13. A river 20 feet deep and 100 yards wide is flowing at the rate of 3 miles an hour; find how many tons of water run into the sea per minute, it being given that a cubic foot of water weighs 1,000 ounces.

14. Assuming that the ratio of the perimeter of a circle to its diameter is that of 3.1416 to unity, find the volume and the surface of a cylinder, the radius of which is 4 feet and the height 100 feet.

Supposing that a wedge is cut out of this cylinder by two planes through its axis inclined to each other at an angle, which is half a right angle, what will be the volume left ?

MATHEMATICS. (GEOMETRY AND TRIGONOMETRY). 3 hrs. 1. If a straight line be bisected, and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced.

2. Divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part.

Prove that the given straight line has also been divided so that the squares on the whole and one of the parts are together equal to three times the square on the other part.

3. Equal straight lines in a circle are equally distant from the centre.

If two parallel chords AB, CD be drawn in a circle then AC and BD are equally distant from the centre.


4. Describe a circle about a given triangle.

If the triangle be right-angled, prove that the sum of the two sides containing the right angle is equal to the sum of the diameters of the inscribed and circumscribing circles.

5. Inscribe an equilateral and equiangular hexagon in a given circle.

6. If an angle of a triangle be bisected by a straight line which also cuts the base the segments of the base shall have the same ratio which the other sides of the triangle have to one another.

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

8. A and B are fixed points, and P is a moveable point such that PA always bears a fixed ratio to PB.

Prove that P lies on a circle, and find the centre of that circle.

9. Define the degree, grade, and unit of circular


The number that denotes an angle when expressed in circular measure is less by 1:5 than that denoting the number of degrees in the same angle. Find the angle.

10. Express all the trigonometrical ratios in terms of the sine.

If the cosecant be twice the sine, find the angle.

11. Prove that

cos(A + B)= cos A cos B-sin A sin B, , stating the limitations assumed in the proof as to the values of A and B.

12. Find cos 2 A and sin 3 A.


13. If any arc of a circle be divided into two equal and also into two unequal parts, prove that the rectangle of the chords of the two-unequal arcs, together with the square of the chord of the arc between the points of section, is equal to the square of the chord of half the arc. 14. Prove that in any triangle

sin A = 7e V{s (8 – a) (s – 0) (s—c)}.

s ) b)

15. In the triangle ABC

A=26° 26' b=127 and a=85. Find B.
Given log 1.27 =•1038037,

log 8.5 = .9294189.
log sin 26° 26' = 9:6485124,
log sin 41° 41' = 9.8228302,

log sin 41° 42' =9.8229721. 16. If one angle of a triangle be 65°, and one of the sides containing this angle be four times the other, find the remaining angles. Given log 2 = -3010300, log 3 = .4771213,

log cot 32° 30' = 10.1958127,
log tan 43° 17' = 9:9739602,

, log tan 13° 18' = 9:9742133. 17. The angle of elevation of a tower at a distance of 20 yards from its foot is three times as great as the angle of elevation 100 yards from the same point.

300 Shew that the height of the tower in feet is



May, 1873.



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1. Add together of }, of 4, and 11 of 2 2. Subtract 73 from 2013 3. Multiply together 43, 115, 45, and 175. 4. Divide 216 by 3ş. 5. Add together 12.07043, 7131, 754-5,6:3, and 07385. 6. Subtract 82.6874 from 701.212. 7. Multiply ·0263 by 2:36. 8. Divide •4336218 by 7362.

9. Reduce 16:08 pennyweights to the decimal of a pound troy.

10. Add together 14, 175, and 523. 11. Subtract 3.11 from 83. 12. Multiply together 24 of 141, 2 of $, and 31 of 1o. 13. Divide 43 by 12.

14. Add together 6.2 of a day and .0265 of an hour, and give the answer in minutes and the decimal fraction of a minute.

15. Subtract 6.42 of a furlong from 3•64 of a mile, and give the answer in yards and the decimal fraction of a yard.

16. Multiply .382 by •148.
17. Divide 81.27 by 481 to 3 places of decimals.



18. Reduce .0325 of £5 to the decimal of £3. 6s. 8d.

19. Reduce 1 mile 3 furlongs 10 poles 2 feet to feet.

20. If 1 ton 5 cwt. of iron cost £1. 11s. 3d., what will 1 cwt. 2 qrs.

cost? 21. Find (by Practice) the dividend on £8,976. 58. at 13s. 3d. in the pound.

22. Find the simple interest on £1,760 for 9 years at 34 per cent. per annum.

23. In 3 acres 2 roods 10 square yards how many

square feet?



24. If either 5 oxen or 7 horses will eat up the

grass of a field in 87 days, in what time will 4 oxen and 6 horses eat up the same?

25. Find (by Practice) the value of 3 tons and 21 lbs. at £12. 10s.


cwt. 26. Find the amount of £6,500 in 3 years at 5 per cent. compound interest (neglecting fractions of a penny).

27. Compute by means of the tables the value of (2.4806) (1.2701)

to four places of decimals. 28. Given log 2=:3010300, log 95-9542425, find, without using the tables, log 5, log 6, log •0216, and log(+375).

29. Extract the square root of 3915380329 and of 83193

30. Find the present value of £2,587. 188. 9d., due 5 years hence, at 34 per cent.

31. By selling goods for £817. 198. a person lost 9 per cent. on his outlay; find at what price he should have sold them in order to have gained 104 per cent.

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