SATURDAY, JUNE 2, from 2.30 to 5.30 P.M. SECTION IV. Mathematics. 2. Elementary Geometry. 1. If from a point on a circle two right lines be drawn, one touching and the other cutting the circle, prove that the angle between them is equal to the angle in the alternate segment of the circle. Divide a circle into two segments, such that the angle in one of them shall be five times the angle in the other. 2. Shew that the bisector of the vertical angle of a triangle cuts the base in the ratio of the sides. If A, B, C be three points in a right line, and P a point at which the segments AB and BC subtend equal angles, shew that the locus of P is a circle. 3. Shew how to draw a pair of tangents to a parabola from a given point in the directrix. 4. Shew that a tangent to a hyperbola forms with the asymptotes a triangle of constant area. 5. Given the base of a triangle and the point where the inscribed circle touches the base, find the locus of the vertex. 6. Find the equation of the normal at a point P on the 202 y2 = a2 — b2. ellipse a2 + = 1, in the form ax by cos o sin axis major in G, and if in If the normal at P meet the PG, a point Q be taken dividing PG in a given ratio, find the locus of Q. 7. From the definition of a parabola as the locus of a point whose distance from a fixed point is equal to its distance from a fixed line, deduce the condition that the equation ax2+2hxy+by2+2gx+2fy+c= o, shall represent a parabola. Find the equation of the axis of the parabola 8. Shew that the sum of two focal chords of a central conic drawn parallel to two conjugate diameters is constant. Discuss this theorem in the case of an equilateral hyperbola. 9. Shew that the equation of the chord which joins the two points (aλ,2), (aμ,~) on the hyperbola ay = a2, is 2+λμy = (λ +μα. xy Find the equations of the cominon tangents to the curves xy = a2 and x2 + y2 = 2a2. 10. Trace the following loci : 11. Shew that the volume of a cone on a plane base is one third of that of a cylinder on the same base and of the same altitude. MONDAY, JUNE 4, from 9.30 A.M. to 12.30. SECTION IV. Mathematics. 3. Elementary Mechanics. 1. Define Force, and hence shew how to determine the direction and magnitude of the acceleration produced by two forces acting on the same particle. A moving particle is attracted to one centre of force A, and repelled from another B, both forces varying as the distance, and being of the same absolute intensity. Prove that the particle has a constant velocity in a direction perpendicular to AB, and a constant acceleration parallel to AB. 2. The moment of any forces in a plane about any axis is equal to the moment of their resultant. Shew that the moment of a couple is the same about all axes perpendicular to the plane of the couple. 3. A mass M lies on an inclined plane and is connected by a thread which passes over a smooth pulley at the top of the plane with another mass m, which when the plane is smooth is supposed to be sufficiently great to pull M up the plane. (1) The plane being smooth, shew that the tension on the thread is constant, and find the velocity when the particles have moved over a given space from rest. (2) If the plain is so rough as just to produce equilibrium, find the mass which must be added to m in order that M may be dragged up the plane at the same rate as in case (1). 4. Shew how to compound two parallel forces, and state what is meant by a centre of parallel forces. Find the position of the centre of mass of a number of particles on a plane. 5. Explain the terms 'kinetic energy' and 'work,' and the relation between them. Prove that if a heavy particle slide from rest at A down a smooth curved tube to a point B, the velocity at B is independent of the shape of the tube. A smooth tube AB coincides with the hypotenuse of an isosceles right-angled triangle, the side OA being horizontal and OB turned vertically downwards. A heavy particle P1 is placed in the tube at A and slides down it. Another equal particle p is let fall freely from 0. What interval of time must there be between the starting of the two particles from A and O respectively in order that p, may strike p1 just as it issues from the tube at B? And if the two particles then stick together, in what direction will they move off? 6. Prove that a heavy particle projected in vacuo will describe a parabola in a vertical plane, and that the velocity at any point of the path is that due to a fall from the directrix to the point. A particle is projected from a given point in a fixed direction. Find the locus of the intersection of the axis of the parabolic path with the directrix, when the velocity of projection varies. 7. What is meant by the 'modulus of elasticity' in the case of elastic strings? A heavy rod, the weight of which is sufficient to stretch an elastic string to twice its length, rests with its lower end on a smooth horizontal plane, and its upper end, to which the elastic thread is fastened, on a smooth inclined plane. The other end of the thread is fastened to a given point 0, on the inclined plane. The system is in equilibrium when the rod makes equal angles with the horizontal and inclined planes. Determine the natural length of the string. 8. Prove that the whole pressure on an area, plane or curved, immersed in a heavy fluid, varies as the area and as the depth of its centre of gravity below the surface of the fluid. Shew also that the centre of pressure on a narrow rectangular slip immersed in the fluid with its upper end in the surface divides the length of the strip in the ratio 2 1. 9. A vessel containing heavy fluid revolves uniformly about a vertical axis. Shew that the free surface will assume the form of a paraboloid of revolution. 10. A conical vessel with its axis vertical and its vertex turned downwards is continued upwards in the form of a cylinder, the axes of the cylinder and cone being equal. A piston of given weight, but the thickness of which may be neglected, fits into the cylinder and compresses the air within, and when the water barometer stands at H, sinks half down the cylinder. But when the external atmosphere changes so that the water barometer rises through h, the piston can be made to sink two thirds down the cylinder by filling the space above it with water. Find the length of the axis and the vertical angle of the cone. MONDAY, JUNE 4, from 2.30 to 5.30 P.M. SECTION IV. Mathematics. 4. Elementary Optics and Astronomy. Newton's Principia. 1. In curves of finite curvature the limiting ratio of the subtenses equals that of the squares of the conterminous arcs. 2. If a body move in any orbit about a fixed centre of force, the areas, described by lines drawn from the centre to the body, lie in one plane, and are proportional to the times of describing them. 3. A body moves in a parabola; find the law of force tending to the focus. Optics and Astronomy. 4. Find the geometrical focus of a pencil of rays after direct refraction at a spherical surface, and trace the relative changes of position of the conjugate foci. 5. A plane surface touches a self-luminous sphere; find the illumination of the surface at any point. Find the total light received by a narrow circular band traced on the plane with its centre at the point of contact of the sphere and plane. 6. Find the focal length of a lens equivalent to a combination of two lenses on the same axis. Two equiconvex lenses equal in every respect are made of glass, of which the refractive index is 1.52. A planoconvex lens made of glass, of which the refractive index is 1.54, is equal in power to the combination of the other two when they are separated by an interval equal to the radius of their curved faces. Find the relation between this radius and that of the curved face of the plano-convex lens. 7. Describe Galileo's telescope, and find its magnifying power. Why will there be a ragged edge' to the field of view? 8. Find the length of the day at a given place, the declination of the sun being known. 9. Find the latitude of a place from two equal altitudes of the sun, observed before and after noon. 10. Express celestial latitude and longitude in terms of right ascension and declination. If is the latitude and a the right ascension of the earth, and if cos w = 1-x, where w is the obliquity of the ecliptic and x a number so small that all powers above the first may be neglected; show that tan (-a) = x sin l cos l. TUESDAY, JUNE 5, from 9.30 A.M. to 12.30. SECTION IV. Mathematics. 5. Algebra and Trigonometry. 1. (1) Find the relation between a, B, y that the following series may be convergent, viz. |