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27 Mar 2020, 05:45
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
45% (02:05) correct
55%
(02:15)
wrong
based on 264
sessions
History
Date
Time
Result
Not Attempted Yet
Let S be the set of all points (x, y) in the xy plane such that |x| + |y| ≤ 2 and |x| ≥ 1. Then what is the area of the region represented by S?
A. 1/2
B. 1
C. 2
D. 3
E. 4
A. 1/2
B. 1
C. 2
D. 3
E. 4
30 Jul 2023, 04:23
Kudos
Bookmarks
↧↧↧ Detailed Video Solution to the Problem ↧↧↧
Given that S be the set of all points (x, y) in the xy plane such that |x| + |y| ≤ 2 and |x| ≥ 1 and we need to find what is the area of the region represented by S
Plotting |x| + |y| ≤ 2
We will have four case for values of x and y each being ≥ 0 and ≤ 0 respectively (in four quadrants)
Mod x + Mod y LTEQ 2.jpg [ 32.92 KiB | Viewed 6887 times ]
Plotting |x| ≥ 1
We will have two cases x≥0 and x<0
Mod x + Mod y LTEQ 2 intersection with mod x GTEQ 1.jpg [ 21.96 KiB | Viewed 6708 times ]
Point of intersection of both the graphs
There will be four points of intersections
So, common area between the two graphs will be given by the two shaded triangles shown below:
Common area.jpg [ 23.42 KiB | Viewed 6567 times ]
Area of each triangle = \(\frac{1}{2}\) * Base * Height = \(\frac{1}{2}\) * 2 * 1 = 1
=> Total area of two triangles= 2*1 = 2 units
So, Answer will be C
Hope it helps!
Watch the following video to learn Basics of Absolute Values
Given that S be the set of all points (x, y) in the xy plane such that |x| + |y| ≤ 2 and |x| ≥ 1 and we need to find what is the area of the region represented by S
Plotting |x| + |y| ≤ 2
We will have four case for values of x and y each being ≥ 0 and ≤ 0 respectively (in four quadrants)
- 1. x≥0, y≥0 => |x| = x and |y| = y => x + y ≤ 2 (Quadrant 1). We will plot the line x + y = 2 and will consider the area below the line as we have ≤ sign
- 2. x≤0, y≥0 => |x| = -x and |y| = y => -x + y ≤ 2 (Quadrant 2). We will plot the line -x + y = 2 and will consider the area below the line as we have ≤ sign
3. x≤0, y≤0 => |x| = -x and |y| = -y => -x - y ≤ 2 (Quadrant 3) => x + y ≥ -2. We will plot the line x + y = -2 and will consider the area above the line as we have ≥ sign
4. x≥0, y≤0 => |x| = x and |y| = -y => x - y ≤ 2 (Quadrant 4) => y ≥ x - 2. We will plot the line x - y = 2 and will consider the area above the line as we have ≥ sign
Attachment:
Mod x + Mod y LTEQ 2.jpg [ 32.92 KiB | Viewed 6887 times ]
Plotting |x| ≥ 1
We will have two cases x≥0 and x<0
- 1. x≥0 => |x| = x => x ≥ 1 and Intersection of x≥0 and x ≥ 1 equals x ≥ 1 => Area will be to the right of x = 1
2. x<0 => |x| = -x => -x ≥ 1 => x ≤ - 1 and Intersection of x<0 and x ≤ - 1 equals x ≤ - 1 => Area will be to the left of x = -
Attachment:
Mod x + Mod y LTEQ 2 intersection with mod x GTEQ 1.jpg [ 21.96 KiB | Viewed 6708 times ]
Point of intersection of both the graphs
There will be four points of intersections
- 1. x=1 and x+y=2 will intersect at (1,1)
2. x=-1 and -x+y=2 will intersect at (-1,1)
3. x=-1 and -x-y=2 will intersect at (-1,-1)
4. x=1 and x-y=2 will intersect at (1,-1)
So, common area between the two graphs will be given by the two shaded triangles shown below:
Attachment:
Common area.jpg [ 23.42 KiB | Viewed 6567 times ]
Area of each triangle = \(\frac{1}{2}\) * Base * Height = \(\frac{1}{2}\) * 2 * 1 = 1
=> Total area of two triangles= 2*1 = 2 units
So, Answer will be C
Hope it helps!
Watch the following video to learn Basics of Absolute Values
General Discussion
27 Mar 2020, 07:13
Kudos
Bookmarks
for the condition this to be true lxl +lyl <=2, the area will be a square with vertices (-2,0)(2,0)(0,-2)(0,2)
Now we need to subtract the the area which does not satisfy the condition lxl >=1
Area of Set will be the area of Non Overlapping Part
2 x (1/2) x base x height
2 x (1/2) x 1 x 2 = 2
Now we need to subtract the the area which does not satisfy the condition lxl >=1
Area of Set will be the area of Non Overlapping Part
2 x (1/2) x base x height
2 x (1/2) x 1 x 2 = 2
Attachments
Area.PNG [ 11.53 KiB | Viewed 11172 times ]
