I. VII. PART III. SECOND PAPER. [Full marks will be given for about two-thirds of this paper.] Prove that the semi-latus rectum of any conic is a harmonic mean between the segments of any focal chord. Prove also that the length of any focal chord of a parabola is equal to 4SP, S being the focus, and P the extremity of the diameter which bisects the chord. 2. Give a construction for drawing two tangents to an ellipse from a given point. Defining an asymptote of a hyperbola as a tangent whose point of contact is infinitely distant, show that the lines joining the centre of a hyperbola to the points where a directrix cuts the auxiliary circle are asymptotes according to this definition. 3. Interpret the constants in the equation of a straight line expressed in the form y = mx+b; (i.) (ii.) in the form x = ny+a. If the axis of y is measured vertically upwards, prove that y' - mx' – b represents the vertical height of the point (x', y') above the line whose equation is y = mx+b. Deduce the length of the perpendicular from the point (x', y') on the line y = mx+b. 4. Show that the equation af2+b2 - 2fgh =0 ax2+2hxy+by2+2gx+2fy +' represents two straight lines. If these are inclined to the axis of x at angles 01 and 02, express tan (02-01) and tan (02+01) in terms of a, h, b. 5. Find the co-ordinates of the middle point of the chord which the circle x2+12-2x+2y=2 cuts off on the line y = x - I. Find also the locus of the middle points of all chords of the circle which are parallel to the line y = x - I. 6. Prove that the equations y2 = 4a (x+a) and y2= -4b(x − b) represent a pair of parabolas having the same focus and axis. Find the co-ordinates of their points of intersection, and prove that at each of these points the tangents to the two parabolas are at right angles. 7. Define the eccentric angle of any point on an ellipse. Find the equation of the chord joining the points on the ellipse whose eccentric angles are 0, 4, and deduce the equation of the tangent at the point whose eccentric angle is 0. represents, for different values of a, a series of ellipses all inscribed in a certain rhombus. 8. A uniform ladder of length / and weight w is being gradually tilted off the ground by a man who can reach up to a height h. If the ground be perfectly smooth where the lower end touches it, and if l>2h, prove that the ladder cannot be raised into a vertical position unless its lower end be weighted, and find the least weight that will suffice. If the ground be rough, show by a diagram that as soon as the inclination of the ladder to the vertical is less than the angle of friction the ladder may be supported at a point below its middle point, provided that a force of suitable magnitude and direction be applied to the point of support, and show how to determine this force when equilibrium is limiting, having given its point of application. 9. State Newton's Third Law of Motion, and show how it affords a means of comparing the masses of different bodies. A body moving with velocity 5 feet per second overtakes another body moving with velocity I foot per second. The first body rebounds with velocity I foot per second, its direction of motion being reversed, while the second body has its velocity increased to 2 feet per second. Compare the masses of the bodies, and find the coefficient of restitution. 10. A stone is being whirled in a circle in a vertical plane at one end of a light string of which the other end is fixed. When the stone is at the highest point of the circle, the tension in the string just vanishes. Find the tension in the string when the stone is at the lowest point of the circle and when it is at the end of a horizontal diameter, expressing the result in terms of the weight of the stone. If the string be cut at the instant when the stone is ascending vertically, determine in terms of the radius of the circle the height to which the stone will rise. II. Prove that the work done in raising a number of weights through different heights is equal to the work which would be done in raising a single weight, equal to the sum of the weights, through a height equal to the height through which the centre of gravity of the weights is raised. A lake whose superficial area is half a square mile has its surface at a height of 95 feet above the sea level. If the water is run off till the surface is lowered by ten feet, calculate the value of the work which can be done in driving machinery by this water in its descent to the sea, a cubic foot of water being assumed to weigh 62 lb., and work being valued at a halfpenny per horse-power per hour. 12. A number of particles are placed on a rectangular sheet of paper. Given the masses of the particles and their distances from two adjacent edges of the sheet, find the distances of their centre of gravity from these edges. A series of cubes of the same material are piled one above the other on a horizontal plane. The lengths of the sides of the cubes, beginning with the lowest, are 2a, 2ar, 2ar2, 2ar3, and so on in geometrical progression. If there are n cubes in the pile, find the height of the centre of gravity of the whole system above the horizontal plane. If the number of cubes is infinite, being less than unity, prove that the height of the centre of gravity of the pile is I +23 a I MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, JUNE, 1900. I. 2. OBLIGATORY EXAMINATION. I. PART I. FIRST PAPER. [Great importance will be attached to accuracy.] Express as single decimals (1) 0*125 × 09152÷0'0715. What income will a man receive from investing £1000 in a 21 per cent. stock at 112? If he sells out his stock at 105, what loss of capital will he sustain ? 3. Find the true discount on £3570. 18s. 4d., due 50 days hence, at 3 per cent. per annum. 4. A beam, having a square section, is 9 ft. long, and weighs 3 cwt. A cubic foot of the substance of the beam weighs 32 lbs. What is the thickness of the beam? 5. A dishonest tradesman marks his goods at an advance of 5 per cent. on the cost price, but uses a fraudulent balance, whose beam is horizontal when the weight in one scale is one-fifteenth more than the weight in the other. What is his actual gain per cent.? 6. A man walks a third as far again as a boy in a given time. His stride is 5 centimetres longer than the boy's, and he takes 125 strides per minute, while the boy takes only 100. At what rate in kilometres per hour does each walk? 7. Multiply 7x2 - 23x+6 by 6x2 - 35x+11, and divide the result by 3x2-10x+3. 10. Having given the first term and the constant ratio of a geometric progression, find the expression for the sum of the first ʼn terms. Find the sum of ten terms of the geometric series 3+6+12+24+... What is meant by saying that the sum to infinity of the series is 2? 1 + 1 + 1 + 1 + ... II. Find the number of combinations of n things, r at a time, not assuming the formula for the number of permutations. In how many ways could a party of five scouts be selected from 20 available men? In how many ways could the twenty men be formed into four parties of five scouts, to proceed in different directions? 12. Prove the binomial expansion, ʼn being a positive integer. find the value, when n=9, of 1 − n2+ n − ng+ng. |