MATHEMATICAL TRIPOS Part IA Thursday 31st May 2007 9 am to 12 noon PAPER 1 Before you begin read these instructions carefully. The examination paper is divided into two sections. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt all four questions from Section I and at most five questions from Section II. In Section II, no more than three questions on each course may be attempted. 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, marked A, B, C, D, E and F according to the code letter affixed to each question. Include in the same bundle all questions from Section I and II with the same code letter. Attach a gold cover sheet to each bundle; write the code 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 REQUIREMENTS 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 1A Algebra and Geometry (i) The spherical polar unit basis vectors er, eand ein R3 are given in terms of the Cartesian unit basis vectors i, j and k by er = i cos sin + j sin sin + k cos , e= i cos cos + j sin cos − k sin , e= −i sin + j cos . Express i, j and k in terms of er, eand e. (ii) Use suffix notation to prove the following identity for the vectors A, B, and C in R3: (A × B) × (A × C) = (A · B × C)A. 2B Algebra and Geometry For the equations px + y + z = 1, x + 2y + 4z = t, x + 4y + 10z = t2, find the values of p and t for which (i) there is a unique solution; (ii) there are infinitely many solutions; (iii) there is no solution. Paper 13 3F Analysis Prove that, for positive real numbers a and b, 2pab 6 a + b . For positive real numbers a1, a2, . . ., prove that the convergence of 1Xn=1 an implies the convergence of 1Xn=1 pan n . 4D Analysis Let P1n=0 anzn be a complex power series. Show that there exists R 2 [0,1] such that P1n=0 anzn converges whenever |z| < R and diverges whenever |z| > R. Find the value of R for the power series 1Xn=1 zn n . Paper 1 [TURN OVER4 SECTION II 5B Algebra and Geometry (i) Describe geometrically the following surfaces in three-dimensional space: (a) r · u = |r|, where 0 < || < 1; (b) |r − (r · u)u| = , where > 0. Here and are fixed scalars and u is a fixed unit vector. You should identify the meaning of , and u for these surfaces. (ii) The plane n · r = p, where n is a fixed unit vector, and the sphere with centre c and radius a intersect in a circle with centre b and radius . (a) Show that b − c = n, where you should give in terms of a and . (b) Find in terms of c, n, a and p. 6C Algebra and Geometry Let M: R3 ! R3 be the linear map defined by x 7! x0 = ax + b(n × x) , where a and b are positive scalar constants, and n is a unit vector. (i) By considering the effect of M on n and on a vector orthogonal to n, describe geometrically the action of M. (ii) Express the map M as a matrix M using suffix notation. Find a, b and n in the case M = 0@ 2 −2 2 2 2 −1 −2 1 21A. (iii) Find, in the general case, the inverse map (i.e. express x in terms of x0 in vector form). Paper 15 7C Algebra and Geometry Let x and y be non-zero vectors in a real vector space with scalar product denoted by x · y. Prove that (x · y)2 (x · x)(y · y), and prove also that (x · y)2 = (x · x)(y · y) if and only if x = y for some scalar . (i) By considering suitable vectors in R3, or otherwise, prove that the inequality x2 + y2 + z2 yz + zx + xy holds for any real numbers x, y and z. (ii) By considering suitable vectors in R4, or otherwise, show that only one choice of real numbers x, y, z satisfies 3(x2 + y2 + z2 + 4) − 2(yz + zx + xy) − 4(x + y + z) = 0, and find these numbers. 8A Algebra and Geometry (i) Show that any line in the complex plane C can be represented in the form ¯cz + c¯z + r = 0 , where c 2 C and r 2 R. (ii) If z and u are two complex numbers for which z + u z + ¯u= 1 , show that either z or u is real. (iii) Show that any M¨obius transformation w = az + b cz + d (bc − ad 6= 0) that maps the real axis z = ¯z into the unit circle |w| = 1 can be expressed in the form w = z + k z + ¯k , where , k 2 C and || = 1. Paper 1 [TURN OVER6 9F Analysis Let a1 = p2, and consider the sequence of positive real numbers defined by an+1 = q2 + pan , n = 1, 2, 3, . . . . Show that an 6 2 for all n. Prove that the sequence a1, a2, . . . converges to a limit. Suppose instead that a1 = 4. Prove that again the sequence a1, a2, . . . converges to a limit.Prove that the limits obtained in the two cases are equal. 10E Analysis State and prove the Mean Value Theorem. Let f : R ! R be a function such that, for every x 2 R, f00(x) exists and is non-negative. (i) Show that if x y then f0(x) f0(y). (ii) Let 2 (0, 1) and a < b. Show that there exist x and y such that fa + (1 − )b= f(a) + (1 − )(b − a)f0(x) = f(b) − (b − a)f0(y) and that fa + (1 − )bf(a) + (1 − )f(b) . 11E Analysis Let a < b be real numbers, and let f : [a, b] ! R be continuous. Show that f is bounded on [a, b], and that there exist c, d 2 [a, b] such that for all x 2 [a, b], f(c) f(x) f(d). Let g : R ! R be a continuous function such that lim x!+1g(x) = lim x!−1g(x) = 0 . Show that g is bounded. Show also that, if a and c are real numbers with 0 < c g(a), then there exists x 2 R with g(x) = c. Paper 17 12D Analysis Explain carefully what it means to say that a bounded function f : [0, 1] ! R is Riemann integrable. Prove that every continuous function f : [0, 1] ! R is Riemann integrable. For each of the following functions from [0, 1] to R, determine with proof whether or not it is Riemann integrable: (i) the function f(x) = x sin 1x for x 6= 0, with f(0) = 0; (ii) the function g(x) = sin 1x for x 6= 0, with g(0) = 0. END OF PAPER Paper 1