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(2) How many florins, cents, and mils are there in £3. 48. 9 d.

5. The 3 per cent. consols are at 98, and certain railway securities are at 26 for original £20 shares. What rate per cent. ought these last to pay that it may be as profitable to invest in the one as the other?

6. Define Discount. Shew how to find the discount of £35, 10s. 9d. due seven months hence at 41 per cent. per annum.

What approximation to the true rule for calculating discount is made use of in practice ? and on which side does the advan

tage lie?

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7. Find the quantity of carpeting required for the central portion of a room, this portion being 14 ft. 5 in. wide, and 18ft. 7 in. long. Also the cost; the carpet being only three quarters of a yard wide, and 4s. 6d. per yard.

If between the edge of the carpet and the walls there is a distance all round of 27 feet, how much of the area of the floor will remain uncovered ?

8. Workmen can perform a certain labour in a week if they work 11 hours a day for six days; how many hours a day must they work to perform the same in the same time if they take half of Saturday as a holiday, but do a twelfth more work each hour ?

9. Prove that a (b c) = ab ac. On what grounds do we believe this when bsc?

Add 70 - 4d te, 6c + 3d - 5e, 4e - 12c.

Multiply x* – 3ax by x + 3a, and find the value of

X-V 2

when x=8.

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10. State and prove the rule for finding the Greatest Common Measure of two algebraical quantities. Find the G. C. M. of (1) (a - b), a? ,

(2) ** - 3a*x - 2a", 2C9 - ax - 4a".

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y 12. A and B can perform a certain task in 30 days, working together. After 11 days however B is called off, and A finished it by himself 28 days after. How long would each take to do the work alone?

13. Why are two entries required in every transaction under the system of double entry? Distinguish between “ Bills Receivable” and “ Bills Payable;" “ Balance Sheet" and " Trial

“ ' Balance,” with examples.

Give a specimen of a “ Merchandise account.”

FRIDAY, Dec. 17, 1858. 9 to 12.
II. 7. Pure Mathematics.

(HIGHER PAPER.) 1. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

If segments of a circle be described with the same radius on the three sides of a triangle, shew that if they pass through one point within the triangle the arcs will together form a complete circumference.

2. The straight line drawn at right angles to the diameter of a circle from the extremity of it falls without the circle.

If AB be a fixed chord of a circle, AC and AD any two other chords making equal angles with AB; prove that the tangents at C and D always intersect in a fixed straight line.

3. Shew how to inscribe an equilateral and equiangular hexagon in a given circle.

If an irregular hexagon be such that a circle can be inscribed in it touching all its sides, then the sum of the first, third, and fifth sides is equal to the sum of the second, fourth, and sixth.

4. How does Euclid define Ratio ; when are two ratios said to be equal?

Shew that triangles and parallelograms of the same altitude are one to another as their bases.

Let ABC be any triangle, and AD any straight line drawn from the angle A to any point D in the base. It is required to produce AB to E so that the triangle AED may be any given multiple of the triangle ADC.

5. From the expansion of (x +a)”, deduce the ordinary rule for " completing the square" in the solution of a quadratic equation. Solve the equations:

(1) 2004 – 1= 5x + 2. (2) 200 - 3.00 +5=0.
(3) 2V** + x + 2x=1-V-71+x.
+

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(4) The difference of two numbers is 3, and the differ-

ence of their cubes is 279, find the numbers. 6. Explain how to find the sum of a series of n terms in arithmetical progression, whose first term is a, and last term 1. Sum the series : 6, -2, ,

6

&c. to infinity.

3 9 7. State the algebraical definition of proportion. If a : b :: 0 :d, prove that a : 4-6 :: c:c-d, 1

1 1 1 (a 3 and 26 3c 4d ad 14

- 2 8. If A B when C is constant; and ACC when B is constant; then Acc BC when B and C both vary.

Shew the application of this in the following example: A beam 56 feet long, 2 broad, 11 thick, costs £11, what will be the cost of a beam 159 feet long with a uniform section of 14

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9. Define an angle, and explain the difference between the English and Continental divisions of a right angle.

Compare the values of 39°2', and 3982'.

Trace the changes in the value and sign of sin A, as A in-. creases from 900 to 2700.

Shew that sin A= +11 – cos’A. Explain the double sign.

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10. Shew that

sin (A + B) = sin A cos B+cos A sin B,
where A and B are two angles whose sum is less than 90°.

Given the values of sin 45°, and cos 60°; deduce sin 15o.
11. Shew that the area of a circle of radius a is ma'.

Two circles touch each other and a common tangent is drawn. Supposing their radii to be respectively r and 3r, prove that the area of the curvilinear triangle bounded by the two

11 circles and the common tangent is (4 73

6 12. In any triangle, assuming the expression for the cosine of an angle in terms of the sides, prove that

(s – a)

bc where a, b, c are the three sides of the triangle, 28= a +b+c, and A is the angle opposite to the side a.

Ex. If a = 222, b = 318, c=406, find A, having given log 473 = 2.6748611, log 406 = 2.6085260, log 318 = 2.5024271, tab log cos (16°28') = 9.9818117, log 251 = 2.3996737, tab log cos (16°29') = 9.9817744.)

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SATURDAY, Dec. 18, 1858. 6 to 8 P.M.

II. 8. Mechanics and Hydrostatics.
1. DEFINE force, weight, density.
A force may be properly represented by a straight line.

How are forces usually measured in Statics? and explain the expression “a force of 10 lbs."

2. Assuming that the resultant of two forces, acting at a point, is represented in direction by the diagonal of a parallelogram the sides of which represent the forces in direction and magnitude; shew that the diagonal will also represent the resultant in magnitude.

(1) Forces represented in magnitude and direction by the diagonals of a parallelogram act at one of the angles, what single force will counteract them?

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(2) Can three forces which are as 3 : 4 : 1 keep a point at rest? 3. Find the centre of gravity of (1) A triangle.

(2) A circle with two-fifths of a regular inscribed pentagon cut away.

(3) How, practically, may the centre of gravity of a heavy beam be found of which one end is heavier than the other?

If it be made up of two uniform cylinders whose lengths are as 3 : 2 and weights as 3 : 5, where is the centre of gravity ?

4. Find the ratio of the power to the weight in a system of pullies where all the strings are attached to an uniform bar from which the weight is suspended, the weights of the pullies being neglected. From what point of the bar ought the weight to be suspended that the bar may rest in a horizontal position ?

A is a fixed pully, B, C heavy moveable pullies. An inextensible string without weight is thrown over A. One end of it passes under C and is fastened to the centre of B, the other end passes under B and is fastened to the centre of A. Compare the weights of B and C that the system may be in equilibrium; the strings being all parallel.

5. A force P acting up an inclined plane supports a weight W on it. If R be the reaction of the plane, prove that

P:W:R:: height of plane : length : base. A rod AB is fixed at an inclination of 60° to a vertical wall, and a heavy ring (W) slides along it. The ring is supported by a tight string which is attached to the wall. Shew that the tensions of this string, when the ring is respectively pulled up

W and pulled down the rod by a force acting along the rod,

4 are as 1 : 3.

6. What is a fluid? On what experimental facts is the science of Hydrostatics usually founded? How is the pressure at a point measured?

Compare the pressures on two horizontal areas which are as 3:5, at depths of 15 feet and 17 feet below the surface of an incompressible fluid at rest.

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