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III. PLANE TRIGONOMETRY AND MENSURATION.

(Including the Solution of Triangles.)

[N.B.-Great importance will be attached to accuracy.]

Given that sin A = 3; find the other trigonometrical ratios of A.
Trace the changes in magnitude and sign of

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4. Obtain a formula giving the value of sin in terms of sin A.

2

Show how to get rid of the ambiguities of sign when A lies between 270° and 450°.

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(i.)

sin A sin(B-C) + sin B sin(C -- A) + sin C sin (A – B) = 0,

(ii.)

sin(B-C) sin (B+C −2A) + sin (C – A) sin (C + A − 2B) + sin(A - B) sin (A + B − 2C) = 0.

6. Find an expression for sin 72°, and calculate its value correctly to three places of decimals.

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8. Explain how to solve a triangle, having given two sides and the

included angle.

Given that b = 23, c = 32, A = 63°; find a as nearly as you can, by aid of the logarithm tables supplied.

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IO. The angle of elevation of the top of a tower at a certain spot is 55°, and at a spot, in the same horizontal plane, 25 yards further from the tower the angle of elevation is 47°. Find the height of the tower.

II.

Find the area of the surface (including the ends) of a hexagonal prism, whose height is 8 ft., the base being a regular hexagon with a side of length 3 ft.

12. The radii of the internal and external surfaces of a hollow spherical shell of metal are 3 ft. and 5 ft. respectively. If it be melted down and the material formed into a cube, find an approximate value for the length of an edge of the cube.

IV. STATICS AND DYNAMICS.

[The acceleration due to gravity may be taken as equal to 32 foot-second units.]

I.

Explain how the resultant of two known forces acting on a particle may be determined by a geometrical construction.

If D be the middle point of the base BC of a triangle ABC, and the resultant of the forces represented by BA, BD be equal to the resultant of those represented by CA, CD, show that the triangle ABC is isosceles.

2. A, B, C are three points in a straight line ABC. Draw a diagram representing the directions of three parallel forces P, Q, R in equilibrium acting at these points respectively. State also the necessary and sufficient

conditions of equilibrium.

A uniform rod, 2 feet long and weighing 3 lbs., lies on a horizontal plane; find the least force which, applied 5 inches from one end, will raise that end above the plane.

3. Define the moment of a force about a point, and show how it can be geometrically represented.

The side BC of a triangle ABC is bisected in E and produced to F. Show that the sum of the moments about F of the forces represented by AB, AC is equal to twice the moment about F of the force represented by AE.

4. Show that the centre of gravity of a triangular lamina coincides with that of three equal weights placed at its angular points.

A lamina in the form of a right-angled triangle ABC is suspended from the right angle C. If CA = 2 feet, and CB = 3 feet, find the weight which must be suspended from A in order that AB may be horizontal in the position of equilibrium, the weight of the lamina being 12 lbs.

5. A smooth rod BC is passed through a small ring and placed upon a horizontal plane, with its ends attached to a fixed point A in the plane by two strings AB, AC, which are tight. A horizontal force being applied to the ring, find its direction, and also the position of the ring on the rod, in order that equilibrium may not be disturbed, the lengths of BC, CA, AB being 25, 20, and 15 inches respectively.

6. Define a machine, and point out its use as regards work done by it.

Find the greatest vertical height through which a force of 150 lbs. can raise a weight of 4 cwt. by drawing it up a smooth, sloping plank 20 feet in length.

7. Prove that, if an insect crawl along the minute hand of a clock with a velocity equal to that of the extremity of the hand, it will pass from one end to the other in 9 minutes 33 seconds nearly.

8. Explain, with examples, what is meant by a particle having two or more velocities in different directions at the same instant.

A particle has two velocities, 3u in a direction from A to B, and 8u in a direction from C to A, ABC being an equilateral triangle. Find the magnitude of its resultant velocity.

9. State the expression for the space described in time by a particle projected with velocity u, and subject to an acceleration a in the direction of projection. If v be the final velocity, find the space in terms of v, t, and a.

If u = 20 and a = 12, the units of time and space being one second and one foot, find the space described in the 4th second.

IO.

What is meant by the statement that g, the acceleration due to gravity, is equal to 32 nearly?

Show how you would roughly obtain the value of g by observing the motion of two weights P, Q, connected by a string passing round a fixed smooth pulley.

Describe fully a single observation which would give the value of g; P, Q being 47 and 49 ounces respectively.

II.

A particle is projected in any direction not vertical; explain clearly why it does not proceed to describe a straight line.

The engine driver of an express train throws a ball vertically upwards; show in a diagram the actual path of the ball.

FURTHER EXAMINATION.

V. PURE MATHEMATICS.

No

[Full marks may be obtained for about three-fifths of this paper. Candidate must attempt more than ten questions. Great importance will be attached to accuracy.]

I. A certain council consists of a chairman, two vice-chairmen, and twelve other members. How many different committees of six members can be chosen, including always the chairman and at least one of the vicechairmen ?

Show that 8190 is the total number of different committees which can be formed consisting of the chairman, one (and only one) of the vicechairmen, and at least one of the other twelve members.

2. Express as a continued fraction the quotient obtained by dividing

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3. There are six balls in a bag, each of which is known to be either white or black. The first five balls drawn are three white and two black. What is the chance that the other ball is white?

If the five balls were replaced, and a second drawing gave also three white and two black. What would the chance then be?

4. A hill in the shape of a right cone stands on a horizontal plane. At a certain point in the plane the circular base of the cone subtends a right angle, and the elevation of the summit is half a right angle. Show that the slant side of the hill, as seen against the sky, subtends 60° at the same point.

5. The lengths of the sides of a right-angled triangle are in arithmetical progression. Find the ratios of the sides, and show that the diameters of the escribed circles are in harmonical progression.

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