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JUNE 1894. FROM QUADRATICS.

1. Solve the equation 12x2 + 5x + 1 = 0.

2. Determine by inspection the roots of the equation x2 - 2ax (b+ a)(b − a),

and state the theorem employed.

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4. (a) Derive the formula for the number of arrangements of m different elements taken n at a time.

(b) How many combinations can be made of 10 different things taken in sets of 7?

5. Prove the following propositions: (a) log (mn)

I

log m+log n; (b) log mã =

loge NX loga e.

log m

11

(c) loga N =

6. (a) Given log10 3 0.47712 to determine the number of figures in 3100. (b) Determine from the datum of (a) log10 V0.3. (c) Indicate the manner of finding the value of x from the equation 5* = 8, when you have given a table of logarithms.

7. Expand

2

3x2 4x

into a series of ascending powers

of x to five terms, by the method of undetermined coefficients.

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3. Resolve 8x2 + 18x 5 into two factors.

4. Solve the simultaneous equations,

x' + y3 504, and x2- xy + y2

84.

5. Find the sum of the series 3+1 + + etc., to infinity.

6. (a) Derive the formula for the number of combinations of m things taking n at a time.

(b) Show that the number of combinations of m things taken n at a time is the same as the number taken m at a time.

7. (a) Prove that loga b X logь a I

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(b) Given log 0.11 = 1.04139, to find log /0.11.

11

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3. Solve the simultaneous equations (a + c)x

bc, x+ya+b.

by

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5. Expand (2x3±1)5 by the binomial formula.

6. Simplify (a) 21/128/18; (b) (xm+n)m-n

+

(x") +m ÷ (xm)n+m; (c)

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7. Which is the greater, the cube root of two, or the seventh root of five? Prove your answer.

SEPTEMBER 1895. To QUADRATICS.

I. Resolve into lowest factors: (a) y-y-6; (b) x' + x3y + xy2 + y3; (c) (2a - b)2 — (2b+a)2; (d) x4n

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4. Prove that the sum of the squares of any two different integers is greater than twice their product.

5. Extract the square root of 9x-12x2+10x2 — 4x + 1. 6. Simplify the following expressions.

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JUNE 1895. FROM QUADRATICS.

1. (a) Find by inspection the sum and the product of the roots of the equation ax2 + bx + c o, and state the theorem used. (b) For what value of a will the equation ax2 + 6x + 3 = o have equal roots?

2. Solve the equation (4a3 — b')(x2 + 1)

4a3 + b2

3. Solve the simultaneous equations

x2 + xy = 12, xy — 2y2 = I.

= 2X.

4. Derive a formula for the amount of P dollars in n years at r per cent, compound interest, interest being payable annually.

I

X

5. Convert

into a series of ascending powers of

2+3x

x, by the method of undetermined coefficients. Five terms will suffice.

6. Prove the following: (a) log (mn) = log m— log n; (b) log mPp log m; (c) log, ex loge 10 = 1.

7. (a) Write equivalents for loga a; loga 1; loga (1 ÷ a); logo .001; log, 8.

I

(b) Indicate the solution of 3* = IO.

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