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7. In a circle, the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

ALGEBRA.

I.

Specimen Paper.

(Time, 3 hours.)

1. Find (1) the sum, (2) the difference, of the two expressions

1

(x−2) (x−1)x(x+1)'

1

(x-1)x(x+1)(x+2)°

(3) the quotient when the former is divided by the latter.

2. Reduce to their simplest forms the expressions—

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3. Prove the rules for pointing in the multiplication and division of

decimals.

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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

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7. Prove that ao=1, and aTM=√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 arithmetical progession.

In the series of − +

(2) when n= infinity.

&c., find s (1) when n = = 5,

9. When does one quantity vary (1) directly, (2) inversely as another? lf x varies as y, prove that x2 + y2 will vary as x2

10. Expand

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y2

in a series of ascending powers of x by means of

the Binomial theorem, writing down the first four terms and the rth term.

11. Express the numbers 957 and 23·125 in the septenary scale?

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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 — c2 + (a + c)2 — b2 + (b + c)2 — α2.

6. Find the greatest common measure of

4x4

12x3 + 5x2 + 14x - 12 and 6x4 — 11x3 +9x13x + 6.

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8. Investigate an expression for the sum of a geometrical_series ; find the sum of 30 terms of the series 2 ++7+ &c.

and

9. A number consists of 2 digits; if it be multiplied by 2, and the product diminished by four, the digits are inverted; and if 19 be subtracted from it, the remainder is equal to 3 times the sum of the digits; find the number.

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 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.

5. Reduce to its lowest terms the expression

6. Solve the following equations

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+ =n. C d

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3x4 +14x3 + 9x+2
2x4+9x3 + 14x + 3

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In equation (2) investigate the conditions that the values of x may be two positive integers.

7. Write down the (r2)th term and the middle term of the expansion of (a - b)27.

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

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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, is there ambiguity?

10. Given log,.71968-4.8571394; diff. for 1=60, find the value of

0719686 to seven places of decimals.

11. If three sides of a triangle are x2 + x + 1, 2x + 1, and x2-1, 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.

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 inclination 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. Determine 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 immersed 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 mechanical power in a machine of given dimensions.

Is the force attractive

11. A particle moves in a circle under the action of a central force resident in an external point. Find the law of force. or repulsive?

PRACTICAL GEOMETRY, BUILDERS' WORK, &c.

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 24 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?

5. Compare the contents of a triangular and hexagonal pyramid, one side of the base in each being 5 feet, and their altitude 31 feet.

6. Construct a plain scale of 171⁄2 feet to an inch, and pen in the lines in Indian ink.

7. A semicircular arch of 16 feet span is intersected by an arch springing from the same level, and of the same rise, but of 10 feet span, forming groins. Draw the plan of their intersections, and a section of the smaller arch.

8. Before commencing work with a Theodolite, of which the good adjustment is doubtful, how would you test it?

Builders' Work.

1. State what you know of the different kinds of limestone in this country, and of their economic value.

What are the objections to "shelly" limestone, and when it is used for building what precaution should be observed with respect to the portion exposed?

2. State how the quality of granite is affected by an excess of quartz or mica, or feldspar. Should you consider, generally, a find or coarsegrained granite to be preferable? What effect has sulphuret of iron on the granite in which it occurs?

3. State the materials with which you are acquainted suitable for lining furnaces or limekilus.

4. What points would you consider in selecting materials for a road covering? Name, in the order of their fitness for this purpose, the materials with which you are acquainted.

5. Certain limes, if not properly treated, will swell after they have been made into mortar and introduced into work. How does this arise, and what precautions would you observe in the preparation of the morter, to prevent it?

6. What do you consider to be affair proportion of sand, gravel, and lime, for a foundation of a building under ordinary circumstances? What are the conditions under which you consider that these proportions should be varied?

5. What are the characteristics of a good brick? What crushing force per square inch should such a brick be capable of supporting? 8. What thickness of mortar would you allow in joints in ashlar work,

and in brick work?

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