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Question

# The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0,0),(0,21) and (21,0) is

A
133
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B
190
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C
233
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D
105
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Solution

## The correct option is B 190Let the vertices of the triangle be A(21,0),B(0,21) and C(0,0)Thus, any point in the interior of the triangle lies in first quadrant.∴a>0 & b>0Point (a,b) lies on the same side of the AB where O liesFor (0,0)x+y−21=−21<0therefore,a+b−21<0a+b<21For a=1;b<21−1⇒b<20:b∈[1,19] total 19 integral valuesFor a=2;b<21−2⇒b<19:b∈[1,18] total 18 integral values............SimilarlyFor a=19;b<21−19⇒b<2:b=1 1 integral valuesThus, Number of integral points =19+18+.....+1 =19(19+1)2 =190Thus there are total 190 integral points, which lies inside the triangle.

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