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MATHEMATICAL TRIPOS Part IB
Tuesday, 2 June, 2009 9:00 am to 12:00 pm
PAPER 1
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
1G Linear Algebra
(1) Let V be a finite-dimensional vector space and let T : V → V be a non-zero endomorphism of V. If ker(T ) = im(T ) show that the dimension of V is an even integer. Find the minimal polynomial of T. [You may assume the rank-nullity theorem.]
(2) Let Ai, 1 6 i 6 3, be non-zero subspaces of a vector space V with the property that
V = A 1 ⊕ A 2 = A 2 ⊕ A 3 = A 1 ⊕ A 3.
Show that there is a 2-dimensional subspace W ⊂ V for which all the W ∩ Ai are one-dimensional.
2G Geometry What is an ideal hyperbolic triangle? State a formula for its area. Compute the area of a hyperbolic disk of hyperbolic radius ρ. Hence, or otherwise, show that no hyperbolic triangle completely contains a hyperbolic circle of hyperbolic ra- dius 2.
3D Complex Analysis or Complex Methods
Let f (z) = u(x, y) + iv(x, y), where z = x + iy, be an analytic function of z in a domain D of the complex plane. Derive the Cauchy–Riemann equations relating the partial derivatives of u and v.
For u = e−x^ cos y, find v and hence f (z).
Part IB, Paper 1
6C Numerical Analysis The real non-singular matrix A ∈ Rm×m^ is written in the form A = AD + AU + AL, where the matrices AD, AU , AL ∈ Rm×m^ are diagonal and non-singular, strictly upper- triangular and strictly lower-triangular respectively.
Given b ∈ Rm, the Jacobi iteration for solving Ax = b is
ADxn = −(AU + AL)xn− 1 + b, n = 1, 2 ...
where the nth iterate is xn ∈ Rm. Show that the iteration converges to the solution x of Ax = b, independent of the starting choice x 0 , if and only if the spectral radius ρ(H) of the matrix H = −A− D^1 (AU + AL) is less than 1.
Hence find the range of values of the real number μ for which the iteration will converge when
A =
1 0 −μ −μ 3 −μ − 4 μ 0 4
Part IB, Paper 1
7H Statistics What does it mean to say that an estimator ˆθ of a parameter θ is unbiased? An n-vector Y of observations is believed to be explained by the model
Y = Xβ + ε,
where X is a known n × p matrix, β is an unknown p-vector of parameters, p < n, and ε is an n-vector of independent N (0, σ^2 ) random variables. Find the maximum-likelihood estimator ˆβ of β, and show that it is unbiased.
8H Optimization Find an optimal solution to the linear programming problem
max 3x 1 + 2x 2 + 2x 3
in x > 0 subject to 7 x 1 + 3x 2 + 5x 3 6 44 , x 1 + 2x 2 + x 3 6 10 , x 1 + x 2 + x 3 > 8.
Part IB, Paper 1 [TURN OVER
12F Metric and Topological Spaces Given a function f : X → Y between metric spaces, we write Γf for the subset {(x, f (x)) | x ∈ X} of X × Y. (i) If f is continuous, show that Γf is closed in X × Y. (ii) If Y is compact and Γf is closed in X × Y , show that f is continuous. (iii) Give an example of a function f : R → R such that Γf is closed but f is not continuous.
13D Complex Analysis or Complex Methods
Consider the real function f (t) of a real variable t defined by the following contour integral in the complex s-plane:
f (t) =
2 πi
Γ
est (s^2 + 1)s^1 /^2
ds,
where the contour Γ is the line s = γ + iy, −∞ < y < ∞, for constant γ > 0. By closing the contour appropriately, show that
f (t) = sin(t − π/4) +
π
0
e−rtdr (r^2 + 1)r^1 /^2 when t > 0 and is zero when t < 0. You should justify your evaluation of the inversion integral over all parts of the contour. By expanding (r^2 + 1)−^1 r−^1 /^2 as a power series in r, and assuming that you may integrate the series term by term, show that the two leading terms, as t → ∞, are
f (t) ∼ sin(t − π/4) +
(πt)^1 /^2
[You may assume that
0 x − 1 / (^2) e−xdx = π 1 / (^2) .]
Part IB, Paper 1 [TURN OVER
14B Mathematical Methods Find a power series solution about x = 0 of the equation
xy′′^ + (1 − x)y′^ + λy = 0,
with y(0) = 1, and show that y is a polynomial if and only if λ is a non-negative integer n. Let yn be the solution for λ = n. Establish an orthogonality relation between ym and yn (m 6 = n).
Show that ymyn is a polynomial of degree m + n, and hence that
ymyn =
m∑+n
p=
apyp
for appropriate choices of the coefficients ap and with am+n 6 = 0.
For given n > 0, show that the functions
{ym, ymyn : m = 0, 1 , 2 ,... , n − 1 }
are linearly independent.
Let f (x) be a polynomial of degree 3. Explain why the expansion
f (x) = a 0 y 0 (x) + a 1 y 1 (x) + a 2 y 2 (x) + a 3 y 1 (x)y 2 (x)
holds for appropriate choices of ap, p = 0, 1 , 2 , 3. Hence show that ∫ (^) ∞
0
e−xf (x) dx = w 1 f (α 1 ) + w 2 f (α 2 ) ,
where
w 1 = y 1 (α 2 ) y 1 (α 2 ) − y 1 (α 1 )
, w 2 = −y 1 (α 1 ) y 1 (α 2 ) − y 1 (α 1 )
and α 1 , α 2 are the zeros of y 2. You need not construct the polynomials y 1 (x), y 2 (x) explicitly.
Part IB, Paper 1
16A Electromagnetism Suppose the region z < 0 is occupied by an earthed ideal conductor. (a) Derive the boundary conditions on the tangential electric field E that hold on the surface z = 0.
(b) A point charge q, with mass m, is held above the conductor at (0, 0 , d). Show that the boundary conditions on the electric field are satisfied if we remove the conductor and instead place a second charge −q at (0, 0 , −d).
(c) The original point charge is now released with zero initial velocity. Ignoring grav- ity, determine how long it will take for the charge to hit the plane.
17D Fluid Dynamics
A canal has uniform width and a bottom that is horizontal apart from a localised slowly-varying hump of height D(x) whose maximum value is Dmax. Far upstream the water has depth h 1 and velocity u 1. Show that the depth h(x) of the water satisfies the following equation:
D(x) h 1
h h 1
F
h^21 h^2
where F = u^21 /gh 1.
Describe qualitatively how h(x) varies as the flow passes over the hump in the three cases
(i) F < 1 and Dmax < D∗ (ii) F > 1 and Dmax < D∗ (iii) Dmax = D∗,
where D∗^ = h 1
1 − 32 F 1 /^3 + 12 F
Calculate the water depth far downstream in case (iii) when F < 1.
Part IB, Paper 1
18H Statistics What is the critical region C of a test of the null hypothesis H 0 : θ ∈ Θ 0 against the alternative H 1 : θ ∈ Θ 1? What is the size of a test with critical region C? What is the power function of a test with critical region C? State and prove the Neyman–Pearson Lemma. If X 1 ,... , Xn are independent with common Exp(λ) distribution, and 0 < λ 0 < λ 1 , find the form of the most powerful size-α test of H 0 : λ = λ 0 against H 1 : λ = λ 1. Find the power function as explicitly as you can, and prove that it is increasing in λ. Deduce that the test you have constructed is a size-α test of H 0 : λ 6 λ 0 against H 1 : λ = λ 1.
Part IB, Paper 1 [TURN OVER
