Examination Instructions and Sample Questions from a Mathematics Paper, Exams of Mathematics

Download Examination Instructions and Sample Questions from a Mathematics Paper and more Exams Mathematics in PDF only on Docsity! MATHEMATICAL TRIPOS Part IB Thursday 9 June 2005 9 to 12 PAPER 3 Before you begin read these instructions carefully. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt at most four questions from Section I and at most six questions from Section II. Complete answers are preferred to fragments. Write on one side of the paper only and begin each answer on a separate sheet. Write legibly; otherwise, you place yourself at a grave disadvantage. At the end of the examination: Tie up your answers in separate bundles labelled A, B, . . . , H according to the examiner letter affixed to each question, including in the same bundle questions from Sections I and II with the same examiner letter. Attach a completed gold cover sheet to each bundle; write the examiner letter in the box marked ‘Examiner Letter’ on the cover sheet. You must also complete a green master cover sheet listing all the questions you have attempted. Every cover sheet must bear your examination number and desk number. STATIONERY REQUIRMENTS SPECIAL REQUIREMENTS Gold cover sheet None Green master cover sheet You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 SECTION I 1C Groups, Rings and Modules Define what is meant by two elements of a group G being conjugate, and prove that this defines an equivalence relation on G. If G is finite, sketch the proof that the cardinality of each conjugacy class divides the order of G. 2A Geometry Write down the Riemannian metric on the disc model ∆ of the hyperbolic plane. Given that the length minimizing curves passing through the origin correspond to diameters, show that the hyperbolic circle of radius ρ centred on the origin is just the Euclidean circle centred on the origin with Euclidean radius tanh(ρ/2). Prove that the hyperbolic area is 2π(cosh ρ− 1). State the Gauss–Bonnet theorem for the area of a hyperbolic triangle. Given a hyperbolic triangle and an interior point P , show that the distance from P to the nearest side is at most cosh−1(3/2). 3B Analysis II Let f : R2 → R be a function. What does it mean to say that f is differentiable at a point (a, b) in R2? Show that if f is differentiable at (a, b), then f is continuous at (a, b). For each of the following functions, determine whether or not it is differentiable at (0, 0). Justify your answers. (i) f(x, y) = { x2y2(x2 + y2)−1 if (x, y) 6= (0, 0) 0 if (x, y) = (0, 0). (ii) f(x, y) = { x2(x2 + y2)−1 if (x, y) 6= (0, 0) 0 if (x, y) = (0, 0). Paper 3 5 SECTION II 10B Linear Algebra Let S be the vector space of functions f : R → R such that the nth derivative of f is defined and continuous for every n > 0. Define linear maps A,B : S → S by A(f) = df/dx and B(f)(x) = xf(x). Show that [A,B] = 1S , where in this question [A,B] means AB −BA and 1S is the identity map on S. Now let V be any real vector space with linear maps A,B : V → V such that [A,B] = 1V . Suppose that there is a nonzero element y ∈ V with Ay = 0. Let W be the subspace of V spanned by y, By, B2y, and so on. Show that A(By) is in W and give a formula for it. More generally, show that A(Biy) is in W for each i > 0, and give a formula for it. Show, using your formula or otherwise, that {y,By,B2y, . . .} are linearly indepen- dent. (Or, equivalently: show that y,By,B2y, . . . , Bny are linearly independent for every n > 0.) 11C Groups, Rings and Modules (i) Define a primitive polynomial in Z[x], and prove that the product of two primitive polynomials is primitive. Deduce that Z[x] is a unique factorization domain. (ii) Prove that Q[x]/(x5 − 4x+ 2) is a field. Show, on the other hand, that Z[x]/(x5 − 4x+ 2) is an integral domain, but is not a field. Paper 3 [TURN OVER 6 12A Geometry Describe geometrically the stereographic projection map π from the unit sphere S2 to the extended complex plane C∞ = C∪{∞}, positioned equatorially, and find a formula for π. Show that any Möbius transformation T 6= 1 on C∞ has one or two fixed points. Show that the Möbius transformation corresponding (under the stereographic projection map) to a rotation of S2 through a non-zero angle has exactly two fixed points z1 and z2, where z2 = −1/z̄1. If now T is a Möbius transformation with two fixed points z1 and z2 satisfying z2 = −1/z̄1, prove that either T corresponds to a rotation of S2, or one of the fixed points, say z1, is an attractive fixed point, i.e. for z 6= z2, Tnz → z1 as n→∞. [You may assume the fact that any rotation of S2 corresponds to some Möbius transfor- mation of C∞ under the stereographic projection map.] 13B Analysis II Let f be a real-valued differentiable function on an open subset U of Rn. Assume that 0 6∈ U and that for all x ∈ U and λ > 0, λx is also in U . Suppose that f is homogeneous of degree c ∈ R, in the sense that f(λx) = λcf(x) for all x ∈ U and λ > 0. By means of the Chain Rule or otherwise, show that Df |x(x) = cf(x) for all x ∈ U . (Here Df |x denotes the derivative of f at x, viewed as a linear map Rn → R.) Conversely, show that any differentiable function f on U with Df |x(x) = cf(x) for all x ∈ U must be homogeneous of degree c. 14A Complex Analysis State the Cauchy integral formula, and use it to deduce Liouville’s theorem. Let f be a meromorphic function on the complex plane such that |f(z)/zn| is bounded outside some disc (for some fixed integer n). By considering Laurent expansions, or otherwise, show that f is a rational function in z. Paper 3 7 15H Methods Obtain the power series solution about t = 0 of (1− t2) d 2 dt2 y − 2t d dt y + λ y = 0 , and show that regular solutions y(t) = Pn(t), which are polynomials of degree n, are obtained only if λ = n(n+ 1), n = 0, 1, 2, . . .. Show that the polynomial must be even or odd according to the value of n. Show that ∫ 1 −1 Pn(t)Pm(t) dt = knδnm , for some kn > 0. Using the identity( x ∂2 ∂x2 x+ ∂ ∂t (1− t2) ∂ ∂t ) 1 (1− 2xt+ x2) 12 = 0 , and considering an expansion ∑ n an(x)Pn(t) show that 1 (1− 2xt+ x2) 12 = ∞∑ n=0 xnPn(t) , 0 < x < 1 , if we assume Pn(1) = 1. By considering ∫ 1 −1 1 1− 2xt+ x2 dt , determine the coefficient kn. Paper 3 [TURN OVER