1 (2 sec -1)" 21 + 2 n (2) If Up is the nth term of a series of positive terms, prove that the series is convergent if u," tends to a limit which is < I as r tends to infinity. Is the series in which w, ( convergent or not? 2. The number of integers less than N and prime to it is N 1 1 , 7 prime numbers by which N is divisible. 3. If dì, 22, Qgg ... An and P1, P2, P3... Pn are two sets of positive numbers each arranged in increasing order of magnitude, prove that Piaza + P2a,? + ... +Puan? ? > (91+2, +...+(n) P1+P2 + ... + Pn 4. If a, b, c, d are the sides and 28 the perimeter of a convex quadrilateral inscribed in a circle of radius R, prove that 16 R2 (s—a)(s—6)(s—c)(s—d) = (ab +cd) (ac + bd) (ad + bc). 5. Sum the series : x2 sin 20 23 sin 30 (1) 2 sin 0 + + +...ad inf.; 1.2.3 + ...ad inf.; 5 3 + 3* +...to n terms. 3.5 5.7 6.8 6. (1) If V is the determinant formed of the first minors of A, a determinant of the nth order, prove that V = (2) If each element of a determinant of the nth order is the algebraic sum of p numbers, A can be expressed as the sum of pr determinants. Prove this, and hence shew that, when all the elements of the principal diagonal of A are unity and all the others are 1-2, then A = næa-1-(n-1).x". 7. In an algebraic equation, if each negative coefficient be taken positively and divided by the sum of all the positive coefficients which precede it, the greatest of all the fractions thus formed increased by unity, is a superior limit of the positive roots. 1.2 I (2) + 202 + 32 2 + + 3* 4.6 8. If f(x) = Pox" + P, 01-1+...Prx*-* + ... +Pn = o is an algebraic equation of which the roots are an, ag, ... Om, prove that £am may be expressed in terms of the coefficients of the first m + 1 terms of f(x). Express in the form of a determinant the relation between these coefficients when Eam is zero. 9. Shew how to break up into its partial fractions a fraction of which the numerator and denominator are rational and integral algebraical expressions. If m is even, prove that (22 – a2) 2 x + a" x2. r=m-1 .am I ta2 10. Define the Eliminant of f (x) and "(x), and shew that the Discriminant of f (w) is proportional to the product of the squares of the differences of the roots of f(x) = 0. 11. In a spherical triangle, of which E is the spherical excess, (1) cos a sin basin a cos b cos C sin c cos A; {sin s sin(s—a) sin(s—b) sin(s—c)}#; (2) sin 2 a 6 с 2 COS COS COS 2 (3) From (2) deduce the area of a plane triangle in terms of its sides. TUESDAY, JUNE 5, from 2.30 to 5.30 P.M. SECTION IV. Mathematics. 6. Geometry. 1. Shew how to find the centre of a conic which passes through five given points in a plane. 2. Shew that the anharmonic ratio of a pencil, whose sides pass through four fixed points of a conic section, and whose vertex is any variable point of it, is constant. Find the locus of the vertex of a triangle on a given base when the intercept made by the sides on a fixed right line is of given length. a 3. Give a brief account of the method of Projection, and indicate by examples the kind of theorems which lend themselves most easily to generalization by projection. Prove both by projection and polar reciprocation that the line joining the focus of a conic to the pole of a focal chord is at right angles to the chord. 4. If the distances x, x', of two points on a right line, measured from any origin on the line, be connected by a relation of the form Aca' + Bx+Cx'+D=0, shew that the o points form two homographic systems. Shew that pairs of tangents drawn from a fixed point to a variable conic touching four fixed right lines form a pencil in Involution. 