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4. Sum the following series to infinity: 1, 1⁄2 4, etc. 5. Expand to four terms by the binomial theorem:

(x2 — 2ay).

6. Expand to four terms by indeterminate coefficients :

I + 2x

I

3X

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3. A sum of $54 is divided among three men. John gets six times as much as Charles. Henry and Charles together have as much as John. How much does each get?

4. Multiply 2-1-5 by 3 + 41-5.

5. Rationalize the denominator of

I y + √2 X

In which case will this fraction be real? Imaginary?

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JUNE 1895.

(b)

1. The base of a rectangle is four times its altitude. Find the number of feet in its altitude, if the sum of the number measuring the area (in square feet) and the number measuring the base (in linear feet) is three.

2. Solve: 3x+2y2 =40, 2x + 2y = 12.

3. Express 31 as a continued fraction.

4. Resolve into partial fractions

X - I

x(x

2)

5. What is the value of the sum of the series

1 + 1 + 1 + 27+ etc., to infinity?

6. How many odd numbers of four figures each can be formed with the digits 3, 5, 6, 8, 9; it being understood that in each number the four figures are all different?

JUNE 1896.

(a)

1. Remove the parentheses and reduce to its simplest form x - (2x y [3x 2y- (4x 3y)]).

2. Resolve each of the following expressions into factors: (a) x 4x3y3z2 + 4y*z*; (b) (a + b)2 — c2; (c) 8(x + y)* (2xy).

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4. The sum of three numbers taken two by two, are 20, 29, and 27. What are the numbers?

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a'b' (a+b), to find x and y.

3. Convert 112 into a continued fraction and find the

153

second convergent.

4. Expand by the binomial theorem (a2 + x)

terms.

to four

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