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SEPTEMBER 1895. FROM QUADRATICS.

I. Find the roots of the equation ax2 bx + co, and show what relation must exist between the coefficients in order (1) that the roots may be real and unequal, (2) real and equal, (3) that they may be imaginary.

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4. Find the value of the recurring decimal 0.123535 +.

5. (a) Derive the formula for the number of permutations of m different things taken n at a time?

(b) In how many different ways may a base-ball nine be arranged, the pitcher and catcher being always the same but the others playing in any position?

6. Define logarithm, characteristic, mantissa. Transform 3 a' d2 into a form adapted to computation.

log

7. (a) Of what number is 2 the logarithm in a system whose base is 3?

(b) Find log, 1000, having given log10 5 = 0.699.

JUNE, 1896. TO QUADRATICS.

1. Resolve into lowest factors:

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(a)x — y; (b) m2 + 5m 50; (c) a'(bc) + b2(c-a) + c'(a - b).

2. Solve the equation (xa)(x — b) + (a + b)2 = (xa)(x+b).

3. Solve the simultaneous equations 5x + 3y = 65, 2y z=11, 3x + 4z = 57.

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4. The width of a room is two-thirds of its length. If the width had been three feet more, and the length three feet less, the room would have been square: find its dimensions.

5. Extract the square root of x* +16.

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6. Simplify the following expressions:

(a) 21/175 31/63 + 51/28; (b) x- y +¶ × x2-4a × (x2)¶−2r ÷ x4p—8r;

2T

(c) x3p
(d) 1/12 + 21/11; (e) (1- 1)° (f) ∞

a

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yx;

7. Transform

31/5 +51/3
√5- √3

into an equivalent expres

sion having a rational denominator.

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2X

bxx ax

3. Solve the simultaneous equations, x + 2y + 32 == 17, 3y+ z = 0, 3x + y — 5% = 15.

4. A. has $15 more than B, B has $5 less than C, and they have $65 between them. How much has each?

5. Expand (2x-2y)' by the binomial formula. 6. Simplify the following expressions:

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JUNE 1896. FROM QUADRATICS.

1. What relation do the roots of the equation x2 + px + q = o, bear to the coefficients p and q? What will be the value of ૧ if the roots are reciprocals of each othei?

2. Solve the equations: (a) x' — 3x2 = 88,

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3. Solve the simultaneous equations

x3 — y3 = 98, x y = 2.

4. (a) Deduce the formula for the sum of the first n terms in a geometric progression.

(b) Find the limit of the sum of an infinite number of terms in the series 9

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-6+4

2x2

etc.

into a series of ascending powers

of x, by the method of indeterminate coefficients. (b) What will be the nth term of this series?

6. Prove the following: (a) log mn log m+ log n;

(b) log m=

log m

==

; (c) loga I = 0. r

7. (a) Give the numerical value of the sum

loga a

log10 .001 + 2/3 log, 8.

(b) Show that log10 5 = I - log10 2.

SEPTEMBER 1896.

FROM QUADRATICS.

1. (a) For what value of c are the roots of the equation 3x2+4x+co, equal?

(b) Construct the equation whose roots are

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4. (a) Deduce the formula for the number of combinations of m different things taken n at a time.

(b) In how many different ways may the letters in the word Yale be arranged?

5. If a b c d, prove that

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computation. (b) What is the base of a system of logarithms in which the logarithm of 81 is 4 ?

2x

7. Given log10 2 0.301, find x from the equation

100.

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