6. AHK is an equilateral triangle, and ABCD is a rhombus whose sides are equal to the sides of the triangle, and BC, CD pass through H, K respectively. Prove that the angle A of the rhombus is ten-ninths of a right angle. 7. Lines are drawn joining the angular points A, B, C of a triangle to any point O in its plane. Prove that the lines from the middle points of the sides BC, CA, AB respectively parallel to OA, OB, OC, meet in a point. 8. Show that the straight line joining the points (24, 16) and (− 21, - 14) passes through the origin, and determine the co-ordinates of the points of trisection of this line. 9. Explain how to find the length of the perpendicular from the point (h, k) on the line x cos a + y sin a = p. Find the equations of the lines bisecting the angles between the straight lines 3x+4y = 5, 12x-5y = 41. IO. Prove that x2+ y2 - 4x-2y+4=0 represents a circle, and find the length of its radius. Show that the lines x = 1, y = 2 each touch the circle, and find the other co-ordinates of the points of contact. in terms of the eccentric angle of the point of contact. Prove that two tangents to an ellipse which are at right angles to each other intersect on a fixed circle concentric with the ellipse. 12. Find the equation to the tangent at any point of the rectangular hyperbola xy = c2. If c tan 0, ccot be the co-ordinates of a point on the curve, show that the chord through the points and , where + is constant, passes through a fixed point on the conjugate axis of the hyperbola. 13. Given a chord of a parabola and the direction of the axis, show that the locus of the focus is a hyperbola whose foci are at the extremities of the given chord. 14. Through one of the vertices A, and the extremities P, P', of a double ordinate of an ellipse or hyperbola, a circle is drawn cutting the axis again in K. If G be the foot of the normal at P, prove that GK is of constant length. VII. MECHANICS. [Full marks may be obtained for about two-thirds of this paper. importance is attached to accuracy. N.B.-g may be taken = 32.] Great I. Find the magnitude of the resultant of two forces P, Q which act at a point, the angle between their directions being 0. Find the resultant of two forces 3 lbs. and 5 lbs. which act at an angle of 60°; and show that its magnitude will be unaltered if either of the given forces be replaced by a force of 8 lbs. acting in the opposite direction. 2. State the necessary and sufficient conditions for the equilibrium of three parallel forces acting upon a rigid body. the other consists of A bookshelf supported at its extremities is just filled by two sets of books, the books of each set being placed together. One set consists of 14 volumes, each 1 inches thick and weighing 2 lbs. ; 12 volumes, each 1 inches thick and weighing 2 lbs. Find the pressures on the supports, the weight of the shelf being 8 lbs. 3. The weight and centre of gravity of a body, and also of a portion of the body, being known, show how to determine the centre of gravity of the remainder. A figure is formed by taking away from a square the triangle whose angular points are the middle points of three of the sides. Find the position of its centre of gravity. 4. State the laws of Limiting Friction, and explain what is meant by the "Coefficient of Friction." A uniform beam AB whose length is 12 feet rests with one extremity A on a rough horizontal plane AC and is kept from falling forwards by a cord BC, 20 feet long, whose extremity is attached to a fixed point C in the plane, directly behind the beam. If the beam be on the point of slipping when AC = AB, find the coefficient of friction. 5. Find the relation between the power and the weight in that system of pulleys in which all the strings are attached to the weight, the weights of the pulleys being equal. If there be one fixed, and two moveable, pulleys, find how far the weight can be practically raised, if it be initially 24 feet below the lower moveable pulley. 6. Explain how velocity is measured, and if u be the measure of a velocity when s feet and t seconds are the units of space and time, find its measure when the units are s' feet and t' seconds. Compare the velocities of two particles, one of which describes 9 miles in two hours and the other II feet in 4 seconds. 7. Express the space passed over by a particle moving subject to uniform acceleration in terms of its initial and final velocities and the time occupied. An engine-driver reduces the speed of a train (at a uniform rate) from 40 to 30 miles per hour in a quarter of a minute. Find the distance passed over in this time, and also the velocity of the train when half this distance has been described. 8. A ball of mass m impinges directly upon a ball at rest of the same size but of mass m'. Show that after impact the balls will move in the same direction or in opposite directions according as m is > or < em', e being the coefficient of elasticity. 9. A particle is projected from O with velocity u in a direction inclined to the horizon at an angle a. Prove that the equation of its path is the axes of x and y being the horizontal and vertical lines drawn in the plane of the motion through the point of projection. Find the velocity and angle of projection of a particle which being thrown from the level of the ground just clears a wall 18 feet high at a distance of 36 feet from the point of projection, and strikes a wall parallel to the former and 60 feet beyond it, at a point 8 feet above the ground. The plane of projection is perpendicular to the walls. IO. A particle of weight W, attached by a string of length Z to the vertex of a smooth cone whose axis is vertical and semi-vertical angle a describes a horizontal circle on the surface of the cone with uniform velocity v ; find the tension of the string and the pressure on the surface. MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, NOVEMBER, 1895. 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. If two triangles ABC, DEF, have the sides AB, AC of the one respectively equal to the sides DE, DF of the other, and the angle BAC equal to the angle EDF, the triangles are equal in all respects. 2. Prove that triangles on equal bases and between the same parallels are equal to one another. If two triangles have equal bases, but the height of one be double the height of the other, prove, by Euclid's methods, that one of the triangles is double the other. |