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25 Aug 2021, 08:00
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Difficulty:
95%
(hard)
Question Stats:
37% (02:27) correct
63%
(02:43)
wrong
based on 243
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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 ?
(1) ab = -4
(2) a + 4b = 0
(1) ab = -4
(2) a + 4b = 0
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24 Sep 2021, 03:03
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Official Solution:
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 ?
Let's check which set(s) of integers \((a, \ b)\), where \(a > 0\) and \(b < 0\), satisfy \(-b + 2a < 4\). Notice that since \(b\) must be negative, then \(-b\) will be positive, so \(-b + 2a =(positive \ integer) + (even \ positive \ integer)\), which is at least 3. \(-b + 2a\) is 3, when \(b=-1\) and \(a=1\). For all other values, \(-b + 2a\) will be \( \geq 4\), not \(< 4\). So, the question basically asks whether \(a=1\) and \(b=-1\). (In other words, point \((1, \ -1)\) is the ONLY point which satisfies \(a > 0\), \(b < 0\), and \(-b + 2a < 4\).)
(1) \(ab = -4\). Since \(a=1\) and \(b=-1\) is not a solution of this equation, then point \((a, \ b)\) is not in the region. Sufficient.
(2) \(a + 4b = 0\). Since \(a=1\) and \(b=-1\) is not a solution of this equation, then point \((a, \ b)\) is not in the region. Sufficient.
For better understanding check the image below:
The question asks whether point \((a, \ b)\) is in the yellow region. As you can see the only point in this region which has integer coordinates is point (-1, 1), so the question aks whether \(a=1\) and \(b=-1\)
Answer: D
General Discussion
25 Aug 2021, 08:39
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On solving the equation -y + 2x - 4 = 0 we get the coordinates of this line as (0,-4) and (2,0)
Therefore, for the inequality to be valid, the integer points should be less than these two points and should also satisfy x > 0 and y > 0.
Statement 1: ab = -4
1) a = 1, b= -4 => Not in the region
2) a = -1, b= 4 => Not in the region
3) a = 4, b= -1 => Not in the region
4) a = -4, b= 1 => Not in the region
5) a = 2, b= -2 => Not in the region
6) a = -2, b= 2 => Not in the region
Hence we get a definite answer and Statement 1 is sufficient.
Statement 2: a = -4b
1) a = 0, b= 0 => Not in the region
2) a = -4, b= 1 => Not in the region
3) a = 4, b= -1 => Not in the region
4) a = 8, b= -2 => Not in the region
5) a = -8, b= 2 => Not in the region
The rest of the values of a and b will lie outside the bounded region.
Hence we get a definite answer and Statement 2 is sufficient.
Hence the answer is D.
Therefore, for the inequality to be valid, the integer points should be less than these two points and should also satisfy x > 0 and y > 0.
Statement 1: ab = -4
1) a = 1, b= -4 => Not in the region
2) a = -1, b= 4 => Not in the region
3) a = 4, b= -1 => Not in the region
4) a = -4, b= 1 => Not in the region
5) a = 2, b= -2 => Not in the region
6) a = -2, b= 2 => Not in the region
Hence we get a definite answer and Statement 1 is sufficient.
Statement 2: a = -4b
1) a = 0, b= 0 => Not in the region
2) a = -4, b= 1 => Not in the region
3) a = 4, b= -1 => Not in the region
4) a = 8, b= -2 => Not in the region
5) a = -8, b= 2 => Not in the region
The rest of the values of a and b will lie outside the bounded region.
Hence we get a definite answer and Statement 2 is sufficient.
Hence the answer is D.
