6. Two particles start simultaneously from different points, in given directions, with uniform velocities. Show, by geometrical construction, how to find their relative distance at the end of any time; and determine when this distance is least. 7. Prove that the path of a projectile (in vacuo) is a parabola, and that the velocity at any point is equal to that due, under the action of gravity, to the vertical distance of the point from the directrix. A train is moving with a velocity of 30 miles an hour when a ball is dropped from the roof of one of the carriages 10 feet above the earth. Show how to find the focus and directrix of the parabolic path described by the ball, relatively to the earth. 8. Enunciate accurately Newton's Laws of Motion. A train is moving on a horizontal railroad. Assuming the weight of the train (exclusive of the engine) to be 120 tons, and the resistance arising from friction, etc., to be 10 lbs. per ton, find the tension of the couplings of the carriage which is attached to the engine, (1) when the velocity of the train is uniform, (2) when it is moving with an acceleration of 4 feet per second, per second. 9. Two imperfectly elastic spherical bodies, whose centres are moving in the same straight line with given velocities, impinge on each other: show how to find their velocities immediately after the impact. Two spheres meet directly with equal and opposite velocities, find the ratio of their masses in order that one of them should be brought to rest by the collision, (1) when perfectly elastic, (2) for coefficient of resilience e. IO. A ball moving with a velocity of 500 feet per second has its velocity reduced by 50 feet after penetrating 1 inch into a plank. Find how far it will penetrate into the plank before being stopped, assuming the resistance of the plank to be uniform. MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, JUNE, 1894. OBLIGATORY EXAMINATION. 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. The opposite sides and angles of a parallelogram are equal to one another. 2. The complements of the parallelograms which are about the diameter of any parallelogram, are equal to one another. A point E is taken in the side AB of the parallelogram ABCD and ED and EC are joined; prove that, if the line HK parallel to AB cuts ED and EC in F and G respectively, the parallelogram AHKB will be double of either of the triangles EDG or ECF. 3. 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. 4. Divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part. The line AB is bisected in C and produced to D, so that the square on CD is equal to the sum of the squares upon AB and BC, prove that the rectangle AD, DB is equal to the square on AB. Express the ratio of BD to AB algebraically. 5. State, without proof, the difference between the square on one side of a triangle and the sum of the squares of the two remaining sides. A point P is taken on the circumference of the circle APB, whose centre is O, and with P as centre, another circle, QAB, is described, cutting the former in A and B. If QM be drawn from Q, any point on QAB outside of APB, perpendicular to AB produced, prove that twice the rectangle of QM and OP is equal to the difference of the squares on OQ and OA. 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 arc. 7. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles. 8. Describe a circle about a given triangle. The straight line AB of given length moves so that its extremities are respectively upon the two fixed straight lines OC and OD meeting at O. Prove that the centre of the circle circumscribing the triangle OAB lies upon the circumference of a circle whose centre is 0. 9. Inscribe an equilateral and equiangular pentagon in a given circle. IO. If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another, and if the segments of the base have the same ratio which the other sides of the triangle have to one another, the straight line drawn from the vertex to the point of section shall bisect the vertical angle. II. If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means; and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines are proportionals. 12. The diameter BCA of the circle APQB whose centre is C, is produced through A to O, and from O the line OPQ is drawn, cutting the circle in P and Q, prove that the triangles OPA and OQB are similar, and prove that if the circle circumscribing the triangle PCQ meets OC in D then (1) The point D will be fixed for all directions of the line OPQ; (2) The ratios OA: AD and OC : CP will be equal. [N.B.-The working as well as the answers must be shown.] 3. annum? What is the interest on £775 for 3 years at 2 per cent. per 6. Find by Practice the value of 13 lbs. 7 ozs. 5 dwts. 8 grs. of silver at 3s. 9d. per oz. 7. 8. Find the least common multiple of 132, 165, 220. A cube of metal, each edge of which measures § of an inch, weighs 625 lb. What is the length of each edge of a cube of the same metal which weighs 40 lbs. ? 9. If 9 men dog of a piece of work in 14 days, working 10 hours a day, how many extra men must be employed to finish the work in 5 days more if all of them are to work only 8 hours a day? IO. (N.B. Prove that the difference of the squares of any two odd numbers is divisible by the double of their sum, and that, if the numbers are consecutive odd numbers, the difference of their squares is a multiple of 8. You may take any two odd numbers which you like to select, show that the propositions are true for those numbers, and then extend your reasoning to all odd numbers.) II. A's money is to B's money as 3:7; and if B pays A £345, the proportion will be 3: 5. How much money has each? 12. If 9 men and 6 boys can do in 2 days what 5 men and 7 boys could do in 3 days, in what time could 2 men and 5 boys do the same? |