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MATHEMATICAL TRIPOS Part IA
Monday 6 June 2005 9 to 12
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 C and E 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 gold cover sheet to each bundle; write the code letter in the box marked ‘EXAMINER LETTER’ on the cover sheet.
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 REQUIRMENTS 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
1E Numbers and Sets
Find the unique positive integer a with a ≤ 19, for which
17! · 316 ≡ a (mod 19).
Results used should be stated but need not be proved.
Solve the system of simultaneous congruences
x ≡ 1 (mod 2), x ≡ 1 (mod 3), x ≡ 3 (mod 4), x ≡ 4 (mod 5).
Explain very briefly your reasoning.
2E Numbers and Sets
Give a combinatorial definition of the binomial coefficient
( (^) n m
for any non-negative integers n, m.
Prove that
( (^) n m
( (^) n n−m
for 0 ≤ m ≤ n.
Prove the identities
( n k
k l
n l
n − l k − l
and ∑k
i=
m i
n k − i
n + m k
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SECTION II
5E Numbers and Sets
What does it mean for a set to be countable? Show that Q × Q is countable, and R is not countable.
Let D be any set of non-trivial discs in a plane, any two discs being disjoint. Show that D is countable.
Give an example of a set C of non-trivial circles in a plane, any two circles being disjoint, which is not countable.
6E Numbers and Sets
Let R be a relation on the set S. What does it mean for R to be an equivalence relation on S? Show that if R is an equivalence relation on S, the set of equivalence classes forms a partition of S.
Let G be a group, and let H be a subgroup of G. Define a relation R on G by a R b if a−^1 b ∈ H. Show that R is an equivalence relation on G, and that the equivalence classes are precisely the left cosets gH of H in G. Find a bijection from H to any other coset gH. Deduce that if G is finite then the order of H divides the order of G.
Let g be an element of the finite group G. The order o(g) of g is the least positive integer n for which gn^ = 1, the identity of G. If o(g) = n, then G has a subgroup of order n; deduce that g|G|^ = 1 for all g ∈ G.
Let m be a natural number. Show that the set of integers in { 1 , 2 ,... , m} which are prime to m is a group under multiplication modulo m. [You may use any properties of multiplication and divisibility of integers without proof, provided you state them clearly.]
Deduce that if a is any integer prime to m then aφ(m)^ ≡ 1 (mod m), where φ is the Euler totient function.
7E Numbers and Sets
State and prove the Principle of Inclusion and Exclusion.
Use the Principle to show that the Euler totient function φ satisfies
φ(pc 11 · · · pc rr ) = pc 11 −^1 (p 1 − 1) · · · pc rr −^1 (pr − 1).
Deduce that if a and b are coprime integers, then φ(ab) = φ(a)φ(b), and more generally, that if d is any divisor of n then φ(d) divides φ(n).
Show that if φ(n) divides n then n = 2c 3 d^ for some non-negative integers c, d.
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8E Numbers and Sets
The Fibonacci numbers are defined by the equations F 0 = 0 , F 1 = 1 and Fn+1 = Fn + Fn− 1 for any positive integer n. Show that the highest common factor (Fn+1, Fn) is 1.
Let n be a natural number. Prove by induction on k that for all positive integers k,
Fn+k = FkFn+1 + Fk− 1 Fn.
Deduce that Fn divides Fnl for all positive integers l. Deduce also that if m ≥ n then (Fm, Fn) = (Fm−n, Fn).
9C Dynamics
A particle of mass m and charge q moving in a vacuum through a magnetic field B and subject to no other forces obeys
m ¨r = q r˙ × B,
where r(t) is the location of the particle.
For B = (0, 0 , B) with constant B, and using cylindrical polar coordinates r = (r, θ, z), or otherwise, determine the motion of the particle in the z = 0 plane if its initial speed is u 0 with ˙z = 0. [Hint: Choose the origin so that r˙ = 0 and r¨ = 0 at t = 0.]
Due to a leak, a small amount of gas enters the system, causing the particle to experience a drag force D = −μr˙, where μ qB. Write down the new governing equations and show that the speed of the particle decays exponentially. Sketch the path followed by the particle. [Hint: Consider the equations for the velocity in Cartesian coordinates; you need not apply any initial conditions.]
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11C Dynamics
A puck of mass m located at r = (x, y) slides without friction under the influence of gravity on a surface of height z = h(x, y). Show that the equations of motion can be approximated by ¨r = −g∇h ,
where g is the gravitational acceleration and the small slope approximation sin φ ≈ tan φ is used.
Determine the motion of the puck when h(x, y) = αx^2. Sketch the surface
h(x, y) = h(r) =
r^2
r
as a function of r, where r^2 = x^2 + y^2. Write down the equations of motion of the puck on this surface in polar coordinates r = (r, θ) under the assumption that the small slope approximation can be used. Show that L, the angular momentum per unit mass about the origin, is conserved. Show also that the initial kinetic energy per unit mass of the puck is E 0 = 12 L^2 /r^20 if the puck is released at radius r 0 with negligible radial velocity. Determine and sketch ˙r^2 as a function of r for this release condition. What condition relating L, r 0 and g must be satisfied for the orbit to be bounded?
12C Dynamics In an experiment a ball of mass m is released from a height h 0 above a flat, horizontal plate. Assuming the gravitational acceleration g is constant and the ball falls through a vacuum, find the speed u 0 of the ball on impact.
Determine the speed u 1 at which the ball rebounds if the coefficient of restitution for the collision is γ. What fraction of the impact energy is dissipated during the collision? Determine also the maximum height hn the ball reaches after the nth^ bounce, and the time Tn between the nth^ and (n + 1)th^ bounce. What is the total distance travelled by the ball before it comes to rest if γ < 1?
If the experiment is repeated in an atmosphere then the ball experiences a drag force D = −α |u| u, where α is a dimensional constant and u the instantaneous velocity of the ball. Write down and solve the modified equation for u(t) before the ball first hits the plate.
END OF PAPER
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