BRITISH HISTORY (Senior Associate Grade).

Tuesday, June 27th, 1922.- Afternoon, 2.30 to 5.30.

No candidate is allowed to answer more than EIGHT of the thirteen

questions.

The questions are of equal value.

1. Show, from the history of the seventeenth century, what reasons led men to leave England in order to found colonies.

2. How did religious differences affect political affairs in England under the Stuarts ?

3. Write an account of Oliver Cromwell's policy after he became Protector.

4. Explain why men of all parties joined together to invite William of Orange to England.

5. Why did Great Britain enter upon war with France (a) under William III and Anne, (b) under George III ?

6. Write an account of either (a) the East India Company, or (b) the Hudson's Bay Company.

7. Explain the importance of Rodney's battle of the Saints, Aboukir Bay, Plassey, Paardeberg.

8. What led to the war between Great Britain and the United States in 1812? Give a short account of the war.

9. What advantages to (a) Scotland and (b) the British Empire have been secured by the Union between England and Scotland?

10. Write an account of the Reform Act of 1832, and explain the objects of the Factory Acts.

11. What were the principal events in Europe in 1848 ?

12. What were the demands of the Chartists in England, and how far have they been secured by later legislation ?

13. Describe briefly the system of government in Newfoundland at the present time.

MATHEMATICS (a) and (d).

(Senior Associate Grade.)

Tuesday, June 27th, 1922.-Morning, 9 to 12.

Squared paper may be obtained from the Presiding Examiner.

1. Prove, by any method, that, if the same quantity h is added to x, to y, and to z, the expression a2 + y2+z2—xy—yz-zx remains unaltered; also that a2-4x+1 is a factor of x5-209x+56.

2. Find the factors of

{(a+b)(a+c)+2a (b+c)}2 — { (a−b) (a–c)}2,

and show that, if y-a, x-a, and z-a are in Geometrical Progression, then

(ii) 2√(2x2+4x+9) +2 √(2x2−3x+7)=7x+2.

4. The natural numbers being arranged in groups, 1, 2+3+4, 5+6+7+8+9, and so on, find the sum of the numbers in the nth group.

5. How many terms will there be in the expansion of (x2+2x+2)5 arranged according to descending powers of a?

What will be the coefficient of a7, and the sum of the coefficients of the even powers of x, in the expansion?

6. Obtain the first four terms in the expansion of

8. Express cosec+cot and sec+tan in terms of tan, and

hence, or otherwise, find single values of A and B which satisfy

the equations

7 -1

17

17

8

Also prove, by any method, that tan ̄ +cos

value of which satisfies the equation

9. In a triangle ABC, prove that

10. Find the least angle of the triangle whose sides are 17, 21, and 24 units long respectively..

Given, in a triangle ABC, that A = 23° 47', a = 18, and b25, find the respective areas of the two possible triangles.

11. Prove that the radius of the inscribed circle of the triangle whose sides are a+b, b+c, and c+a is a mean proportional between the radius of the circumscribed circle and diameter of the inscribed circle of the triangle whose sides are a, b, and c.

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MATHEMATICS (b) and (c).

(Senior Associate Grade.)

Wednesday, June 28th, 1922.-Morning, 9 to 12.

Recognized abbreviations may be used.

1. Prove that triangles on the same base and between the same parallels are equal in area.

A straight line parallel to the diagonal DB of a parallelogram ABCD meets CD and CB in E and F respectively. Show that the triangle ABF is equal to the triangle ADE.

2. Prove that perpendiculars drawn to the sides of a triangle through their middle points are concurrent.

A convex quadrilateral ABCD is divided by its diagonals into four triangles AOB, BOC, COD, and DOA, and the centres of the circumscribing circles of those triangles are E, F, G, and H respectively. Prove that EFGH is a parallelogram, and that HE: EF = BD: AC.

3. A is the vertex of an isosceles triangle ABC, D is the middle point of AB, and E is in AB, produced, such that BE is equal to AB. Prove that CE = 2CD.

4. Prove that in any triangle the sum of the squares on two sides is equal to twice the square on half the third side together with twice the square on the median which bisects the third side.

Prove that three times the difference between the squares on the lines drawn from the vertex of a triangle to the points of trisection of the base is equal to the difference between the squares on the two sides of the triangle.

5. Show, with proof, how to construct a square equal in area to a given rectangle.

Divide a straight line 17 units long into two parts so that the area of the rectangle contained by the two parts may be equal to that of a square whose side is 7 units long. Test your work by measurement and calculation.

6. E and F are two points in two intersecting lines AB and AC respectively, such that AE. AB = AF.AC. Prove that E, B, C, and F are concyclic.

AC and AD are two chords of a circle on the same side of a diameter AB, and when produced they meet the tangent to the circle at B, in E and F respectively. Show that the angle FDE is equal to the angle FCE.

7. Prove that the areas of triangles of equal altitude are to one another as their bases.

In the sides AB, BC, and CA of a triangle, points E, F, and G are taken, such that AE, BF, and CG are n times EB, FC, and GA respectively. Compare the areas of the triangles EGF and ABC.

Hence deduce, or show otherwise, that when n = 2 the triangle DEF is one third of the triangle ABC.

8. Obtain, accurately, a third proportional to AB and BC, sides of a scalene triangle ABC.

Through C draw CD parallel to AB, and equal to the third proportional, so that BD cuts AC in E. Through E draw EF parallel to AB to meet BC in F, and prove that EF is a mean proportional between BF and FC.

Test your work by measurement and calculation.

9. In a trihedral angle, prove that the sum of any two of the face angles is greater than the third.

Prove that if the opposite edges of a tetrahedron are equal, then the three face angles at each corner are together equal to two right angles.

10. Obtain an expression (i) for the volume of a pyramid on a triangular base; and (ii) the area of the curved surface of the frustum of a right cone, in the form of a product.