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MATHEMATICAL TRIPOS Part IB
Wednesday, 2 June, 2010 1:30 pm to 4:30 pm
PAPER 2
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.
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.
SECTION I
1F Linear Algebra Suppose that φ is an endomorphism of a finite-dimensional complex vector space.
(i) Show that if λ is an eigenvalue of φ, then λ^2 is an eigenvalue of φ^2.
(ii) Show conversely that if μ is an eigenvalue of φ^2 , then there is an eigenvalue λ of φ with λ^2 = μ.
2H Groups Rings and Modules Give the definition of conjugacy classes in a group G. How many conjugacy classes are there in the symmetric group S 4 on four letters? Briefly justify your answer.
3G Analysis II Let c > 1 be a real number, and let Fc be the space of sequences a = (a 1 , a 2 ,... ) of real numbers ai with
r=1 c −r (^) |ar| convergent. Show that ‖a‖c = ∑∞ r=1 c −r (^) |ar| defines a norm on Fc.
Let F denote the space of sequences a with |ai| bounded; show that F ⊂ Fc. If c′^ > c , show that the norms on F given by restricting to F the norms ‖. ‖c on Fc and ‖. ‖c′^ on Fc′^ are not Lipschitz equivalent.
By considering sequences of the form a(n)^ = (a, a^2 ,... , an, 0 , 0 ,... ) in F , for a an appropriate real number, or otherwise, show that F (equipped with the norm ‖. ‖c) is not complete.
4H Metric and Topological Spaces On the set Q of rational numbers, the 3-adic metric d 3 is defined as follows: for x, y ∈ Q, define d 3 (x, x) = 0 and d 3 (x, y) = 3−n, where n is the integer satisfying x − y = 3nu where u is a rational number whose denominator and numerator are both prime to 3.
(1) Show that this is indeed a metric on Q. (2) Show that in (Q, d 3 ), we have 3n^ → 0 as n → ∞ while 3−n^6 → 0 as n → ∞. Let d be the usual metric d(x, y) = |x − y| on Q. Show that neither the identity map (Q, d) → (Q, d 3 ) nor its inverse is continuous.
Part IB, Paper 2
8E Statistics A washing powder manufacturer wants to determine the effectiveness of a television advertisement. Before the advertisement is shown, a pollster asks 100 randomly chosen people which of the three most popular washing powders, labelled A, B and C, they prefer. After the advertisement is shown, another 100 randomly chosen people (not the same as before) are asked the same question. The results are summarized below.
A B C before 36 47 17 after 44 33 23
Derive and carry out an appropriate test at the 5% significance level of the hypothesis that the advertisement has had no effect on people’s preferences.
[You may find the following table helpful:
χ^21 χ^22 χ^23 χ^24 χ^25 χ^26 95 percentile 3. 84 5. 99 7. 82 9. 49 11. 07 12. 59
]
9E Optimization Consider the function φ defined by
φ(b) = inf{x^2 + y^4 : x + 2 y = b}.
Use the Lagrangian sufficiency theorem to evaluate φ(3). Compute the derivative φ′(3).
Part IB, Paper 2
SECTION II
10F Linear Algebra (i) Show that two n × n complex matrices A, B are similar (i.e. there exists invertible P with A = P −^1 BP ) if and only if they represent the same linear map Cn^ → Cn^ with respect to different bases. (ii) Explain the notion of Jordan normal form of a square complex matrix. (iii) Show that any square complex matrix A is similar to its transpose. (iv) If A is invertible, describe the Jordan normal form of A−^1 in terms of that of A. Justify your answers.
11H Groups Rings and Modules For ideals I, J of a ring R, their product IJ is defined as the ideal of R generated by the elements of the form xy where x ∈ I and y ∈ J.
(1) Prove that, if a prime ideal P of R contains IJ, then P contains either I or J.
(2) Give an example of R, I and J such that the two ideals IJ and I ∩ J are different from each other.
(3) Prove that there is a natural bijection between the prime ideals of R/IJ and the prime ideals of R/(I ∩ J).
Part IB, Paper 2 [TURN OVER
14F Geometry Suppose that a > 0 and that S ⊂ R^3 is the half-cone defined by z^2 = a(x^2 + y^2 ), z > 0. By using an explicit smooth parametrization of S, calculate the curvature of S.
Describe the geodesics on S. Show that for a = 3, no geodesic intersects itself, while for a > 3 some geodesic does so.
15D Variational Principles Describe briefly the method of Lagrange multipliers for finding the stationary points of a function f (x, y) subject to a constraint φ(x, y) = 0.
A tent manufacturer wants to maximize the volume of a new design of tent, subject only to a constant weight (which is directly proportional to the amount of fabric used). The models considered have either equilateral-triangular or semi-circular vertical cross– section, with vertical planar ends in both cases and with floors of the same fabric. Which shape maximizes the volume for a given area A of fabric? [Hint: (2π)−^1 /^23 −^3 /^4 (2 + π) < 1.]
16B Methods Explain briefly the use of the method of characteristics to solve linear first-order partial differential equations.
Use the method to solve the problem
(x − y) ∂u ∂x
= αu,
where α is a constant, with initial condition u(x, 0) = x^2 , x > 0.
By considering your solution explain:
(i) why initial conditions cannot be specified on the whole x-axis;
(ii) why a single-valued solution in the entire plane is not possible if α 6 = 2.
Part IB, Paper 2 [TURN OVER
17D Quantum Mechanics A particle of mass m moves in a one-dimensional potential defined by
V (x) =
∞ for x < 0 , 0 for 0 6 x 6 a, V 0 for a < x,
where a and V 0 are positive constants. Defining c = [2m(V 0 − E)]^1 /^2 /ℏ and k = (2mE)^1 /^2 /ℏ, show that for any allowed positive value E of the energy with E < V 0 then c + k cot ka = 0.
Find the minimum value of V 0 for this equation to have a solution.
Find the normalized wave function for the particle. Write down an expression for the expectation value of x in terms of two integrals, which you need not evaluate. Given that 〈x〉 =
2 k
(ka − tan ka),
discuss briefly the possibility of 〈x〉 being greater than a. [Hint: consider the graph of −ka cot ka against ka.]
18C Electromagnetism A steady current I 2 flows around a loop C 2 of a perfectly conducting narrow wire. Assuming that the gauge condition ∇ · A = 0 holds, the vector potential at points away from the loop may be taken to be
A(r) = μ 0 I 2 4 π
C 2
dr 2 |r − r 2 |
First verify that the gauge condition is satisfied here. Then obtain the Biot-Savart formula for the magnetic field
B(r) =
μ 0 I 2 4 π
C 2
dr 2 × (r − r 2 ) |r − r 2 |^3
Next suppose there is a similar but separate loop C 1 with current I 1. Show that the magnetic force exerted on loop C 1 by loop C 2 is
F 12 =
μ 0 I 1 I 2 4 π
C 1
C 2
dr 1 ×
dr 2 × r 1 − r 2 |r 1 − r 2 |^3
Is this consistent with Newton’s third law? Justify your answer.
Part IB, Paper 2
