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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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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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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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Let ABCD be a rectangle with sides 15 and 20 as shown. Let m be a circle centered at A of the circle passing through F, C, and E.

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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?

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

- The number of straight lines that can be formed by joining them is
^{n}C_{2}. - The number of triangles that can be formed by joining them is
^{n}C_{3}. - The number of polygons with k sides that can be formed by joining them is
^{n}C_{k}.

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

- The number of straight lines that can be formed by joining them is
^{n}C_{2}–^{m}C_{2}+ 1. - The number of triangles that can be formed by joining them is
^{n}C_{3}–^{m}C_{3}. - The number of polygons with k sides that can be formed by joining them is
^{n}C_{k}–^{m}C_{k}.

(c) The number of diagonals of an n sided polygon is ^{n}C_{2} – n = n × (n – 3)/2

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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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The number of divisors the form 4n +1, n ≥0, of the number 10^{10} × 11^{11} × 13^{13}

__________

A. 256

B. 924

C. 1024

D. 512

E. None of these

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In the figure, ABCD is a rhombus. Point E is on the diagonal AC such that AE = 41, EC = 9 and measure of ∠ABE = 90º. Find the area of this rhombus.

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