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EXAMINATION FOR FIRST APPOINTMENTS TO THE CAVALRY AND INFANTRY.

January, 1874.

EUCLID (BOOK I.)

1. If two angles of a triangle be equal to one another, the sides also which subtend, or are opposite to, the equal angles shall be equal to one another.

2. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.

3. If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.

4. Prove that the angle between the bisectors of two consecutive angles of a quadrilateral is half the sum of the two remaining angles.

5. If the square described on one of the sides of a triangle be equal to the squares described on the other two sides of it, the angle contained by these two sides is a right angle.

6. If any point P be joined to A, B, C, D, the four angular points of a rectangle, prove that the squares on PA and PC are together equal to the squares on PB and PD.

ARITHMETIC.

1. Add together 4, 18, 1, and 11.

2. Subtract 6 from 8.

3. Multiply 4 by 18.

4. Divide 17 by 3.

5. Add together 70·152, 00628, 1·905, and 3281-6013. 6. Subtract 47.6125 from 350·013.

7. Multiply 501-32 by 0063.

8. Divide 17.465 by 98.6 to three places of decimals.

9. Find the value of 6·125 of £1. 13s. 4d.

10. In 3706529 inches how many miles, furlongs, poles, &c. are there?

11. If 69 lbs. of salt cost 7s. 24d., what will be the value of 4 cwt. 1 qr. 7 lbs.?

12. Find the simple interest on £5,781 for 10 years at 4 per cent. per annum.

13. Add together 53, 3112, and 14.

14. Subtract 7 from 10.

15. Multiply together 101,,, and 11.

16. Divide 6 by 2.

17. Add together 1.375 gallons and 3.5 quarts.

18. Subtract 33.75 of a minute from 1.95 of an hour.

19. Multiply 401296 by 805.

20. Divide 2054.95 by 0563.

21. Express 7s. 93d. as the decimal of £8.

22. In 57625318 square inches how many acres, roods, &c., are there?

23. A person after paying an income-tax on £515. has £484. 19s. 7d. left, how much did he pay in the £?

L

24. A bankrupt's estate amounts to £910. 3s. 1d., and his debts to £1,875., what can he pay in the £?

25. At what rate per cent. will £425. amount to £573. 15s. in 5 years at simple interest?

26. Find by practice the value of 5cwt. 3 qrs. 21lbs. at £1. 5s. 8d. per cwt.

27. A person completes a journey of 1,056 miles in 12 days, travelling 11 hours a day; in how many days would he complete 480 miles, going 6 hours a day at the same rate?

MATHEMATICS.

(ALGEBRA, MENSURATION, LOGARITHMS).

Logarithm books will be supplied.

1. Multiply the factors (x − 3), (x+3), (x − 5), and find the values of the product when x=5, and when

x=

9.

Find the numerical value of the expression

{25+ (49)* — (64) *}.

2. Perform the operations indicated in the following examples, and in each case express the result in its simplest form :

(a) (x3- 2x2 - 23x + 60) × (x + 1)3.

(B) {x−2+√(3)} × {x − 2 − √√(3)}.

(y) (8x3 – 36x2 + 54x − 27) ÷ (2x – 3).

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4. Find the algebraical expression which is called

the least common multiple of

(1 − x3), (1 − x3), and (1+x)*.

, express the relation between a, b, c

Form a quadratic equation whose two roots shall be

- 3+2 √(-1), 3-2 √(-1).

8. A grocer finds that by mixing black and green tea together in the ratio of two pounds of black to one of green he can sell the mixture without loss at 3s. 3d. a pound, but if the mixture is made in the ratio of seven pounds of black to two of green he can sell at 3s. 2d. a pound; what was the cost per pound of the black and green tea respectively?

9. A grazier offered to sell a flock of sheep for £262. 10s., but before a purchase was completed 10 of the sheep died, and it was found that the remainder of the sheep must be each sold at an advance of 12s. 6d. in order that the grazier may receive the same sum by the sale; how many sheep were there in the flock?

10. A and B have each the same sum of money: A buys equal amounts of 3 per cent. stock at 91 and of 3 per cent. stock at 97; B invests his money equally in the purchase of the same stocks; A's income being one shilling more than B's; how much money had each?

11. In any system of logarithms shew that log 1 = 0, and that the logarithms of all fractions less than unity are negative.

Explain why to a base 10 the decimal parts of the logarithms of the numbers 500, 5, 005 may be expressed by the same digits; write down these logarithms, it being given that log,,S=9030900.

Find by means of the tables (005718)*.

12. The greatest angle of a parallelogram is 125° 30', the sides containing the angle are 95-5 and 46-5 feet respectively; find the area of the parallelogram in square feet.

13. Given one side of an equilateral triangle, find its area. Hence find the area of an equilateral and equiangular hexagon inscribed in a circle whose radius is 10 feet.