Find the nth differential coefficients of 7. Prove that under certain conditions ƒ (a + h) = f (a) + h ƒ' (a + 0h), and state the conditions. Expand tan x in ascending powers of x as far as the term in x5. 8. State and prove the rule for finding the value of an expression which takes the indeterminate 9. State and prove the rule for integration by parts. Integrate the expressions x2 sin x, x2 tan -1x. 10. Find the whole area of the curve a2y2 = x2 (a2 — x2). 11. Find the volume generated by the revolution about the axis of y of the curve xy2 = (2a-x)3. PURE MATHEMATICS.-PART III. The Board of Examiners. 1. If u be a function of x, y, z where x, y, z are functions of t, state and prove the rule for finding the differential coefficient of u with respect to t. If u be a homogeneous function of x, y, z of n dimensions prove that , from the equation ƒ{x, y, z, p(a), ¥(a)} = 0, where a is a given function of x, y, z. where p, q, r, s, t denote the differential coefficients of z with respect to x, y. 3. Shew how to find the asymptotes of the curve Find the general equation of a curve of the third degree which has for asymptotes the axes of coordinates and the line 7. Find an expression for the shortest distance between two straight lines whose equations are given in the symmetrical form. Find the shortest distance between a diagonal of a cube and one of the edges which it does not meet. 8. Find the equation of the normal at any point of an ellipsoid. Shew that six normals can be drawn from a given point to an ellipsoid. 9. Find the circular sections of an ellipsoid. Shew that no two parallel circular sections of a conicoid, which is not a surface of revolution, are on a sphere. 10. Shew how to integrate the homogeneous differential equation 11. Shew how to find the complementary function of the differential equation 1. Define work, and express an English horse-power in ergs per second. A mass m is pulled up a smooth plane inclined at an angle a to the horizon by a constant horizontal force Fin a vertical plane containing a line of greatest slope. Find the work of the force in t seconds, the particle having an initial velocity u up the plane. 2. A small smooth ring of mass m is slung on a light string which goes up vertically on both sides of the ring, passes over two small smooth pegs, and sustains two masses M, M' hanging freely at its two ends. If the system is allowed to move, shew that the acceleration of the ring is half the algebraic sum of the accelerations of the masses in the opposite direction, and find these accelerations. |