9. Write down the expression for the cosine of an angle of a triangle in terms of its sides, and prove that, for a triangle ABC, Having given that the sides of a triangle are 35 ̊3, 34°5, and 1°2 find its greatest angle. IO. An observer standing at the distance of 52 feet from the foot of a tower finds that the angle of elevation of the top of the tower is 42° 15′ 15′′; find the height of the tower, the observer's eye being 5 feet from the ground. II. A piece of wood is in the form of a regular pyramid on a square base; the side of the base is 6 inches, and the perpendicular distance of the vertex from the base is 8 inches; find the number of cubic inches in the volume of the wood, and the number of square inches in its surface. 12. A cylindrical boiler is hemispherical at its two ends; its radius is 2 feet, and its total length is 8 feet; assuming that a cubic foot of water weighs 62.5 lbs., find the number of tons of water which will fill the boiler. (Take π = 3*14.) IV. STATICS AND DYNAMICS. [The acceleration due to gravity may be taken as equal to 32.] I. State and prove the proposition known as the " Polygon of Forces." Forces 1, 2, 3 and 2/2 act on a point in the directions of the sides AB, BC, CD and the diagonal DB of a square ABCD respectively; determine their resultant, graphically or otherwise. 2. Each of a pair of sculls has four-fifths of its length outside the rowlock, and a man sculling pulls at the handle of each with the force P. Another man thrusts an oar over the stern against the bottom of the water with the force 2P at an angle of 60° to the horizon. Compare their effects in propelling the boat. 3. State the conditions of equilibrium of any number of forces acting on a body in one plane. A smooth wall is inclined at an angle of 60° to the horizon; a heavy uniform rod AB 4 √6 feet long, is in equilibrium at an angle of 45° to the wall, its lower end, A, rests on the wall, and a point in it, C, rests on a smooth horizontal rail parallel to the wall. Draw a diagram showing how the forces act, and find the distance of C from the wall. 4. Find the centre of mass of a uniform triangular board. The sides AB, BC, CD, DA, of a trapezium are of lengths 54, 36, 27, 45 respectively, AB being parallel to CD. Prove that its centre of gravity is at a distance 16 from AB. 5. Find the ratio of P to Win a smooth screw. If a power of I cwt. acting horizontally and at right angles to the extremity of an arm 8 ft. 4 in. long will raise 5 tons by means of a screw whose axis is vertical and diameter 2 inches, find the inclination to the horizon of the thread of the screw. 6. Find the relation between the power and weight in that system of pulleys in which each string is attached to the weight, the strings being parallel and the weights of the pulleys being neglected. If there be three pulleys, two of them moveable, and the weight of each lb., what will be the value of P when W= 8 lbs. ? 7. What is meant by uniform acceleration ? A stone is thrown vertically upwards, with a velocity 36 feet per second. After what times will its velocity be 12 feet a second? 8. A particle moving in a straight line with a uniform acceleration a passes a certain point A with velocity u. State (without proof) the formulae which give (1) the distance s subsequently moved over in time t, and (2) the velocity v with which the particle will be moving when at a distance d from A. A railway train, moving with velocity 48 miles an hour, has its velocity reduced to 16 miles an hour in 5 minutes. Find the space passed over in the interval, the retardation being assumed to be uniform. 9. Two smooth inclined planes have a common altitude, and their inclinations are 30° and 60° to the horizon. Two masses start simultaneously from the common vertex to fall one down each plane. Compare (1) their times of falling to the bottom, and (2) their final velocities. IO. A particle is projected horizontally from the top of a vertical wall 16 feet high with a velocity 32/3 feet per second. Find its range on the horizontal ground, and prove that when it strikes the ground its velocity is 64 feet per second. II. State the law of Newton which tells us how to measure force. If the unit of force were that which acting on I ton of mass would in I minute generate a velocity of 1 mile a minute, how many units of force would the weight of 1 ton contain ? A man 10 stone in weight, who pulls with a force equal to his own weight, drags a 10 ton railway carriage from rest on a smooth horizontal line of rails. How far will he move it in one minute? 12. A mass (m), falling vertically, draws a mass M along a smooth horizontal table by means of a fine string which passes over a smooth pulley at the edge of the table; find the tension of the string at any time and the acceleration produced. If the tension is equal to half the weight of m what is the ratio of M to m? [Full marks may be obtained for about four-fifths of this paper. Great importance is attached to accuracy.] I. Explain the meanings of a, b, a, p, m, c in the following equations representing a straight line : x a + = I, x cos a + y sin a-p=0, y = mx +c. What are the relations between these constants, if all three equations represent the same straight line? 2. Find the equations of the sides of a triangle, the coordinates of whose vertices are (1, 4), (2, 3), (− 1, − 2), respectively. Also find the equations of the bisectors of the angles between the lines 12x+5y-4=0, 3x+4y+7 = 0. 3. Find the locus of a point P, so that when PM and PN are drawn respectively perpendicular to two fixed lines, OM and ON, the sum of OM and ON is given. 4. Find the locus of a point P (1) when the tangents from P to two given circles are in a given ratio, (2) when the sum of the tangents is constant. 5. Being given the focus and two points on a parabola, find the position of its directrix, and state the number of solutions of the problem. 6. Prove that the normal at any point on an ellipse bisects the angle between the focal vectors of the point. Draw the normals that pass through a given point on the axis minor of an ellipse, and find when a solution (other than the minor axis itself) is impossible. 7. Show that an ellipse can be projected orthogonally into a circle. Hence prove that any triangle can be projected orthogonally into an equilateral triangle. 8. In the transformation of the general equation of the second degree from one rectangular system of coordinate axes to another, show that the quantities a+b and ab-h2 remain unaltered, where a, b, 2h are the coefficients of x2, y2, xy respectively. Find the lengths of the axes of the conic that is represented by the equation 12x2 - 12xy +712 = 48. 9. Find what curve is represented by the equation = 1, a and give a figure of the curve. IO. Prove that the sum of the squares of any pair of conjugate diameters of an ellipse is constant. Being given in magnitude and position two conjugate diameters of an ellipse, find, by geometrical construction, the position and magnitude of its axes. |