√

the time of an

7. Given (7) the length of a simple pendulum, π oscillation, show how to find, approximately, the height of a mountain, when a seconds pendulum, by being taken from sea-level to its summit, loses (n) beats in 24 hours. If (n) = 15 what is the height of the mountain, the radius of the earth being taken as 4000 miles?

8. A weight of 200 lbs. is to be raised through a height of 40 feet by a cord passing over a fixed smooth pulley; it is found that a constant force P pulling the cord at its other end for three-fourths of the ascent communicates sufficient velocity to the weight to enable it to reach the required height; find P.

9. What is the "vis-viva" of a body in motion? How is the kinetic energy of a body expressed in terms of the vis-viva?

I

A train runs down an incline of 1 in 100 for the distance of a mile; how far will it be carried along the level at the foot of the incline, supposing all the resistances on the train to amount to 8 lbs. per ton: express the result in miles.

IO.

If a railway carriage without flanges to its wheels move on a circular curve, show how the effect of the centrifugal force may be counteracted by a rise of the outer rail, and find what the rise of the outer above the inner rail should be if the radius of the circle be 1320 feet, the velocity of the train be 30 miles an hour, and the breadth of the rail 5 feet.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

APRIL, 1885.

PRELIMINARY EXAMINATION.

EUCLID (Books I.-IV. and VI.).

[Ordinary abbreviations may be employed; but the method of proof must be geometrical. Great importance will be attached to accuracy.]

I. If a side of a triangle be produced, the exterior angle is equal to the two interior and opposite angles.

Through the vertices of a triangle ABC straight lines, falling within the triangle, are drawn making equal angles BAL, CBM, ACN: if these lines intersect, two and two, in points A', B', C', prove that the triangle A'B'C' is equiangular to ABC.

2. To a given straight line apply a parallelogram equal to a given triangle and having an angle equal to a given rectilineal angle.

3. If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line and the several parts of the divided line.

If A, B, C, D are points in a straight line, taken in order, prove that the rectangle AC. BD=AB.CD+BC.AD.

W. P.

I

I I

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

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

Through the middle points of two equal chords of a circle a third chord is drawn: show that the portions of this chord intercepted between the middle points and the circumference are equal.

6. The angles in the same segment of a circle are equal to one another.

AB is a diameter of a circle, and C a given point in AB: find a point in the circumference at which AC, CB will each subtend half a right angle.

7. Inscribe a circle in a given triangle.

If O is the centre of the circle inscribed in the triangle ABC, and AO produced meets the circumference of the circle circumscribed about the triangle in H, prove that HB and HC are each equal to HO.

8. Inscribe an equilateral and equiangular hexagon in a given circle. Prove that the area of the hexagon is of the equilateral and equiangular hexagon circumscribed about the circle.

9. The sides about the equal angles of equiangular triangles are proportionals.

Straight lines AOD, BOE, intersecting at O, being drawn so that A0=20D and BO=20E, AE and BD are drawn and produced to meet in C: prove that AC and BC are bisected at E and D.

IO. Define compound ratio.

Equiangular parallelograms have to one another the ratio which is compounded of the ratio of their sides.

11. The rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle is equal to the two rectangles contained by the opposite sides.

[N.B. Great importance will be attached to accuracy.]

that he had remaining of what A then had.

How much had B at first?

4. What fraction multiplied by 3 of produces 2 of ?

5.

Add together 3'40765, 537 063 and 84379.

from 6010473.

Subtract the result

6. Find the area of a square slab, the side of which measures 2868 of

a metre.

7. Divide 157 505 by '03706.

8. Subtract of 2o of £3. 6s. 6d. from *0475 of £100.

9. Divide 27 by 75'75 and express the answer as a decimal.

10. Express the difference between 3 guineas and £17625 as the decimal of 5 shillings.

II. If with sales amounting to £1000 a tradesman gains

100 in

7 months, what sales would he have to make in order to gain £60. 105. in II months?

12.

What would be the amount of a tax on £575 at 11d. in the £? 13. At what rate per cent. per annum must £5,750 be invested so that the simple interest on it may amount to £646. 175. 6d. in 2 years?

14. Find the square root of the least integer that is a common multiple of 1191, 2211, and 1.

15. A person (C) buys an estate, and some time afterwards disposes of it to another (D) for £5,400. At this price it yields D 23 per cent. C previously obtained 3 per cent. for his investment, but his income, owing to less careful management was £36 less than D's. What did C originally pay for it?

16. A druggist buys a certain commodity by avoirdupois weight and sells it by troy weight. The buying price is 3d. per oz., and he sells at 6d. What is his gain per cent?

17. Strong spirit is mixed with inferior spirit, valued at 5s. per gallon, in the proportion of 6 to 1. The mixture is then worth 9s. the gallon. Find the value of a gallon of the strong spirit.

18. The annual average depth of rainfall for the three years 1879, 1880, 1881, at a certain place was 24'98 inches; the succeeding three years it was 29.62. The year 1883 was the rainiest, when there fell 48 inches more than in 1882, 6.36 inches more than in 1884, and 7°47 more than in 1880. The year 1881 was short of the preceding year by only 17 inches. Find the depth of rain that fell in each of the six years.

19. A fundholder directs his broker to purchase eight £100 shares in a certain mine, quoted at 272 per share. To accomplish this he authorises him to sell out £850 stock in the 3 per cents. at 95§, and £1300 stock in the 4 at 1053. The broker's charge on each of the three transactions is th per cent. What had the broker to receive upon the whole?

23. A person borrowed £11,000 for two months at 5 per cent. per annum. At the end of the time the interest was added on and the debt renewed for another two months. This was continually repeated till at the end of two years the debt and interest were paid. Find by logarithms what this debt and interest together amounted to.

Given logarithms

log 2=0*3010300

log 7=0*8450980 log 13=11139434 log 12151=20846120

Diff. 357.

logo11=20413927 log 90=19542425 log 217°47=23373994 Diff. 200.