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MATHEMATICAL TRIPOS Part IA
Monday 2 June 2003 9 to 12
PAPER 4
Before you begin read these instructions carefully.
Each question in Section II carries twice the credit of each question in Section I. In Section I, you may attempt all four questions. 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 bundles, marked C and E according to the code letter affixed to each question. Attach a blue cover sheet to each bundle; write the code letter 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 you have attempted.
Every cover sheet must bear your examination number and desk number.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
SECTION I
1C Numbers and Sets
(i) Prove by induction or otherwise that for every n > 1,
∑^ n
r=
r^3 =
( (^) n ∑
r=
r
(ii) Show that the sum of the first n positive cubes is divisible by 4 if and only if n ≡ 0 or 3 (mod 4).
2C Numbers and Sets What is an equivalence relation? For each of the following pairs (X, ∼), determine whether or not ∼ is an equivalence relation on X:
(i) X = R, x ∼ y iff x − y is an even integer;
(ii) X = C \ { 0 }, x ∼ y iff xy¯ ∈ R ; (iii) X = C \ { 0 }, x ∼ y iff xy¯ ∈ Z;
(iv) X = Z \ { 0 }, x ∼ y iff x^2 − y^2 is ±1 times a perfect square.
3E Dynamics
Because of an accident on launching, a rocket of unladen mass M lies horizontally on the ground. It initially contains fuel of mass m 0 , which ignites and is emitted horizontally at a constant rate and at uniform speed u relative to the rocket. The rocket is initially at rest. If the coefficient of friction between the rocket and the ground is μ, and the fuel is completely burnt in a total time T , show that the final speed of the rocket is
u log
M + m 0 M
− μgT.
Paper 4
SECTION II
5C Numbers and Sets
Define what is meant by the term countable. Show directly from your definition that if X is countable, then so is any subset of X.
Show that N × N is countable. Hence or otherwise, show that a countable union of countable sets is countable. Show also that for any n > 1, Nn^ is countable.
A function f : Z → N is periodic if there exists a positive integer m such that, for every x ∈ Z, f (x + m) = f (x). Show that the set of periodic functions f : Z → N is countable.
6C Numbers and Sets
(i) Prove Wilson’s theorem: if p is prime then (p − 1)! ≡ −1 (mod p).
Deduce that if p ≡ 1 (mod 4) then (( p − 1 2
≡ − 1 (mod p).
(ii) Suppose that p is a prime of the form 4k + 3. Show that if x^4 ≡ 1 (mod p) then x^2 ≡ 1 (mod p).
(iii) Deduce that if p is an odd prime, then the congruence
x^2 ≡ − 1 (mod p)
has exactly two solutions (modulo p) if p ≡ 1 (mod 4), and none otherwise.
7C Numbers and Sets
Let m, n be integers. Explain what is their greatest common divisor (m, n). Show from your definition that, for any integer k, (m, n) = (m + kn, n).
State Bezout’s theorem, and use it to show that if p is prime and p divides mn, then p divides at least one of m and n.
The Fibonacci sequence 0, 1, 1, 2, 3, 5, 8,... is defined by x 0 = 0, x 1 = 1 and xn+1 = xn + xn− 1 for n > 1. Prove:
(i) (xn+1, xn) = 1 and (xn+2, xn) = 1 for every n > 0;
(ii) xn+3 ≡ xn (mod 2) and xn+8 ≡ xn (mod 3) for every n > 0; (iii) if n ≡ 0 (mod 5) then xn ≡ 0 (mod 5).
Paper 4
8C Numbers and Sets
Let X be a finite set with n elements. How many functions are there from X to X? How many relations are there on X?
Show that the number of relations R on X such that, for each y ∈ X, there exists at least one x ∈ X with xRy, is (2n^ − 1)n.
Using the inclusion–exclusion principle or otherwise, deduce that the number of such relations R for which, in addition, for each x ∈ X, there exists at least one y ∈ X with xRy, is ∑n
k=
(−1)k^
( (^) n k
(2n−k^ − 1)n.
9E Dynamics
Write down the equation of motion for a point particle with mass m, charge e, and position vector x(t) moving in a time-dependent magnetic field B(x, t) with vanishing electric field, and show that the kinetic energy of the particle is constant. If the magnetic field is constant in direction, show that the component of velocity in the direction of B is constant. Show that, in general, the angular momentum of the particle is not conserved.
Suppose that the magnetic field is independent of time and space and takes the form B = (0, 0 , B) and that A˙ is the rate of change of area swept out by a radius vector joining the origin to the projection of the particle’s path on the (x, y) plane. Obtain the equation d dt
m A˙ +
eBr^2 4
where (r, θ) are plane polar coordinates. Hence obtain an equation replacing the equation of conservation of angular momentum.
Show further, using energy conservation and (∗), that the equations of motion in plane polar coordinates may be reduced to the first order non-linear system
r˙ =
v^2 −
2 c mr
erB 2 m
θ^ ˙ = 2 c mr^2
eB 2 m
where v and c are constants.
Paper 4 [TURN OVER
11E Dynamics
State the parallel axis theorem and use it to calculate the moment of inertia of a uniform hemisphere of mass m and radius a about an axis through its centre of mass and parallel to the base.
[You may assume that the centre of mass is located at a distance 38 a from the flat face of the hemisphere, and that the moment of inertia of a full sphere about its centre is 25 M a^2 , with M = 2m.]
The hemisphere initially rests on a rough horizontal plane with its base vertical. It is then released from rest and subsequently rolls on the plane without slipping. Let θ be the angle that the base makes with the horizontal at time t. Express the instantaneous speed of the centre of mass in terms of b and the rate of change of θ, where b is the instantaneous distance from the centre of mass to the point of contact with the plane. Hence write down expressions for the kinetic energy and potential energy of the hemisphere and deduce that ( (^) dθ
dt
15 g cos θ (28 − 15 cos θ)a
12E Dynamics
Let (r, θ) be plane polar coordinates and er and eθ unit vectors in the direction of increasing r and θ respectively. Show that the velocity of a particle moving in the plane with polar coordinates
r(t), θ(t)
is given by
x˙ = ˙rer + r θ˙eθ ,
and that the unit normal n to the particle path is parallel to
r θ˙er − r˙eθ.
Deduce that the perpendicular distance p from the origin to the tangent of the curve r = r(θ) is given by r^2 p^2
r^2
( (^) dr
dθ
The particle, whose mass is m, moves under the influence of a central force with potential V (r). Use the conservation of energy E and angular momentum h to obtain the equation 1 p^2
2 m
E − V (r)
h^2
Hence express θ as a function of r as the integral
θ =
hr−^2 dr √ 2 m
E − Veff (r)
where
Veff (r) = V (r) +
h^2 2 mr^2
Evaluate the integral and describe the orbit when V (r) =
c r^2
, with c a positive constant.
Paper 4
