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ALGEBRA.

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Set to candidates

for the Admiralty who selected Algebra as a subject

of Examination.
(Time allowed, 3 hours.)

No. 1. 1. Find (1) the sum, (2) the difference, of the two expressions— 1

1

; (2-2)(x - 1)x(+ 1)' (x - 1)2(x + 1)(x+2)

x (3) the quotient when the former is divided by the latter. 2. Reduce to their simplest forms the expressions

** - 5.38 + 4x+3x+9 (a)

4.03 - 15x2 +8.3+3

1- x - 2x + x2 (B)

+

1-x+ 2x+x2 1-x + 2x+? 1 - 3-V2x+x2 3. Prove the rules for pointing in the multiplication and division of

decimals. 4. Solve the following equations

n

+ (a)

y

m
+ 1
20


(B) 2x2 +11x+15=0

x++y4=82

x +y = 4. 5. Find a number of two digits, which is three times the sum of its

digits, and such that the difference between the digits is 5. 6. If any number of fractions be equal, show that each of them

sum of any multiples of the numerators

sum of the same multiples of the denominators
If
Y

show that
b+c -a cta

b
(6-c)x+(ca)y+(a−b)x=0.

m

-=1

n

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(v) = 4

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=

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a+b

7. Prove that a'=1, and a"="/a.

Is any assumption necessary in order that this may be true? 8. Investigate a rule for finding the sum of n terms of an arith

metical progression.

In the series of }-+-&c., find s (1) when n=5,

(2) when n= infinity. 9. When does one quantity vary (1) directly, (2) inversely, as

another?
If x varies as y, prove that x2 + y2 will vary as x2 —yo.

a3 10. Expand

in a series of ascending powers of x by means (a3 — 23) of the Binomial theorem, writing down the first 4 terms and

the nth term. 11. Express the numbers 957 and 23.125 in the septenary scale.

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2. Add together a(x2 -* - 2xy) and 5 -7 (21-2%)

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No. 2.

a-x aạy + x3 1. Find the value of axy

when a=2, x=3, y=5.
a2 x2
xy

ху
4

x2 y2
5
3

5 3. Divide 4.3– x3+4x by 2x2+3x+2. 4. Investigate a rule for finding the square root of an algebraical

quantity, and explain how the method of extracting the

square root of a numerical quantity may be deduced. 5. Find the square root of (a+b)2 - 02+(a + c)2 – 62+(6+c)2 – a'. 6. Find the greatest common measure of 424 – 12.23 +5.x2 +14x-12

and 6x4 – 11x3 +9x–13x+6. 7. Solve the equations

-1
2

3
(a)
x+2
у

-4
3 4

y+1

+

+

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8. Investigate an expression for the sum of a geometrical series ;

and find the sum of 30 terms of the series 2+$+7+&c. 9. A number consists of 2 digits; if it be multiplied by 2, and the

product diminished by 4, the digits are inverted; and if 10 be subtracted from it, the remainder is equal to 3 times the sum of the digits; find the number.

MATHEMATICAL QUESTIONS. Set to Candidates for the Colonial Office who selected Mathematics

as a subject of examination.

A.-PURE MATHEMATICS. 1. To a given straight line apply a parallelogram, which shall be

equal to a given triangle, and have one of its angles equal to

a given rectilineal angle. 2. A segment of a circle being given, describe the circle of which

it is the segment. 3. Give Euclid's definition of proportion and of similar figures, and

show that similar polygons may be divided into the same number of similar triangles having the same ratio to one

another that the polygons have. 4. Investigate a rule for extracting the cube root of an algebraical expression.

3x4 + 1423 +9x+2 5. Reduce to its lowest terms the expression

234 +9.x8+14x +3 6. Solve the following equations

у +

=m; 6

(2) ax2+bx+c=o. (1)

y

.

18

+

+

En.

с

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22-23+3 22+2x+4
(3)
+N

=21
2 + 2x+4 x2 - 2x + 3
In equation (2) investigate the conditions that the values of 2

may be two positive integers. 7. Write down the (r+2)th term and the middle term of the expan

sion of (a-6)*

+
+

+

t

8. Find the sum of n terms of the following series

1 5 4
3
+

6 3
1

1 3

+
3 2 4

18 + 23 + 33 + 9. Given two sides of a triangle and the angle opposite one of

them: show how to solve the triangle, and point out when the case is ambiguous.

If a=5, b=7, and A=sin -1, is there ambiguity? 10. Given log1071968=4.8571394; diff. for 1=60, find the value

of 0.0719686 to seven places of decimals. 11. If the three sides of a triangle are x2+x+1, 2x+1, and 22—1,

show that the greatest angle is 120°. 12. A right cone is cut by a plane which meets the cone on both

sides of the vertex; show that the section is a hyperbola. Under what condition is it possible to cut an equilateral

hyperbola from a given cone. 13. In an ellipse prove that the line drawn through the focus at

right angles to the focal distance intersects the tangent and the directrix in the same point.

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B. - MIXED MATHEMATICS. 1. Assuming the truth of the parallelogram of forces as to direction,

prove its truth as to magnitude. 2. Given the sum of two forces and their resultant, and also the

angle which one of them makes with the resultant. Determine

the forces, and the angle at which they act. 3. State the result of any experiments made with reference to

friction. A body weighing 12,000 tons, placed on a plane whose inclina

tion is 1 in 12, and acted on by two chains (each capable of sustaining a strain of 200 tons) in the direction of the plane, is just on the point of moving when the chains break. Find

the coefficient of friction between the body and the plane. 4. An area is cut from one angle of a triangle equal to half the area

of the triangle by a line parallel to the base. Find the centre of gravity of the remainder.

5. Enunciate the first and second laws of motion, and mention any

experiments which seem to suggest their truth. How is their

truth finally established? 6. A body moving uniformly in a straight line is suddenly acted on

by a constant force always acting in a given direction.

Determine the subsequent motion. 7. A body of given elasticity is projected vertically upwards with a

given velocity, and strikes against a horizontal plane. Deter

mine the velocity with which it reaches the ground. 8. Find the line of quickest descent from the focus of a parabola,

whose axis is vertical and vertex upwards, to the curve. 9. Define “specific gravity,” and show that when a solid is im

mersed in a fluid, the weight lost is to the whole weight of the

body as the specific gravity of the fluid is to that of the solid. 10. Explain the principle of the hydraulic press, and find the mecha

nical power in a machine of given dimensions. 11. A particle moves in a circle under the action of a central force

resident in an external point. Find the law of force. Is the force attractive or repulsive?

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QUESTIONS IN PRACTICAL GEOMETRY,

BUILDERS' WORK, &c. Set to Candidates for the Situation of Clerk of the Works in the

Engineering Branch of the War Department. A.-PRACTICAL GEOMETRY AND MENSURATION. 1. How much paper, yard wide, will be required for a room that

is 22 feet long, 14 feet wide, and 9 feet high, if there be 3

windows and 2 doors, each 6 feet by 3 feet? 2. How many square feet are contained in a plank whose length is

10 feet 10 inches, and breadth at the two ends 34 feet and

21 feet? 3. What would be the cost of paving a semicircular alcove with

marble at 2s. 6d. a foot, if the length of its semicircular arc.

was 22:42 feet? 4. A stone 18 inches long, 17 broad, and 7 deep, weighs 278 lbs. how

many cubic feet of this kind of stone will freight a vessel of 230 tons burthen?

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