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II. ALGEBRA.

(Up to and including the Binomial Theorem, the theory and
use of Logarithms.)

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

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3. Define the meaning of the symbol, and from your definition

show that

na

bnb'

where a, b, n are positive integers.

Arrange in order of magnitude, a, b, n being positive integers,

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(ii.) 2ax - by = a2, bxy = a3,

(iii.) √5x-14+√3x-13 = 2√x+√2x+1.

7. Prove that a quadratic equation cannot be satisfied by more than two values of the unknown quantity.

In a certain quadratic the coefficients of x2 and x are 1 and 2 respectively, and the addition of 8 to each of the roots changes the sign but not the magnitude of the third term. Find the original quadratic, and the coefficient of x in the transformed equation.

8. Insert five arithmetic means between a - 2b and 3a + b.

The last term of an arithmetic progression is ten times the first, and the last but one is equal to the sum of the 4th and 5th. Find the number of the terms, and show that the common difference is equal to the first term.

9. Find the number of arrangements that can be made of the letters of the word infinite, (1) when they are taken all together, (2) when they are taken four together so that each arrangement has two vowels and two

consonants.

10.

Write down the first five terms in the expansion of (1 - 4x) by the Binomial Theorem, and show that the coefficient of x" may be written in the form

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and, by means of the logarithm tables supplied, find the fifth root of

66901 × 337
7824

correct to five places of decimals.

III. PLANE TRIGONOMETRY AND MENSURATION.

(Including the Solution of Triangles.)

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

I.

Given that sin A = ; find the other trigonometrical ratios of A.

2. Trace the changes in magnitude and sign of

(i.) cot, and (ii.) cos 0 - √3sin 0,

as e varies from o to 2π.

3. Prove that

4.

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A

= 2;

(iii) tan(+4)-tan(-4) = 2 tan 24.

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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10.

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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.]

1. 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.

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