15. Prove that the area of any triangle whose sides are a, b, and

с

is

√s (sa) (sb) (s—c)},

s being the semi-perimeter.

16. In the triangle ABC

A = 60° 15′, B= 54° 30′, and AB=100 yards,

find AC having given

L sin 54° 30' 9.9106860,

=

L sin 65° 15'9.9581543,

log 896461.9525317.

17. The sides and the included angle of a triangle are 327 feet, 256 feet, and 56° 28' respectively, find the remaining angles given

18. ABC is a triangle and D the middle point of the base BC, prove that the sine of BAD is

b sin A

√(b2 + 2bc cos A + c2) *

EXAMINATION FOR FIRST APPOINTMENTS TO THE CAVALRY AND INFANTRY.

August, 1873.

EUCLID. (BOOK I.). 1 hr.

1. Any two sides of a triangle are together greater than the third side.

2. Construct a triangle of which the sides shall be equal to three given straight lines, but any two whatever of these must be greater than the third.

3. Parallelograms on the same base, and between the same parallels, are equal to one another.

4. To a given straight line apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

5. Define a rhombus, and show that its opposite angles are equal.

6. If a straight line be drawn through A, one of the angular points of a square, cutting one of the opposite sides, and meeting the other produced at F, show that AF is greater than the diagonal of the square.

5. Add together 8.376, 06703, 37·04, and 28·064015. 6. Subtract 28.4532 from 320.6.

7. Multiply 4-2635 by '068.

8. Divide 8.4605 by 248 to three places of decimals. 9. Find the value of 3.75 of £2. 14s. 8d.

10. In 1250784 seconds how many weeks, days, &c. are there?

11. If 3 tons of iron cost £16. 10s., what will be the value of 3 cwt. 2 qrs. 14 lbs.?

12. Find the simple interest on £732 for 8 years at 34 per cent. per annum.

13. Add together 17, 3, and 2.

14. Subtract 5 from 101.

5

15. Multiply together 13, 2, 81 and 22.

16. Divide 218 by 41.

17. Add together 2.16 of a yard and '05 of a mile.

18. Subtract 4:42 of an hour from 3.64 of a day.

19. Multiply 506 by ·632.

20. Divide 13.915 by 575.

21. Express 13s. 4d. as the decimal of £4.

22. In 8156704 cubic inches, how many cubic yards and cubic feet are there?

23. When the income-tax was reduced from 6d. to 4d. in the pound, a person found that he had £42. 3s. 9d. less income-tax to pay. What was his income?

24. In what time will £230. 15s. amount to £305. 14s. 101⁄2d. at 24 per cent. simple interest?

25. Find the dividend on £2,560. 5s. at 12s. 8d. in the £.

26. If 7 men receive £17. 10s. wages for 15 days, what will be the wages of 30 men for 6 days?

C

MATHEMATICS. 3 hrs.

1. Multiply 3-3x2+11x-15 by x2-5x+3, and find the value of the product when x=8. Resolve into factors the expressions

and

and

2. Divide:

(a) (5x+4a)2 - (4x + 5α)2;

(B) x2 - (a + b) x + ab ;

(y) x2-18x+77.

3

(B) x3+y3+z" -3xyz by x+y+z;

(V) x* - 7x*y2+y* by x2+3xy + y2.

3. Find the highest common factor and the least common multiple of the expressions

x3-7x2 + 33x 63 and x3- 6x2 + 30x - 63.

Find the value of r when the two expressions x2 - 9x+20, and x2 - (r + 3) x + 3r have a common factor.

5. Find the square of √(3) − 4 √(5) + √(15), and the square root of 152 — 30 √(15).

Calculate to four places of decimals the value of the expression

6. Prove that the sum of the cubes of three consecutive numbers is divisible by the sum of the numbers, and that the difference between the fourth powers of the greatest and least of the three numbers is divisible by eight times the middle number.

8. A number of two digits is such that, if the digits be inverted, the new number is less by 9 than the original number, and the sum of the squares of the digits is 113; find the number.

9. A body of 684 soldiers is to be drawn up in the form of a hollow square, three deep; find the number of men in the front of each side of the square.

10. Define the logarithm of a number to a given base, and find the logarithms of 10000 to the base 10 and to the base (10), and also of 2401 to the base 7, and to the base

1

(7)*

Find from the tables log, (487 9634), and, by help of the tables, find an approximate value of

(3·7452)3 ÷ (3·7121)*.

11. The sides of a right-angled triangular field, adjacent to the right angle, are 357 feet and 476 feet; find the length of the third side of the field, and the number of square feet in its area.

12. The three different edges of a rectangular parallelopiped are respectively 3 feet, 2·52 feet, and 1·523 feet in length; find the number of cubic feet in the parallelopiped.

Find also the cubic space inside a box of the same external dimensions, constructed of a material one-tenth of a foot in thickness.