5. Find the general equation to a circle in tangential coordinates. Shew that the circle 12++ 1,2 – 2 uv cos A-2vdcos Β – 2λμ cos C = {1 cos(B-C)+u cos(C-A) +v cos(A - B)}? touches each of the circles 12 + m2 +1,2 – 2 uv cos A– 2 vi cos B— 22u cos C = (1 + x + v)?, where A, B, C are the angles of the triangle of reference. 6. If S.= 0 and $ = o represent two conics, shew that kS+S' = o represents a pair of right lines when k is a root of the cubic A k3 + k? + Oʻk+ A' = 0. Explain the geometrical meaning of each of the equations A= 0 and = 0. 7. Find, in the form 2 = 440', the condition that it shall 4A be possible to inscribe in a given conic a triangle which is circumscribed about another given conic. Shew that the two curves y2 = are so related that triangles can be inscribed in either so as also to circumscribe the other. 8. Find the equation to the sphere described on a given segment of a line as diameter. If S, S denote two spheres which cut each other orthogonally, shew that the polar plane of any point P on S with respect to S passes through Q, the diametrically opposite point to P. = 9. Shew that a homogeneous relation, of any degree, between 2, y, 2 represents a cone. Find the equation of the cone whose base is the section x2 y2 of the surface + = 1 by the plane z= 0, and whose a2 72 vertex is the point (ac', ', 2). 10. Find the equation to the surface generated by a right line which always meets the axis of x and also each of the two lines - c y-b y 2-C = 11. Shew that, by a transformation from one set of rectangular axes to another, the equation ax2 + by2 + cz2 + 2 fyz + 2gzx + 2 hxy + d' = 0 can be reduced to the form a' x2 + O'ya +6'z2 + ď = 0, and that the process depends on the solution of a cubic equation. Apply the method in question to classify the surface 22 - y2 + 2yz = 1. WEDNESDAY, JUNE 6, from 9.30 A.m. to 12.30. SECTION IV. Mathematics. 7. Differential Calculus. 1. To what order of differentiation is it in general necessary to proceed to eliminate n arbitrary functions ? Eliminate the arbitrary functions from u = f (x+y)+ xy ♡ (2-y). 2. State and prove Taylor's theorem; and find the oth term in the expansion of cos (m sin-7x). 3. If x = r cos 0, y = r sin 0, and V =f(x, y), prove that d2V d? d2V I d2V + + + dx2 dy? d7.2 r dr 72 d 02 4. If(u, v, w) denote the Jacobian of u, v, w with respect to d u d (x, y, z) x = 2$+3n+75, y = 3$++148, = $+2n+25; then d(u, v, w), 2(x, y, z) w d (, ) d(u, v, w) d (£, n, 5) = 1; 7. d 1(x, y, z) * d(u, v, w) d(x, y, z) d (u, v, w) X, Y, 2; and if Х Х = I. 5. Explain how to find maximum and minimum values of a function of n variables which are connected by m equations. Find the magnitude of the principal axes of a central section of a quadric. 6. If the radius vector OP to a curve S be produced to P. to form a new curve S, PP being always of constant length, shew that the polar subnormals of the curves at P and P respectively are identical. On this found a construction for drawing a normal to the inverse of the hyperbola with respect to a focus. 7. Trace the curves : (1) y (x — 1) (*— 2)(x-3) = x+ (x + 1); (3) 206 — 23 32 —30% y + y2 = 0. Find the envelope of the normal to a cycloid, and shew that the radius of curvature is bisected by the base. 9. Find the equation to the osculating plane at any point of a curve in space, and the radius of absolute curvature at any point of the curve, ao cos ko, y = ao sin ko, ao. 10. Shew how to determine the principal radii of cứrvature at any point of a curved surface; and express in terms of them the radius of curvature of any normal section. 11. Determine the general differential equation to developable surfaces. WEDNESDAY, JUNE 6, from 2.30 to 5.30 P.M. SECTION IV. Mathematics. 8. Integral Calculus. 1. Integrate : dac sec3 Ado. 2x -C) 1.2.3 ... (m-1) n(n + 1)... (n + m-I)' when m and n are positive, and m is an integer. (1) - (3-2) |