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MATHEMATICAL TRIPOS Part IA
Thursday 30 May 2002 9.00 to 12.
PAPER 1
Before you begin read these instructions carefully.
Each question in Section II carries twice the credit of each question in Section I. You may attempt all four questions in Section I. In Section II at most five answers will be taken into account and no more than three answers on each course will be taken into account.
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 three bundles, marked B, C and D according to the code letter affixed to each question. Attach a blue cover sheet to each bundle; write the code in the box marked ‘SECTION’ on the cover sheet. Do not tie up questions from Section I and Section II in separate bundles.
You must also complete a green master cover sheet listing all the questions attempted by you.
Every cover sheet must bear your examination number and desk number.
SECTION I
1B Algebra and Geometry
(a) State the Orbit-Stabilizer Theorem for a finite group G acting on a set X. (b) Suppose that G is the group of rotational symmetries of a cube C. Two regular tetrahedra T and T ′^ are inscribed in C, each using half the vertices of C. What is the order of the stabilizer in G of T?
2D Algebra and Geometry
State the Fundamental Theorem of Algebra. Define the characteristic equation for an arbitrary 3 × 3 matrix A whose entries are complex numbers. Explain why the matrix must have three eigenvalues, not necessarily distinct.
Find the characteristic equation of the matrix
A =
0 0 i 0 −i 0
and hence find the three eigenvalues of A. Find a set of linearly independent eigenvectors, specifying which eigenvector belongs to which eigenvalue.
3C Analysis I
Suppose an ∈ R for n > 1 and a ∈ R. What does it mean to say that an → a as n → ∞? What does it mean to say that an → ∞ as n → ∞?
Show that, if an 6 = 0 for all n and an → ∞ as n → ∞, then 1/an → 0 as n → ∞. Is the converse true? Give a proof or a counter example.
Show that, if an 6 = 0 for all n and an → a with a 6 = 0, then 1/an → 1 /a as n → ∞.
4C Analysis I
Show that any bounded sequence of real numbers has a convergent subsequence.
Give an example of a sequence of real numbers with no convergent subsequence. Give an example of an unbounded sequence of real numbers with a convergent subsequence.
Paper 1
9C Analysis I
State some version of the fundamental axiom of analysis. State the alternating series test and prove it from the fundamental axiom.
In each of the following cases state whether
n=1 an^ converges or diverges and prove your result. You may use any test for convergence provided you state it correctly.
(i) an = (−1)n(log(n + 1))−^1.
(ii) a 2 n = (2n)−^2 , a 2 n− 1 = −n−^2.
(iii) a 3 n− 2 = −(2n − 1)−^1 , a 3 n− 1 = (4n − 1)−^1 , a 3 n = (4n)−^1. (iv) a 2 n+r = (−1)n(2n^ + r)−^1 for 0 6 r 6 2 n^ − 1, n > 0.
10C Analysis I
Show that a continuous real-valued function on a closed bounded interval is bounded and attains its bounds.
Write down examples of the following functions (no proof is required). (i) A continuous function f 1 : (0, 1) → R which is not bounded.
(ii) A continuous function f 2 : (0, 1) → R which is bounded but does not attain its bounds.
(iii) A bounded function f 3 : [0, 1] → R which is not continuous. (iv) A function f 4 : [0, 1] → R which is not bounded on any interval [a, b] with 0 6 a < b 6 1.
[Hint: Consider first how to define f 4 on the rationals.]
Paper 1
11C Analysis I
State the mean value theorem and deduce it from Rolle’s theorem.
Use the mean value theorem to show that, if h : R → R is differentiable with h′(x) = 0 for all x, then h is constant.
By considering the derivative of the function g given by g(x) = e−axf (x), find all the solutions of the differential equation f ′(x) = af (x) where f : R → R is differentiable and a is a fixed real number.
Show that, if f : R → R is continuous, then the function F : R → R given by
F (x) =
∫ (^) x
0
f (t) dt
is differentiable with F ′(x) = f (x).
Find the solution of the equation
g(x) = A +
∫ (^) x
0
g(t) dt
where g : R → R is differentiable and A is a real number. You should explain why the solution is unique.
12C Analysis I
Prove Taylor’s theorem with some form of remainder. An infinitely differentiable function f : R → R satisfies the differential equation
f (3)(x) = f (x)
and the conditions f (0) = 1, f ′(0) = f ′′(0) = 0. If R > 0 and j is a positive integer, explain why we can find an Mj such that
|f (j)(x)| 6 Mj
for all x with |x| 6 R. Explain why we can find an M such that
|f (j)(x)| 6 M
for all x with |x| 6 R and all j > 0.
Use your form of Taylor’s theorem to show that
f (x) =
∑^ ∞
n=
x^3 n (3n)!
END OF PAPER
Paper 1
