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25 Oct 2021, 09:48
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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 \(c > d\).
(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 \(c > d\).
25 Oct 2021, 09:48
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Official Solution:
In the \(xy\)-plane line \(k\) passes through the origin. Is the slope of line \(k\) is greater than 1 ?
A line passing through the origin (through the point (0, 0) has the equation \(y=mx\).
A line passing through the origin AND with the slope greater than 1 must pass through any point from the green region shown below:
(1) Line \(k\) does not pass through any point \((a, \ b)\) where \(a\) and \(b\) are positive and \(a > b\).
This statement says that \(k\) does NOT pass through any point from the red region shown below:
Two cases are possible:
(i) Line \(k\) has a slope of 1, so the equation of line \(k\) is \(y=x\) (blue line). In this case the line won't be in the green region and we'd have a NO answer to the question.
(ii) Line \(k\) has a slope grater than 1, for example \(y=2x\), \(y=10x\), ... In this case the line will be in the green region and we'd have an YES answer to the question.
Not sufficient.
(2) Line \(k\) passes through the point \((c, \ d +1)\) where \(c\) and \(d\) are consecutive integers and \(c > d\).
\(c\) and \(d\) are consecutive integers and \(c > d\) mean that \(c=d+1\). So, this statement says that \(k\) passes through the point \((c, \ c)\)
If we knew that \(c \neq 0\), then we'd have that \(k\) passes through \((0, \ 0)\) and some point \((c, \ c)\) and this would mean that the equation of line \(k\) is \(y=x\). Which would give a NO answer to the question.
But if \(c = 0\), then we'd have that the line passes through the origin, which we already knew from the stem and in this case we could have an YES as well as a NO answers to the questions.
Not sufficient.
(1)+(2) The second statement is basically useless if \(c = 0\) so it adds no additional info to (1), which means that even taken together the statements are not sufficient.
Answer: E
In the \(xy\)-plane line \(k\) passes through the origin. Is the slope of line \(k\) is greater than 1 ?
A line passing through the origin (through the point (0, 0) has the equation \(y=mx\).
A line passing through the origin AND with the slope greater than 1 must pass through any point from the green region shown below:
(1) Line \(k\) does not pass through any point \((a, \ b)\) where \(a\) and \(b\) are positive and \(a > b\).
This statement says that \(k\) does NOT pass through any point from the red region shown below:
Two cases are possible:
(i) Line \(k\) has a slope of 1, so the equation of line \(k\) is \(y=x\) (blue line). In this case the line won't be in the green region and we'd have a NO answer to the question.
(ii) Line \(k\) has a slope grater than 1, for example \(y=2x\), \(y=10x\), ... In this case the line will be in the green region and we'd have an YES answer to the question.
Not sufficient.
(2) Line \(k\) passes through the point \((c, \ d +1)\) where \(c\) and \(d\) are consecutive integers and \(c > d\).
\(c\) and \(d\) are consecutive integers and \(c > d\) mean that \(c=d+1\). So, this statement says that \(k\) passes through the point \((c, \ c)\)
If we knew that \(c \neq 0\), then we'd have that \(k\) passes through \((0, \ 0)\) and some point \((c, \ c)\) and this would mean that the equation of line \(k\) is \(y=x\). Which would give a NO answer to the question.
But if \(c = 0\), then we'd have that the line passes through the origin, which we already knew from the stem and in this case we could have an YES as well as a NO answers to the questions.
Not sufficient.
(1)+(2) The second statement is basically useless if \(c = 0\) so it adds no additional info to (1), which means that even taken together the statements are not sufficient.
Answer: E
