c). Now, (c, c) could be (0, 0) but we already know it passes through origin. Here slope can be anything. Or (c, c) could be something else such as (3, 3) or (-4, -4) etc. In that case slope of line k is 1.
Not sufficient.
Using both statements, we get that slope can be 1 or greater than 1 or negative.
Not sufficient.
Answer (E)
23 May 2019, 08:41
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The line y=x, which has a slope exactly equal to 1, does not pass through any point (a, b) where a > b (since a=b for every point on the line), so using Statement 1, the line could have a slope of 1 exactly. Then the answer to the question is 'no'. But the line could also be, say y = 2x or any other line through the origin with a slope greater than 1, so the answer can also be 'yes'.
Using Statement 2, we know c and d are consecutive integers, and c > d, so c = d + 1. So the point the Statement is talking about is (c, c). Now if c were any nonzero number at all, this would tell us we have the points (0, 0) and (c, c) on our line, and our line would need to be y=x. So it would have a slope of 1. But nothing in the wording rules out the possibility that c=0, so Statement 2 could just be telling us that "the origin is on the line", which we already know. So since it's possible Statement 2 is not giving us any new information whatsoever, Statement 2 can't even be useful when combined with other information, and the answer must be A or E, and since Statement 1 was not sufficient, the answer is E.
Using Statement 2, we know c and d are consecutive integers, and c > d, so c = d + 1. So the point the Statement is talking about is (c, c). Now if c were any nonzero number at all, this would tell us we have the points (0, 0) and (c, c) on our line, and our line would need to be y=x. So it would have a slope of 1. But nothing in the wording rules out the possibility that c=0, so Statement 2 could just be telling us that "the origin is on the line", which we already know. So since it's possible Statement 2 is not giving us any new information whatsoever, Statement 2 can't even be useful when combined with other information, and the answer must be A or E, and since Statement 1 was not sufficient, the answer is E.
