INTEGRAL CALCULUS. TWO HOURS. PASS. 1. Ife is a function of z, and p(z) is what F(x) becomes after substitution, prove that fredo 4. Shew that the area of a curve can be represented by 120 taken between the proper limits. Find the area of a loop of the curve r2=a2 cos 40 5. Find the length of the arc of the curve 4y=2-2log cut off between the ordinates a=1 and x=2. 6. Find the volume of the solid produced by the revolution of the loop of the curve y2+ about the axis of x. SPHERICAL GEOMETRY AND TRIGONOMETRY. TWO HOURS. PASS. 1. Prove the fundamental relations which exist between polar triangles, and point out their application to formula relating to spherical triangles. One side of a spherical triangle is 90°. Find the relation connecting the other two sides and their included angle, either from first principles, or by assuming the proper formula for a right angled triangle. 2. Enunciate what you regard as the three most important relations for spherical triangles in general. Prove any one of them geometrically. 3. Assuming the formula connecting two sides, the included angle, and an angle opposite to one of the sides, of any spherical triangle, deduce two of the formulæ for rightangled triangles. 4. Having given the three sides of a spherical triangle, it is required to find one of the angles by aid of logarithms. State and prove a suitable formula. 5. Having given A=41° 17', B=52° 29′, C=90°, solve the triangle completely. 6. Assuming the earth to be a sphere, find the distance in nautical miles (correct to one mile) between Sydney Observatory (33° 51′ 41′′ S., 151° 12′ 23′′ E.) and Greenwich Observatory (51° 28′ 38′′ N., 0° E.). [A nautical mile is the length of an arc on the surface of the earth subtending l' at the centre.] 7. What is meant by spherical excess? Find the connection between it and the area of a triangle. 8. Prove that the tangent of the radius of the inscribed circle of a triangle is tan A sin(s-a). If you cannot do this, prove that tan A sin(s-a) is equal to each of the two similar expressions. ANALYTICAL GEOMETRY. TWO HOURS. PASS. The same paper as that set in the Second Year Examination. DYNAMICS. TWO HOURS. PASS. 1. Prove the parallelogram of velocities, explaining fully what you mean by a body having two independent velocities. A person walking along a straight road at 5 miles an hour sees a tower a mile distant from his eye, the nearest distance of the tower from the road being half a mile. Find the rate at which he is approaching the tower. 2. A body is projected downwards with velocity u, and at the end of t seconds has a velocity v; shew, from first principles, that the space described is the same as that described by a body moving uniformly for t seconds with a velocity that is the arithmetic mean of u and v. 3. By means of the hodograph or otherwise, find the acceleration, in direction and magnitude, of a particle moving with uniform speed in a circle. Find the difference in the weight of a train, of mass 180 tons, in latitude 60°, moving with a speed of 60 miles an hour, according as it is moving E. or W. Take the earth as a sphere of 4000 miles radius. 4. Discuss briefly the meanings of mass, momentum, force, impulse. Enunciate the second law of motion, and point out its meaning in connection with the motion of a particle in a curved path. 5. Prove that the trajectory of a particle subject to a force constant in magnitude and direction is a parabola (e.g., a projectile falling freely under gravity). Find the connection between the velocity at any point and the distance of that point from the directrix. A particle is projected and reaches a point P in the time t If t be the time it takes after leaving P to reach the horizontal plane through the point of projection, shew that the height of P above the plane is agtť. 6. Investigate the amount of kinetic energy lost when two spheres collide directly. A sphere of mass A moving with velocity u impinges directly on a second B at rest, and then B impinges directly on a third C at rest. Prove that the loss of kinetic energy is 季 (1—62) { 1+(A+C)(B+C) (1+e)2}u2; A+B e being the coefficient of restitution in each collision. 7. Define simple harmonic motion and prove that it is isochro nous. A clock with a pendulum, which at the surface of the earth gains 10 seconds a day, loses 10 seconds a day when taken down a mine. Compare the forces of gravity at the surface and down the mine. For Engineering Students only. 8. State and prove the theorem connecting the moments of inertia of a material system round parallel axes, one of which passes through the centre of gravity of the system. Find the moment of inertia of a uniform circular hoop round an axis through a point in the rim perpendicular to the plane of the hoop. 9. A bullet of mass m is fired with a horizontal velocity V into a pendulum of mass M, which is initially at rest. The line of fire being at a distance x below the axis of support of the pendulum, the distance of the pendulum's centre of gravity from the axis being h, and the angle of the swing being 90°, find the moment of inertia of the pendulum about the axis. SENIOR FRENCH I. AND II. The same papers as those set in the Second Year, with additional passages for translation from Pages choisies de Lesage. SENIOR GERMAN I. AND II. The same papers as those set in the Second Year, with additional passages for translation from Fouqué, Undine; Börne, Aus meinem Tagebuche. LOGIC AND MENTAL PHILOSOPHY. PASS. Not more than SEVEN questions to be attempted. 1. Contrast the teaching of Socrates with that of the Sophists. 2. Note the main points of difference between ancient and modern democracies. 3. Discuss the psychological basis of Plato's system of education. 4. Compare the Aristotelian divisions of the State with those given by Plato. 5. Sketch briefly the main features of Stoicism or Epicureanism as ethical theory. 6. Explain the origin and nature of the opposition between Nominalism and Realism during the middle ages. 7. What do you understand by the moral sense? Examine Butler's account of conscience. 8. Discuss Kant's analysis of the moral consciousness, with special reference to the opposition between reason and passion. 9. State what you know of the different views held with regard to the nature of sovereignty. 10. What were the general characteristics of ethical theory and practise during the eighteenth century? LOGIC AND MENTAL PHILOSOPHY. HONOURS I. 1. Sketch briefly the main points in the Platonic theory of idealism, noting any differences between earlier and later forms. |