5. Prove that in every circle the angle subtended by an arc equal to the radius is an invariable angle. Express that angle in degrees and decimals of degrees, and find the circular measure of one minute. On a circle 10 feet in radius it was found that an angle of 22° 30' was subtended by an arc 3 feet 11 inches in length; hence calculate, to four decimal places, the numerical ratio of the circumference of a circle to its diameter. 6. If a, b, c be the sides subtending respectively the angles A, B, C of a plane triangle,

7. If (7) be the radius of the circle inscribed in the triangle ABC, prove

8. In a right-angled triangle ABC, C being the right angle, find AB, if B=30° and BC= 100 feet.

A'B'C' is also a right-angled triangle, C' the right angle and B' = 30°, find A'C' when the area of the triangle A'B'C' is three times the area of ABC.

9. When two sides and an included angle of a triangle are given, investigate a formula for determining the other two angles. Shew that for this determination it is not necessary to know the absolute lengths of the two sides, provided their ratio is given.

Ex. The included angle is 70° 30′, the ratio of the containing sides is 5 3, find the other angles.

16. When is one number prime to another?

Find the greatest common measure and the least common multiple of 7560, 27720, and 108108.

17. Find the length of the edge of a cubical block of stone containing 46 cubic yards 513 cubic inches, and the number of square inches in its entire surface.

18. One gallon of spirit which contains 11 per cent. of water, is added to 3 gallons containing 7 per cent. of water, and to this mixture half a gallon of water is added. Find the per-centage of water in the mixture.

19. A buys 3 per cent. stock at 89%. He receives one half-year's dividend, and afterwards sells his stock at 949, and finds that he has gained £54. What sum did he originally invest?

20.

Find the true discount on £142. Is. 9d. due 18 months hence at 3 per cent. per annum.

21. Find, correct to a farthing, the present value of £10,000 due 8 years hence at 5 per cent. per annum compound interest.

22.

Find log 10, and calculate to six decimal places the value of

[N.B.-Great importance will be attached to accuracy in numerical results.]

I. Find the Greatest Common Measure of

a (a− 1) x2 + (2a2 − 1 ) x + a (a + 1)

(a2 - 3a+2) x2 + (2a2 − 4a + 1 ) x + a (a − 1).

and

x3- x2+3x+5, when x=1+2=1;

and prove that (√3 + 1)2 - 2 (√2 − 1)2 = √59-24√√6.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

JULY, 1880.

PRELIMINARY EXAMINATION.

I. GEOMETRY.

I. Define a straight line, an angle, and a triangle.

Distinguish between equal triangles and triangles which are equal in all

respects.

2. 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 the one shall be equal to the angle which is contained by the two sides equal to them of the other.

Shew also that the two triangles are equal in all respects.

3. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts are themselves equal and parallel.

Shew also that the straight lines which join the extremities towards opposite parts, bisect each other.

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

Shew that if a parallelogram be a rhombus, each of the parallelograms described about its diameter will also be a rhombus.

5. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by the side on which, when produced, the perpendicular falls and the straight line inter

W. P.

I

I

cepted without the triangle, between the perpendicular and the obtuse angle.

Describe a square whose area shall be three times the area of a given

square.

6. If a straight line touch a circle, the straight line drawn from the centre to the point of contact shall be perpendicular to the line touching the circle.

Prove that the base of any segment of a circle makes equal angles with the diameter drawn through one extremity of the base, and with the perpendicular let fall from that extremity upon the tangent at the other extremity.

7. In equal circles, equal angles stand on equal circumferences, whether they be at the centre or circumference.

Describe a circle cutting the sides of a given square in eight points, such that they shall be the angular points of a regular octagon.

8. Two straight lines AB, CD within a circle, one of which passes, and the other does not pass, through the centre, intersect in O; shew that the rectangle AO, OB is equal to the rectangle CO, OD.

Segments of circles are described on a given base, and from a fixed point in the base produced a tangent is drawn to each segment. Shew that the points of contact all lie on the circumference of a circle.

9. Give constructions (without proof) for

(1) inscribing a circle in a given triangle,

(2) describing a circle about a given square,

(3) inscribing an equilateral and equiangular pentagon in a given circle.

Shew that any equilateral figure inscribed in a circle must also be equiangular.

JO. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those sides produced, proportionally.

Shew that two straight lines drawn from two angular points of a triangle to the middle points of the opposite sides enclose with those two sides a quadrilateral, whose area is one-third of the area of the triangle.

II. If from the vertical angle of a triangle a straight line be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle.

[Name your authority in all cases where you do not follow Euclid.]

II. ARITHMETIC.

(Including the use of Common Logarithms.)

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

I. Add together 8 of 3, 91 of 18, 39 of 1%, and 68 of 15.

3. Multiply together of 1%, by 2% of 6 of 1015.

5. Add together ‘023 of a £, '946 of a shilling, and 3'48 pence, and subtract the sum from 26 of a guinea. Give the answer in pence and the decimal of a penny.

6. Multiply 36.2894 by 00893.

7. Divide 99'994 by 2890.

8. Divide 42 547 by '00542.

9. Express 3 ozs. 7 dwts. 12 grs. as the decimal of a lb. troy.

IO. What is the difference between 038 of a mile and of a furlong? Express the answer as the fraction (vulgar) of a furlong.

II. Find, by Practice, the value of 3 tons 5 cwt. 2 qrs. 21 lbs. at £12 per ton.

12.

Divide £56 between A, B, C, and D in the ratio of the numbers 3, 5, 7, and 9.

13. In what time will £345 amount to £454. 25. 1d. at 2 per cent. per annum simple interest?

14. Two horses can plough in a given time as much as 3 oxen, but the cost of 4 oxen is only equal to that of 3 horses, the daily cost of a horse being 35.

A certain field can be ploughed by 3 horses in 8 days. What would be the cost of ploughing it by oxen in 6 days?

15. The value of a certain house in 1880 has increased 35 per cent. since 1877. The house was rated in 1877 at two-thirds of its value, and in 1880 it is rated at three-fifths of its value, the rate in the £ remaining the same. Compare the rate paid in 1877 with that paid in 1880.