MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Moolwich, JUNE, 1895. OBLIGATORY EXΑΜΙΝΑΤΙON. I. EUCLID (Books I.-IV. AND VI.). [Ordinary abbreviations may be employed, but the method of proof must be geometrical. Proofs other than Euclid's must not violate Euclid's sequence of propositions. Great importance will be attached to accuracy.] I. If ABC, DEF be two triangles which have the sides AB, AC equal to the sides DE, DF, each to each, and also the angle ABC equal to the angle DEF; then shall the angles ACB, DFE be either equal or supplementary. 2. Define parallel straight lines, extreme and mean ratio; and draw some simple figure of a superficies which is not plane. If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles. 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. 4. If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. 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 intercepted without the triangle, between the perpendicular and the obtuse angle. ACB is a straight line, and on AC is described an equilateral triangle DAC; show that the square on DB is equal to the squares on AC and CB, together with the rectangle contained by AC and CB. 6. If two circles touch one another externally, the straight line which joins their centres shall pass through the point of contact. 7. In a circle the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle. BEAC is a semicircle whose diameter is BC; D is any point on BC; AD is perpendicular to BDC; EB is equal to AD; and Fis on DA produced so that DF is equal to AB; show that CE is equal to CF. 8. If from any point without a circle two straight lines be drawn, one of which cuts the circle and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square on the line which touches it. 9. Describe a circle about a given triangle. 10. What is Euclid's method for describing about a circle, a regular pentagon, hexagon, or quindecagon ? Show that this method would not apply to the describing about a circle of a triangle equiangular to a given triangle. II. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another. 12. Similar triangles are to one another in the duplicate ratio of their homologous sides. ADOB is the diameter of a circle whose centre is 0; C is a point on the circumference such that CD is perpendicular to AB; and EC, EA are tangents to the circle; show that triangle ECA: triangle OCB :: AD : DB. II. ARITHMETIC. [N.B. The working as well as the answers must be shown.] 1. Simplify (2 of 5) + (3 of 9) - 101. 2. Divide 7777 by 35 35. 3. In what time will the interest on £250, at 34 per cent. per annum, amount to £60. 18s. 9d.? 4. Reduce 3 cwt. 3 qrs. 21 lbs. to the decimal of 5 cwt. 5. Find the value of 5 of £13. 13s. 6d. 6. What is the rent of a farm of 246 acres, 3 roods, 24 poles, at £2. 5s. per acre? 7. Find the greatest common measure of 8775 and 12025. 8. If the water in a tank, 8 feet long and 7 feet wide, is 4 feet deep; and a cubic foot of water weighs 1000 ozs.; what is the weight in tons of the water in the cistern? 9. A certain field could be reaped by 7 men in a certain time, and 5 boys could do as much as 2 men. Find how many boys would be required, in addition to 30 men, for the reaping of a field of twice the size, in a third part of the time. 10. Show that the difference between any improper fraction and unity is always greater than the difference between unity and the reciprocal of the fraction. [You may take any improper fraction which you like to select, show that the proposition is true for that fraction; and then extend your reasoning to improper fractions generally.] II. The residue of an estate was left to be divided between three persons, A, B, C, in such proportion that A's share was to be to B's share as 4: 5, and B's share to C's as 9: 16; the residue realised £2,415. How much was each person entitled to? 12. If you invest a sum of money in such ways that on one-third of it you gain 3 per cent., on one-fifth you lose 4 per cent., and on the remainder gain 6 per cent.; what average rate per cent. do you make on the whole sum invested? 13. Ten years ago a man was three times as old as his son, and five years hence he will be only twice as old. What are their ages respectively? 14. A farmer bought 6 oxen and 100 sheep for £336; of the sheep, 4 died, and the rest were sold at £2. 7s. 6d. each; and 2 of the oxen fetched £15 each. At what price must the remaining 4 oxen have been sold if the profit on the whole transaction amounted to 5 per cent.? |