Re: GMAT CLUB OLYMPICS: In the xy-coordinate plane, does region bounded by
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26 Aug 2021, 01:44
Answer: D
In the xy-coordinate plane, does region bounded by x > 0, y < 0, and -y + 2x < 4 contain point (a, b), where a and b are integers ?
(a,b) will be in the fourth quadrant.
For third inequality -y + 2x < 4 i.e. line -y + 2x = 4,
x-intercept will be (2,0) and y-intercept will be (0,-4)
Therefore, we need to find the points (a,b) such that they lie in a triangle made x>0, y<0 and -y+2x<4.
Vertices of the triangle will be (0,0), (2,0) and (0,-4)
Statement 1: ab = -4
since a and b are integers, we can have (a,b) as
(1, -4), (-1, 4), (4,-1) (-4, 1) (2,-2) and (-2,2)
(-1, 4), (-4,1) and (-2,2) does not belong in the region of x>0, y<0
For (1, -4), (4,-1) and (2,-2), lets put these values in L.H.S of -y + 2x < 4
(1,-4) --> 4+2 = 6 which is not < 4, hence outside the region.
(4,-1) --> 1+8 = 9 which is not < 4, hence outside the region.
(2,-2) --> 4+4 = 8 which is not < 4, hence outside the region.
Hence, all the points fall outside the region bounded by x > 0, y < 0, and -y + 2x < 4.
Therefore, sufficient.
Statement 2: a + 4b = 0
This expression can have many values, but we only need to check if any values fall inside triangle with vertices (0,0), (2,0) and (0,-4).
therefore, we only need to check for a combination of x = 1 and y = -1, -2, or -3.
i.e. (1,-1), (1,-2) and (1,-3)
None of these satisfy the above expression if we substitute the above values as (a,b) or (b,a).
Hence, any points (a,b) for expression a+4b=0 will lie outside the region.
Therefore, sufficient.
Answer Choice: D (Both statements are sufficient by themselves)