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Previous Year Paper

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CAT-2005-Previous Years Paper

Question
12 out of 30
 

Consider a triangle drawn on the X-Y plane with its three vertices of (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X, Y) coordinates.
The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is



A 780
B 800
C 820
D 741

Ans. A

Equation of line = x + y = 41. If the (x, y) co ordinates of the points are integer, their sum shall also be integers so that x + y = k (l, a variable) as we have to exclude points lying on the boundary of the triangle; k can take all values from 1 to 40 only. K = 0 is also rejected as at k = 0 will give the point A; which can not be taken. Now, x + y = k (k = 1, 2, 3,…40) With k = 40; x + y = 40; taking integral solutions.We get points (1, 39), (2, 38); (3, 37)… (39, 1) i.e., 39 points

1) x + y = 40 will be satisfied by 39 point is similarly x + y = 39 is satisfied by 38 points.

x + y = 38 by 37 point.

x + y = 3 by 2 point.

x + y = 2 is satisfied by 1 point

x + y = 1 by no points.

So, the total number of all such points is:
39 + 38 + 37 + 36 + … 3 + 2 + 1

= = 780 points

CAT-2005-Previous Years Paper Flashcard List

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