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# Algebra

The equation 11x + 67y = 133 has

1. two integral solutions for 0 < x < 50
2. no integral solutions for 10 < y < 20
3. an integral solution for 100 < x < 200
4. no integral solutions for y < 0

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# Arithmetic

A shopkeeper offers 30% off on MRP of his articles. In addition, he gives one article free for every four articles purchased. Even after all these offers, the shopkeeper manages to get a profit of 12%. By what percent has he marked up his articles?

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# Numbers

Among the integers 1, 2,3, 4,……, 2018, what is the maximum number of integers that can be selected such that the sum of any two selected numbers is not a multiple of 7?

# Counting in Geometry (a) In a plane, if there are n points of which no three are collinear, then

1. The number of straight lines that can be formed by joining them is nC2.
2. The number of triangles that can be formed by joining them is nC3.
3. The number of polygons with k sides that can be formed by joining them is nCk.

(b) In a plane, if there are n points out of which m points are collinear, then

1. The number of straight lines that can be formed by joining them is nC2mC2 + 1.
2. The number of triangles that can be formed by joining them is nC3mC3.
3. The number of polygons with k sides that can be formed by joining them is nCkmCk.

(c) The number of diagonals of an n sided polygon is nC2 – n = n × (n – 3)/2

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# Algebra

If a, b and c are positive real numbers such that ab + a + b = bc + b + c = ca + c + a = 35. Find the value of (a + 1) (b +1)(c+1) .

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# Numbers

The number of divisors the form 4n +1, n ≥0, of the number 1010 × 1111 × 1313
__________
A. 256
B. 924
C. 1024
D. 512
E. None of these

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