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09 May 2019, 21:27
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How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line \(y=\pi x\), the line y = -0.1 and the line x = 5.1?
(A) 30
(B) 41
(C) 45
(D) 50
(E) 57
(A) 30
(B) 41
(C) 45
(D) 50
(E) 57
14 May 2019, 03:15
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When x=0, no integral value of y lie in region
When x=1, 0=<y<3.14 Hence 4 points lie in region
When x=2, 0=<y<6.28 Hence 7 points lie in region
When x=3, 0=<y<9.42 Hence 10 points lie in region
When x=4, 0=<y<12.56 Hence 13 points lie in region
When x=5, 0=<y<15.70 Hence 16 points lie in region
First count Total squares with side 1*1
When we choose 2 points on x=1 and 2 points on x=2, squares possible= 4-1=3
When we choose 2 points on x=2 and 2 points on x=3, squares possible= 7-1=6
When we choose 2 points on x=3 and 2 points on x=4, squares possible= 10-1=9
When we choose 2 points on x=4 and 2 points on x=5, squares possible= 13-1=12
Hence total 1*1 squares= 12+9+6+3+0=30
count Total squares with side 2*2=
When we choose 2 points on x=1 and 2 points on x=3, squares possible=4-2=2
When we choose 2 points on x=2 and 2 points on x=4, squares possible=7-2=5
When we choose 2 points on x=3 and 2 points on x=5, squares possible=10-2=8
Hence total 2*2 squares= 2+5+8=15
count Total squares with side 3*3=
When we choose 2 points on x=1 and 2 points on x=4, squares possible=4-3=1
When we choose 2 points on x=2 and 2 points on x=5, squares possible=7-3=4
Hence total 3*3 squares= 1+4=5
There are 0 4*4 squares possible
Total squares lie in the region= 30+15+5=50
When x=1, 0=<y<3.14 Hence 4 points lie in region
When x=2, 0=<y<6.28 Hence 7 points lie in region
When x=3, 0=<y<9.42 Hence 10 points lie in region
When x=4, 0=<y<12.56 Hence 13 points lie in region
When x=5, 0=<y<15.70 Hence 16 points lie in region
First count Total squares with side 1*1
When we choose 2 points on x=1 and 2 points on x=2, squares possible= 4-1=3
When we choose 2 points on x=2 and 2 points on x=3, squares possible= 7-1=6
When we choose 2 points on x=3 and 2 points on x=4, squares possible= 10-1=9
When we choose 2 points on x=4 and 2 points on x=5, squares possible= 13-1=12
Hence total 1*1 squares= 12+9+6+3+0=30
count Total squares with side 2*2=
When we choose 2 points on x=1 and 2 points on x=3, squares possible=4-2=2
When we choose 2 points on x=2 and 2 points on x=4, squares possible=7-2=5
When we choose 2 points on x=3 and 2 points on x=5, squares possible=10-2=8
Hence total 2*2 squares= 2+5+8=15
count Total squares with side 3*3=
When we choose 2 points on x=1 and 2 points on x=4, squares possible=4-3=1
When we choose 2 points on x=2 and 2 points on x=5, squares possible=7-3=4
Hence total 3*3 squares= 1+4=5
There are 0 4*4 squares possible
Total squares lie in the region= 30+15+5=50
Bunuel
________________________
BUMPING FOR DISCUSSION.
BUMPING FOR DISCUSSION.
17 May 2019, 06:48
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Is it really possible to get such lengthy questions in GMAT?
