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MATHEMATICAL TRIPOS Part IA
Monday 31 May 2010 1:30 pm to 4:30 pm
PAPER 4
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 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 sheets 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
1E Numbers and Sets
(a) Find the smallest residue x which equals 28! 13^28 (mod 31).
[You may use any standard theorems provided you state them correctly.]
(b) Find all integers x which satisfy the system of congruences
x ≡ 1 (mod 2) , 2 x ≡ 1 (mod 3) , 2 x ≡ 4 (mod 10) , x ≡ 10 (mod 67).
2E Numbers and Sets
(a) Let r be a real root of the polynomial f (x) = xn^ + an− 1 xn−^1 + · · · + a 0 , with integer coefficients ai and leading coefficient 1. Show that if r is rational, then r is an integer.
(b) Write down a series for e. By considering q!e for every natural number q , show that e is irrational.
3B Dynamics and Relativity
A particle of mass m and charge q moves with trajectory r(t) in a constant magnetic field B = Bˆz. Write down the Lorentz force on the particle and use Newton’s Second Law to deduce that r˙ − ω r × zˆ = c ,
where c is a constant vector and ω is to be determined. Find c and hence r(t) for the initial conditions r(0) = aˆx and r˙(0) = uˆy + vˆz
where a, u and v are constants. Sketch the particle’s trajectory in the case aω + u = 0.
[Unit vectors ˆx, ˆy, ˆz correspond to a set of Cartesian coordinates. ]
Part IA, Paper 4
SECTION II
5E Numbers and Sets
The Fibonacci numbers Fn are defined for all natural numbers n by the rules
F 1 = 1 , F 2 = 1 , Fn = Fn− 1 + Fn− 2 for n > 3.
Prove by induction on k that, for any n ,
Fn+k = Fk Fn+1 + Fk− 1 Fn for all k > 2.
Deduce that F 2 n = Fn(Fn+1 + Fn− 1 ) for all n > 2.
Put L 1 = 1 and Ln = Fn+1 + Fn− 1 for n > 1. Show that these (Lucas) numbers Ln satisfy
L 1 = 1 , L 2 = 3 , Ln = Ln− 1 + Ln− 2 for n > 3.
Show also that, for all n, the greatest common divisor (Fn, Fn+1) is 1, and that the greatest common divisor (Fn, Ln) is at most 2.
6E Numbers and Sets
State and prove Fermat’s Little Theorem.
Let p be an odd prime. If p 6 = 5 , show that p divides 10n^ − 1 for infinitely many natural numbers n.
Hence show that p divides infinitely many of the integers
5 , 55 , 555 , 5555 ,....
Part IA, Paper 4
7E Numbers and Sets (a) Let A, B be finite non–empty sets, with |A| = a, |B| = b. Show that there are ba^ mappings from A to B. How many of these are injective?
(b) State the Inclusion–Exclusion principle.
(c) Prove that the number of surjective mappings from a set of size n onto a set of size k is ∑k
i=
(−1)i
k i
(k − i)n^ for n > k > 1.
Deduce that n! =
∑^ n
i=
(−1)i^
( (^) n i
(n − i)n^.
8E Numbers and Sets What does it mean for a set to be countable?
Show that Q is countable, but R is not. Show also that the union of two countable sets is countable.
A subset A of R has the property that, given ǫ > 0 and x ∈ R , there exist reals a, b with a ∈ A and b /∈ A with |x − a| < ǫ and |x − b| < ǫ. Can A be countable? Can A be uncountable? Justify your answers.
A subset B of R has the property that given b ∈ B there exists ǫ > 0 such that if 0 < |b − x| < ǫ for some x ∈ R, then x /∈ B. Is B countable? Justify your answer.
Part IA, Paper 4 [TURN OVER
10B Dynamics and Relativity A particle of unit mass moves in a plane with polar coordinates (r, θ) and compo- nents of acceleration (¨r − r θ˙^2 , r¨θ + 2˙r θ˙). The particle experiences a force corresponding to a potential −Q/r. Show that
E =
r˙^2 + U (r) and h = r^2 θ˙
are constants of the motion, where
U (r) = h^2 2 r^2
Q
r
Sketch the graph of U (r) in the cases Q > 0 and Q < 0.
(a) Assuming Q > 0 and h > 0, for what range of values of E do bounded orbits exist? Find the minimum and maximum distances from the origin, rmin and rmax, on such an orbit and show that rmin + rmax =
Q
|E|
Prove that the minimum and maximum values of the particle’s speed, vmin and vmax, obey
vmin + vmax =
2 Q
h
(b) Now consider trajectories with E > 0 and Q of either sign. Find the distance of closest approach, rmin, in terms of the impact parameter, b, and v∞, the limiting value of the speed as r → ∞. Deduce that if b ≪ |Q|/v^2 ∞ then, to leading order,
rmin ≈
2 |Q|
v^2 ∞
for Q < 0 , rmin ≈ b^2 v ∞^2 2 Q
for Q > 0.
Part IA, Paper 4 [TURN OVER
11B Dynamics and Relativity
Consider a set of particles with position vectors ri(t) and masses mi, where i = 1, 2 ,... , N. Particle i experiences an external force Fi and an internal force Fij from particle j, for each j 6 = i. Stating clearly any assumptions you need, show that
dP dt
= F and dL dt
= G,
where P is the total momentum, F is the total external force, L is the total angular momentum about a fixed point a, and G is the total external torque about a.
Does the result dL dt
= G still hold if the fixed point a is replaced by the centre of
mass of the system? Justify your answer.
Suppose now that the external force on particle i is −k dri dt
and that all the particles
have the same mass m. Show that
L(t) = L(0) e−kt/m^.
12B Dynamics and Relativity
A particle A of rest mass m is fired at an identical particle B which is stationary in the laboratory. On impact, A and B annihilate and produce two massless photons whose energies are equal. Assuming conservation of four-momentum, show that the angle θ between the photon trajectories is given by
cos θ = E − 3 mc^2 E + mc^2
where E is the relativistic energy of A.
Let v be the speed of the incident particle A. For what value of v/c will the photons move in perpendicular directions? If v is very small compared with c, show that
θ ≈ π − v/c.
[All quantities referred to are measured in the laboratory frame.]
END OF PAPER
Part IA, Paper 4
