MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, JUNE, 1883. PRELIMINARY EXAMINATION. I. EUCLID (Books I.-IV. and VI.). [Great importance will be attached to accuracy.] I. If two straight lines cut one another, the vertical, or opposite, angles shall be equal. ABC is a triangle, BD, CE lines drawn making equal angles with BC, and meeting the opposite sides in D and E and each other in F: prove that if the angle AFE is equal to the angle AFD the triangle is isosceles. 2. Triangles on equal bases, and between the same parallels, are equal to one another. ACB is a triangle, CD, BE parallel lines meeting AB and AC produced respectively in D and E: prove that if the triangles BCE, ACB are equal D is the middle point of AB. 3. In any right-angled triangle, the square which is described on the side subtending the right angle is equal to the squares described on the sides which contain the right angle. If the squares on the first and third sides of a quadrilateral are together equal to those on the second and fourth, the diagonals intersect at right angles. 4. 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. 5. If a straight line drawn through the centre of a circle, cut a straight line in it, which does not pass through the centre, at right angles, it shall bisect it. Two equal circles have a common chord AB. If a chord AC of one of them, equal to AB, produced backwards pass through the centre of the other, AB is equal to the radius of either circle. 6. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same part of the circumference. ABC is an isosceles triangle; on BC as base construct a triangle whose vertical angle shall be equal to either of the equal angles B or C, and which shall be similar to the given triangle. 7. In equal circles, equal angles stand on equal circumferences, whether they be at the centres or circumferences. AB is a fixed chord in a circle APQB, PQ another chord of given length; shew that if AP, BQ meet in R, R will be on the circumference of the same circle for all positions of PQ. 8. Describe a circle about a given triangle. If ABCD is a parallelogram, and BE makes with AB the angle ABE equal to the angle BAD, and meets DC produced in E, the circles described about the triangles BCD, BED will be equal. 9. If the radius AB of a circle is divided in C so that the rectangle AB, BC is equal to the square on AC, and the chord BD is equal to AC, the circle of which AD is a chord and which touches BD will pass through C, and the triangle ACD will be isosceles. IO. 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. Prove the following construction for trisecting a line AB in G and H: On AB as diagonal construct a parallelogram ACBD; bisect AC, BD, in E and F. Join DE, FC, cutting AB in G and H. II. 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. If ABC is a triangle right angled at B, and BD the perpendicular on AC is produced to E so that DE is a third proportional to BD and DC, the triangle ADE will be equal to the triangle BDC. II. ARITHMETIC. (Including the use of Common Logarithms.) [N.B.-Great importance will be attached to accuracy.] divided by 161?? 3. Multiply together 4. 61 710 31 What fraction of 2 is the quotient of 11 5. Add together 5125 of a yard, 62734 of a pole, and 018325 of a furlong; subtract the result from 0049 of a mile, and express the answer as the decimal of a yard. 9. Reduce of half a rood to the decimal off of an acre. 10. Express 9 hours 19 minutes and 3 seconds as the decimal of 9 days. II. 12. Divide 110 tons 11 cwt. 2 qrs. 16 lbs. 8 oz. by 79. Find the dividend on £9,648. 8s. at 16s. old. in the £. 13. At what rate per cent. per annum will £1,885. 15s. amount to £2,569. 6s. 84d. in 7 years and 3 months, simple interest? 14. The French unit of volume is the stère, which is a cube whose side is a mètre. Supposing a linear yard to be 32 of a mètre, find to two places of decimals the difference, in cubic inches, between the volume of a cubic yard and that of a stère. 15. A person bought an estate and subsequently sold it for £625 less than he gave for it, thereby losing 1 per cent. What should he have received in order to gain 12 per cent.? 16. The first of a series of cogged wheels, working into each other in a straight line, has a certain number of teeth; the number of teeth in the second is to that of the first as 6: 7; of the third to the second as 5 : 6; and of the fourth to the second as 2 : 3. If the wheels are set in motion, how many revolutions must each wheel make before they are simultaneously in their original positions? 17. A hare sees a hound 176 yards away from her and scuds off in the opposite direction at a speed of 12 miles an hour; thirty seconds later the hound perceives her and gives chase at a speed of 18 miles an hour. How soon will he overtake the hare, and at what distance from the spot whence the hare took flight? 18. Express, as a decimal, the square root of 6'249; also find the length of the edge of a cube of metal which cost £5,407. 8s. 11d.; one cubic inch being valued at 8s. 4d. 19. Find, by duodecimals, the cubical content of a block 2 ft. 9 ins. long, 1 ft. 8 ins. wide, and 1 ft. 4 ins. deep. If the weight of this block is II cwt., find what would be the length of a bar of the same material, having a sectional area of of a square foot and weighing 18 cwt. 20. If a pipe of 9-inch bore discharges a certain quantity of water in 6 hours, how long would 4 pipes of 6 inches bore take to discharge three times the quantity? (The rates of discharge are as the squares of the diameters of the pipes.) 21. Stock to the amount of £2,700 was sold at 90 and re-invested in. the 5 per cents. at 125; what will the annual income be? (To be worked entirely by logarithms.) 22. Find log 210, 1×log *182, and log√√13√5÷ √7. By means of logarithms find the number which is equal, to |