If P and P' be the points on 12-ax=0 and x2-by=0 the tangents at which are parallel to this common chord, and S, S' be the foci of the parabolæ, then SP is perpendicular to S'P'. 6. If S and S' are the foci, SL the semi-latus rectum of an ellipse, and LS' produced cuts the ellipse at X, show that the length of the ordinate of (1 - 22)2 X is a where 2a is the length of the major axis and e is the eccentricity I + 3e2 of the ellipse. 7. Find the equation of the tangent to the hyperbola point (x, y). If 1, 2 be the ordinates of the points in which this tangent cuts the 8. A smooth pencil, whose cross-section is a circle of radius r, is laid upon a flat and smooth ruler whose breadth is 2c, so that its axis is parallel to the edge of the ruler; the two are now bound together by a string which, passing under the ruler and over the pencil, lies in a plane perpendicular to the axis of the pencil; show that the pressure between the pencil and the ruler is 9. 4rc 22+c2 T where 7 is the tension of the string. Enunciate the laws of statical friction. A pair of step-ladders stands on a rough horizontal plane with each of its ladders on the point of slipping; show that the feet of the ladders are at a distance apart, where μ is the coefficient of friction, and is the length of each ladder. (The centre of gravity of each of the ladders is to be supposed to be half way up the ladder.) 10. Distinguish between stable, neutral, and unstable equilibrium. A uniform beam AB, whose weight is W, is moveable in a vertical plane about a hinge at A; to the end B, one end of a string which carries a weight Wat its other end is attached; the string is passed over a pulley at a height h (h> AB) vertically over A; determine whether the beam is in stable or unstable equilibrium when it is in a vertical position above A. II. Explain the meaning of the terms Work, Kinetic Energy. Find the greatest velocity (in miles per hour) at which an engine of 192 horse power can pull a train of 150 tons mass along the level, the frictional and other resistances to the motion being 16 pounds weight per ton. (A horse-power is 550 foot-pounds per second.) 12. State the laws which determine the velocities, after impact, of two smooth elastic spheres whose velocities before impact are known. A smooth billiard ball strikes another (of equal size) which is at rest, and the direction of the centre of the former makes, just before impact, an angle of 30° with the line joining the centres of the two balls; find the tangent of the angle through which its direction of motion is deflected by the impact, the coefficient of restitution of the balls being o'4, and the motion being supposed to take place upon a smooth billiard table. ་ VII. PART III. SECOND PAPER. [Full marks will be given for about two-thirds of this paper. The accelera tion due to gravity may be taken to be 32 foot-second units.] 1. Having given a focus, and one tangent of a parabola, and the length of the latus rectum, find the point of contact of the given tangent. 2. Prove that any tangent of a hyperbola cuts off from the asymptotes a triangle of constant area. If the tangent at P meets an asymptote in Z, and the perpendicular from the centre C on the tangent FL is equal to the perpendicular from Z on CP, the asymptotes are at right angles. 3. The equation of a straight line being expressed in the form interpret the geometrical meaning of either member of the equation. Straight lines parallel to y=kx are drawn to cut the two given lines y = mx and y = nx; find the equation of the locus of the middle points of the parallel intercepts. 4. Find the equation of the two straight lines drawn through the point (1, 1) which are parallel respectively to the straight lines given by the equation x2+5xy + 2y2 = 0. Find also the equations of the two diagonals of the parallelogram formed by the four lines. 5. Form the equation of a circle of radius a touching both the axes of co-ordinates, and lying in the quadrant in which both co-ordinates are positive. Find the angles which a chord PQ passing through the origin O must make with the axes in order that OQ may be bisected at P. a 6. Find the equation of the tangent to the parabola y2 = 4ax at the point 2a m Prove that, if two tangents are drawn from a point on the line x = their chord of contact subtends a right angle at the vertex. -4a, 7. Form the equation of the diameter of the ellipse x2/a2+y2/b2 = 1 which is conjugate to the diameter y = x tan 0. Chords AP, 4Q parallel to the two diameters are drawn through the vertex (a, o). Prove that PQ makes with the axis of x an angle 8. A rigid body is under the action of any number of forces in one plane. State the conditions of equilibrium. A vertical post is held upright in contact with the ground by two cords, attached to pegs on the ground at equal distances c from the foot of the post, and attached to the post at heights a and b above the ground; the cords being on either side of the post in a vertical plane. Prove that the tangent of the angle which the reaction of the ground makes with the vertical cannot exceed c(a - b)/ab. 9. Two particles of given masses are moving in a plane with given velocities in given directions. Determine the velocity of their centre of mass. If the masses of the particles are m and m', and the velocity of m relative to m' is of magnitude v and makes an angle a with the line drawn from m' to m, determine the magnitude and direction of the velocity of m relative to the centre of mass. IO. A body which weighs 10 lbs. is attached to a cord passing over a fixed smooth pulley, and the cord is drawn over the pulley at a uniform rate of 3 ft. per sec. What is the magnitude of the tension of the cord? If the pulley is rigidly attached to a moving platform, which is ascending with an acceleration of 4 foot-second units, and the cord is drawn over the pulley as before, what change is made in the tension? II. Two particles of equal masses are attached to the ends of a light string of length 7; one of them, A, is placed on a smooth table and the other, B, hangs just over the edge, the string being just tight. Find the magnitudes and directions of the velocities of the two particles (1) just before A leaves the table, (2) just after. 12. A particle is projected from a given point with a given velocity in a given vertical plane; find the envelope of the possible paths. A shot is fired from a gun at the top of a cliff of height h with a velocity u ft. per sec. Prove that, if the range measured from the foot of the cliff is as great as possible, the elevation a is given by the equation COS 2α = =gh/(gh+u2). MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, NOVEMBER, 1900. I. 2. OBLIGATORY EXAMINATION. I. PART I. FIRST PAPER. [Great importance is attached to accuracy.] Express as a fraction in its lowest terms { of 17+18 of 116}×{1 of 144}}. Calculate to five places of decimals (so that the error is less than 0'000005) the value of the expression 14 52, 1575 3. A rectangular field is 330 yards in length and 188 yards in breadth; find the number of acres in the field, and the money which would be obtained by selling half the field for £17. 4s. 6d. an acre, and the other half for £21. 15s. 6d. an acre. 4. A person, having four thousand pounds to invest, places £2000 in 2 per cent. consols at 98, and £2000 in India three per cents. at 98§; find, to the nearest penny, the income thereby obtained. 5. A railway train, 73 metres in length, is travelling at the rate of 60 kilometres an hour on the up line, and another train, 102 metres in length, is travelling in the opposite direction, on the down line, at the rate of 40 kilometres an hour; find the time occupied by the trains in passing each other. |