EXAMINATION OF CANDIDATES FOR COMMISSIONS IN THE ROYAL MARINE ARTILLERY AND LIGHT INFANTRY.

August, 1873.

MATHEMATICS, SECTION I.

ARITHMETIC AND LOGARITHMS. 3 hrs.

5. Add together 3·07, 006, 14·13, and ·00179.

6. From 3.062 subtract 1·0043.

7. Multiply 8.64 by 25•

8. Multiply 3.02 by 1.0201.

9. Divide 0304 by 64.

10. Add together 33, 108, 18, and 11.

14. Express £3. 2s. 6d. as a fraction of 8 guineas.

15. In 2 acres 1 rood 4 perches how many square

yards?

16. Divide 1 by 236 to 5 places of decimals.

17. Convert 965 into a vulgar fraction in the lowest. terms.

18. Express

as a recurring decimal.

19. Find the number of seconds in 869 of a day.

20. Find the value of 565 of a cwt., at 1s. 3d. the pound.

21. Divide 6607 by 228.

22. Find the simple interest on £325 for 8 months at 6 per cent. per annum.

23. If a commission of three pence in the pound on a sum of money is £3. 16s., what is the sum of money? 24. If 2-qrs. 13 lbs. cost £10. 18s. 6d., what will a ton cost?

25. Find the square root of 59136100.

26. Employ logarithms to compute ('030672) to six places of decimals.

27. Find by logarithms the compound interest in 7 years on £326. 15s., at 24 per cent. per annum.

28. By logarithms divide 1237 by '00864.

29. What sum is reduced by a discount of 3 per cent. to £36. 7s. 6d.?

MATHEMATICS, SECTION I. EUCLID. 3 hrs.

1. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal, the angle which is contained by the two sides of one of them shall be equal to the angle contained by the two sides equal to them of the other.

2. Define a parallelogram.

The complements of the parallelograms which are about the diameter of any parallelogram are equal to one another.

In the case of a square, if the squares about the diameter are 169 and 625 square feet in area, what is the area of each complement?

3. If a straight line be bisected and produced to any point, the square on the whole line thus produced and the square on the part of it produced, are together double of the square on half the line bisected, and the square on the line made up of the half and the part produced.

4. Of all straight lines in a circle drawn through a given point within it, the longest is the diameter of the circle, and the shortest is the straight line perpendicular to this diameter.

5. When are segments of circles similar?

Upon the same straight line and upon the same side of it there cannot be two similar segments of circles not coinciding.

6. If from a point without a circle there be drawn two straight lines, one of which cuts the circle and the other meets it, 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 it, the line which meets shall touch the circle.

7. When is a straight line said to be placed in a circle?

In a given circle to place a straight line equal to a given straight line not greater than the diameter of the circle.

8. To inscribe a square in a given circle.

9. If a straight line be drawn to cut two sides of a triangle proportionally, this line shall be parallel to the remaining side of the triangle.

10. Define duplicate ratio.

Similar triangles are to one another in the duplicate ratio of their homologous sides.

11. Define a plane. State when a straight line is perpendicular to a plane, and when two planes are perpendicular to one another.

Planes to which the same straight line is perpendicular are parallel to one another.

MATHEMATICS, SECTION I.-ALGEBRA. 3 hrs.

1. If x = 4 and y = 3, find the value of

(x2-5)2 from (x − 3) (x − 1) (x + 1) (x+3).

3. Divide

(a+b)2 − 3 (a + b) √(ab)+2ab by {√(a) —√√(b)}”.

4. Find the lowest common multiple of the two expressions

x2+6x+5 and x3

7. What is the number which exceeds the sum of

its fourth and fifth parts by 33?

8. Divide the number 20 into two such parts that the square of the greater shall exceed the square of the less by 80.

Prove that if any number be divided into two parts, the sum of the squares of these parts is least when the parts are equal.

9. A railway train travels 288 miles at a certain average speed, and if the speed were 4 miles an hour greater, would perform the journey in an hour less. What is the average speed of the train?

10. How is the ratio of two quantities measured algebraically? If a, b, c, d be in proportion, prove that a2 +3b3, a2 — 2b2, c2 +3d2, c2 — 2d2 are also in proportion.

11. Find the sum of 12 terms in an arithmetical progression where the third term is 4 and the seventh term is - 4.

MATHEMATICS, SECTION I. 3 hrs.

1. Solve the simultaneous equations:

4 (x3 + y3) = 7x2

2 (* +y)=5

2. Prove that it is impossible to take any number of terms of the progression 1+8+6+4+... starting with the first, so that the sum may be 4.

3. In the expansion of (a + x)" by the Binomial Theorem, find the greatest coefficient and the greatest term when a = 8, x = }, and n =

10.

4. Prove that an angle has but one cosine; but if a decimal between 1 and 1 be presented, it is the cosine of any one of an unlimited number of angles. What are called in the tables "natural cosines”?