8. One end of a string is fixed; it then passes under a moveable pulley to which a weight, W, is attached. The string then passes over a fixed pulley, and a (smaller) weight, P, is attached to its other end, all three sections of the string being vertical. Show that, neglecting the masses of the pulleys, the acceleration with which W descends is

Verify this result when P is small compared to W; and when W is small compared to P.

9. A bead is strung on a thread whose two extremities are fixed, and on which it can move without friction; show that if the bead be started so as to move in the elliptic arc to which it is constrained, its velocity will be uniform (gravity not acting).

Show also that the tension of the thread, for different positions of the bead, varies inversely as the product of the two focal radii vectores.

IO.

Define the work done by a force when it continues to act while its point of application moves over a given space in the direction of the force. If that point moves at an angle with the force, what is the work done?

A mass m is moving with velocity u. A constant force acts on it in the direction of the motion until its velocity is increased to v. Prove that the work done by the force is

mv2 -- \mu2.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

JUNE, 1887.

PRELIMINARY EXAMINATION.

I. 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. Upon the same base and on the same side of it there cannot be two triangles which have their sides that are terminated at each extremity of the base equal to one another.

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

In the triangle ABC, AD and AE are drawn from the vertex A to the base BC, making the angle BAD equal to C, and the angle CAE equal to B: prove that the perpendicular from A upon BC bisects DE.

3.

Describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle.

W. P.

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4. If a straight line is divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts, together with the square of the part between the points of section, is equal to the square of half the line.

Show that the greatest right-angled triangle, which has the sum of the sides containing the right angle equal to a given straight line, is isosceles.

5. If a straight line touches a circle, the straight line which joins the centre to the point of contact is perpendicular to the touching line.

ABCD is a straight line; circles are described on AB and CD as diameters; and a common tangent to the circles is drawn, meeting them in E and F. Prove that the triangles AEB and CFD are equiangular to one another.

6. The angle in a semi-circle is a right angle, and the angle in a segment greater than a semi-circle is less than a right angle, and the angle in a segment less than a semi-circle is greater than a right angle.

7. If two straight lines cut one another within a circle, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other. Prove this in the one case only in which one of the straight lines passes through the centre and cuts the other, which does not pass through the centre, not at right angles.

AB, AC are equal chords of a circle; and AED is any chord through A, cutting BC in E; prove that the rectangle AD.AE is equal to the square of AB.

8. Inscribe an equilateral and equiangular pentagon in a given circle.

The five lines, other than the sides of the figure, which can be drawn connecting the angular points of an equilateral and equiangular pentagon include between them a pentagon which is itself equilateral and equiangular.

9. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or these produced, proportionally; and, conversely, if two sides, or these produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle.

Straight lines AOB, COD, intersect in O, and AO: OB :: CO : OD; if P, Q, are the middle points of AB, CD, prove that PQ is parallel to AC and BD.

10. Equal triangles, which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and, conversely, triangles which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.

II. From the vertex A of a triangle ABC, AD and AE are drawn to the base, making the angle BAD equal to the angle CAE: prove that the rectangle BD. BE : the rectangle CE . CD :: AB2 : AC2.

12. Describe a rectilineal figure which shall be equal to one and similar to another given rectilineal figure.

II. ARITHMETIC.

(Including the use of Common Logarithms.)

[N.B.-Great importance is attached to accuracy.]

I. What is the smallest number which, when subtracted from 99099, will make it exactly divisible by 909?

2. What will 37 lbs. of nutmegs cost at 1s. 6d. per oz.?

3. What is meant by (a) the numerator, (b) the denominator of a fraction? Prove that =§, and add together +12+24.

4. If 13 yards of cloth cost £2. os. Id., what must be given for 75 yards? Do this by Rule of Three, and explain the principle on which your statement is made.

5. If £150 gain £3. 7s. 6d. in 9 months, what sum will gain £3 in 12 months?

6. Find by Practice the price of 22 cwt. 3 qrs. 21 lbs. at £2. 9s. 6d. per cwt.

7. Multiply 0047 by '00035, and prove the correctness of your result; and divide 00918 by '018.

8. Find the value of 05854 of a guinea, and reduce 3 weeks 4 days 5 hours 6 minutes 6 seconds to the decimal of a month.

9. Extract the square root of 63409369, and the cube root of 66430*125.

IO. The interest of £25 for 3 years at simple interest was found to be £3.185.9d.; what was the rate per cent.?

II. What is the difference between the compound interest on £100 for 2 years, according as the interest is paid yearly or half-yearly, 4 per cent.?

12. Define Present Worth and Discount: and find the proportion between the interest of £100 for 4 years, at 61 per cent., and the discount of the same sum payable at the end of 4 years, at the same rate of interest.

13. A person sells an estate worth £1200 per annum for 25 years' purchase, and after deducting 1 per cent. for expenses invests the remainder in North-Eastern Consols at 155: allowing 2s. 6d. per cent. for brokerage, find the amount of stock he will receive. If he gets 7 per cent. on his investment what will be the difference of his income, supposing the management of his estate to have cost him 10 per cent. of the rental?

14. What are meant by duodecimal fractions? By means of duodecimals, find the area of a rectangle 2 ft. 3 in. by 5 ft. 7 in., and express your answer in square feet and inches. And find the cost of glazing a window containing 60 panes, each of which is 1 ft. 2′ 3′′ by 11′ 5′′, at 3s. 8d. per square foot.

15. A tradesman forms a mixture of tea by adding 1 lb. at 3s. and 1⁄2 lb. at 4s. 6d. to every 2 lb. at 2s.; what must he sell it at per lb. in order to gain 10 per cent.?

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16. A rectangular wooden box exactly contains 6 iron balls, each 10 inches in diameter, packed in sawdust: the wood is 1 inch thick. Find the weight of the whole; given that a cubic foot of wood or sawdust weighs 38.4 lbs.; a cubic inch of iron weighs 4'56 oz.; and that the volume of a sphere is 52 times the cube of its diameter.

17. What fractions produce circulating decimals? What kind of a decimal is √2 equivalent to? Convert '054 into a fraction, and explain your method of doing so. Multiply together 4, 4, 0004, and find the result to the end of the first recurring period.

18. Two clocks are together at 12 o'clock: one loses 7" and the other gains 8" in 12 hours: when will one be half-an-hour before the other, and what o'clock will it then show?

19. Find the value of 16 √3+1034-4√√12-3√108 to 2 places of

decimals.

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