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2. Multiply also divide

x2-ax3+a3x-a4 by x2+ ax + a2;

p1-9pq3 +18q* by p2-3pq+3 q2.

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4. Find the G. C. M. of

03-6x-4 and 3x3-8x+8;

also the L. C. M. of

(3a2-3ab)2, 18 (a3b2-ab1), and 24 (a3b3—bo).

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7. The sum of two numbers is 35; and their difference exceeds one fifth of the smaller number by 2; find the numbers.

8. After 12%. has been divided equally among a certain number of men, an additional shilling apiece is given to them; and it is then found that each possesses as many shillings as there are men. Find the number of the men. 9. Prove that if b be a mean proportional between a and c, then a2+262: a :: 62 +2c2: c.

10. Sum to 6 terms the series

+11 + 3 + ..... .

Also insert 12 arithmetic means between - and 5.

FRIDAY, MAY 28, from 2 to 4.30 P. M.

III. 6. Mathematics. (Second Paper.)

[N. B. Candidates are reminded that in order to pass in this subject they must satisfy the Examiners in Euclid I, II.]

Euclid I, II.

1. Define a line, a straight line, an angle, a right angle, a triangle, an acute-angled triangle, a parallelogram, and a rectangle.

What is Euclid's criterion of the equality of magnitudes ?

2. Make a triangle of which the sides shall be equal to three given straight lines, any two of which are greater than the third.

3. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square on the whole line.

4. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel.

5. If a straight line be divided into any two parts, the squares on the whole line and on one of the parts are equal to twice the rectangle contained by the whole and that part, together with the square on the other part.

6. 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.

7. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side on which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle.

2. Euclid III, IV, VI.

8. Angles in the same segment of a circle are equal to one another.

9. Define-similar rectilineal figures, and reciprocal figures. If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular to one another, and shall have those angles equal which are opposite to the homologous sides.

10. When is a rectilineal figure said to be described about a circle?

Describe an equilateral and equiangular pentagon about a given circle.

11. Define―ratio, proportionals, and ́ex æquali.'

If four straight lines be proportionals, the similar rectilineal figures similarly described on them shall also be proportionals.

In the quadrilateral figure ABCD, AB: BC :: CD: DA, and angle Bangle D. Prove that ABCD is a parallelogram.

12. If from any point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it, and if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square on the line which meets the circle, the line which meets the circle shall touch it.

13. If two circles cut one another in A and B, and from B straight lines BEC, BFD are drawn cutting the circumferences in E, C and F, D; the triangles AEC, AFD are similar.

SATURDAY, MAY 29, from 2 to 4.30 P. M.

III. 6. Mathematics. (Third Paper.)

[N.B. No credit will be given for any answer, the full working of which is not shewn.]

1. Plane Trigonometry and use of Logarithms. 1. Obtain a formula connecting the circular measure of an angle with the measure of the same angle in degrees.

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Assuming that = 3, find the length of an arc of a circle of radius 3 yards 1 foot which subtends at the centre an angle of 6° 18′.

2. Prove that tan Atan (180°-4).

Determine the values of the trigonometrical functions for an angle of 30°; and find the values of cot 240° and cosec 6oo.

3. Prove that

A

(1) sin (4-B) = sin 4 cos B-cos A sin B;
(2) cos 34 = 4 cos3 A-3 cos A.

I

Given that cos A = find the value of sin

9

4. Establish the identities:

(1) (1+ sec 2 4) (1+ sec 44) = tan 44 cot 4;
(2) cos A+ sino A = 1- sin2 2 A ;

(3) sin 4+ sin B-sin C = 4 sin

where A+B+C = 180°.

A

B
sin COS

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5. Define the characteristic of a logarithm, and shew how the characteristic of a logarithm to base 10 is determined.

25.

Given log 2.301030, log 3 = 477121, calculate to five places of decimals the logarithms of 6.48, 0162, and 3 6. Investigate a formula giving the cosine of an angle of a triangle in terms of the sides.

ABC is a triangle in which = 60°, AB = 3, AC = 2; A AD is the perpendicular drawn from the vertex A on the side BC. Find (1) the length of AD, (2) the difference between BD and CD, (3) the value of cot B cot C.

7. In a triangle ABC, whose sides are a, b, c, shew that

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If A = 33°, C= 130°, b = 25, find a ; having given

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8. The angles of elevation of the top of a tower are observed from two points A and B, respectively due North and due West of the tower, and are found to be a and ß. A and B are on the same level with the base of the tower, and their distance apart is a. Find the height of the tower.

2. Mensuration.
(π = 37).

9. A rectangular field is 62 chains 50 links long and 32 chains wide; by how much does its area exceed that of a square field whose side is 42 chains?

10. Assuming that a cubic foot of water weighs 1000 oz., find the weight of water which can be contained in a hollow sphere whose diameter is 3 ft. 6 in.

11. Find to three places of decimals the volume of a pyramid whose height is 50 feet, and whose base is a regular hexagon of which each side measures 3 yards.

12. A cube of metal, each of whose edges is 9 inches long, is melted down and recast in the form of a circular cylinder whose height is 154 inches; find the radius of the base of the cylinder, and the area of its curved surface.

THURSDAY, MAY 27, from 5.30 to 8 P.M.

III. 7. Mechanics and Mechanism.

[N. B. More credit will be given to a few questions answered fully than to a greater number answered imperfectly. The answers are to be illustrated by diagrams or drawings, where these can be introduced.]

1. Enunciate the 'parallelogram of forces;' and, assuming it to be true for direction, prove it for magnitude.

2. A circular disc, suspended by its centre, has strings from the centre hanging down over the edge and supporting weights of 20 oz., 21 oz., and 29 oz. respectively; the disc rests horizontally. Find the inclination to each other of the strings supporting the lighter weights.

3. Explain-moment of a force about a point.' Compare the moments about a point of a force of 12 oz. acting in a line whose shortest distance from the point is 2 ft. 6 in., and of another of 5 oz. acting at a distance of 3 ft.

4. Define centre of gravity.'

Four particles, A, B, C, D, are placed one at each corner of a square. Two of them, A, C, placed at opposite corners, weigh each twice as much as B; and D weighs three times as much as B. Find the position of the centre of gravity of the particles.

5. Distinguish the three kinds of levers, giving examples of each kind.

6. How are velocities resolved and compounded? A stone is projected horizontally, with a velocity of 16 feet per second, from a steamer which is moving at the rate of 12 miles per hour in a direction perpendicular to that in which the stone is projected. Find the initial velocity of the stone.

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