MATHEMATICAL TRIPOS Part IA 2010
List of Courses
Analysis I
Differential Equations
Dynamics and Relativity
Groups
Numbers and Sets
Probability
Vector Calculus
Vectors and Matrices
Part IA, 2010 List of Questions [TURN OVER
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This is the Exam of Mathematics which includes Belonging, Complexity, Composite Natural Number, Compact Interval, Markov Chains, Coding, Number Theory, Prime Number Theorem etc. Key important points are: Analysis, Differential Equations, Dynamics and Relativity, Groups, Numbers and Sets, Probability, Vector Calculus, Complex Power Series, Radius of Convergence, Power Series
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MATHEMATICAL TRIPOS Part IA 2010
List of Courses
Analysis I
Differential Equations
Dynamics and Relativity
Groups
Numbers and Sets
Probability
Vector Calculus
Vectors and Matrices
Part IA, 2010 List of Questions [TURN OVER
Paper 1, Section I 3D Analysis I Let
n> 0 anz n (^) be a complex power series. State carefully what it means for the power series to have radius of convergence R , with R ∈ [0, ∞].
Suppose the power series has radius of convergence R , with 0 < R < ∞. Show that the sequence | anzn| is unbounded if |z| > R.
Find the radius of convergence of
n> 1 z n/n (^3).
Paper 1, Section I 4E Analysis I Find the limit of each of the following sequences; justify your answers.
(i) 1 + 2 +... + n n^2
(ii) n^ √n ;
(iii) (an^ + bn)^1 /n^ with 0 < a 6 b.
Part IA, 2010 List of Questions
Paper 1, Section II 11D Analysis I Define what it means for a bounded function f : [a, ∞) → R to be Riemann integrable.
Show that a monotonic function f : [a, b] → R is Riemann integrable, where −∞ < a < b < ∞.
Prove that if f : [1, ∞) → R is a decreasing function with f (x) → 0 as x → ∞ , then
n> 1 f^ (n) and^
1 f^ (x)^ dx^ either both diverge or both converge.
Hence determine, for α ∈ R , when
n> 1 n α (^) converges.
Paper 1, Section II 12F Analysis I (a) Let n > 1 and f be a function R → R. Define carefully what it means for f to be n times differentiable at a point x 0 ∈ R.
Set sign(x) =
x/|x|, x 6 = 0, 0 , x = 0. Consider the function f (x) on the real line, with f (0) = 0 and
f (x) = x^2 sign(x)
∣∣cos π x
∣∣ , x 6 = 0.
(b) Is f (x) differentiable at x = 0? (c) Show that f (x) has points of non-differentiability in any neighbourhood of x = 0. (d) Prove that, in any finite interval I, the derivative f ′(x), at the points x ∈ I where it exists, is bounded: |f ′(x)| 6 C where C depends on I.
Part IA, 2010 List of Questions
Paper 2, Section I 1A Differential Equations Find the general solutions to the following difference equations for yn, n ∈ N.
(i) y (^) n+3 − 3 y (^) n+1 + 2 yn = 0, (ii) y (^) n+3 − 3 y (^) n+1 + 2 yn = 2n, (iii) y (^) n+3 − 3 y (^) n+1 + 2 yn = (−2)n, (iv) y (^) n+3 − 3 y (^) n+1 + 2 yn = (−2)n^ + 2n.
Paper 2, Section I 2A Differential Equations Let f (x, y) = g(u, v) where the variables {x, y} and {u, v} are related by a smooth, invertible transformation. State the chain rule expressing the derivatives
∂g ∂u and
∂g ∂v in
terms of ∂f ∂x
and ∂f ∂y
and use this to deduce that
∂^2 g ∂u ∂v
∂x ∂u
∂x ∂v
∂^2 f ∂x^2
∂x ∂u
∂y ∂v
∂x ∂v
∂y ∂u
∂^2 f ∂x ∂y
∂y ∂u
∂y ∂v
∂^2 f ∂y^2
+ H
∂f ∂x
+ K
∂f ∂y
where H and K are second-order partial derivatives, to be determined.
Using the transformation x = uv and y = u/v in the above identity, or otherwise, find the general solution of
x
∂^2 f ∂x^2
y^2 x
∂^2 f ∂y^2
∂f ∂x
y x
∂f ∂y
Part IA, 2010 List of Questions [TURN OVER
Paper 2, Section II 6A Differential Equations (a) By using a power series of the form
y(x) =
∑^ ∞
k=
ak xk
or otherwise, find the general solution of the differential equation
xy′′^ − (1 − x)y′^ − y = 0. (1)
(b) Define the Wronskian W (x) for a second order linear differential equation
y′′^ + p(x)y′^ + q(x)y = 0 (2)
and show that W ′^ + p(x)W = 0. Given a non-trivial solution y 1 (x) of (2) show that W (x) can be used to find a second solution y 2 (x) of (2) and give an expression for y 2 (x) in the form of an integral.
(c) Consider the equation (2) with
p(x) = − P (x) x
and q(x) = − Q(x) x where P and Q have Taylor expansions
P (x) = P 0 + P 1 x +... , Q(x) = Q 0 + Q 1 x +...
with P 0 a positive integer. Find the roots of the indicial equation for (2) with these assumptions. If y 1 (x) = 1 + βx +... is a solution, use the method of part (b) to find the first two terms in a power series expansion of a linearly independent solution y 2 (x), expressing the coefficients in terms of P 0 , P 1 and β.
Part IA, 2010 List of Questions [TURN OVER
Paper 2, Section II 7A Differential Equations (a) Find the general solution of the system of differential equations
x ˙ y ˙ z ˙
x y z
(b) Depending on the parameter λ ∈ R, find the general solution of the system of differential equations
x ˙ y ˙ z ˙
x y z
−λ 1 λ
(^) e 2 t, (2)
and explain why (2) has a particular solution of the form ce^2 t^ with constant vector c ∈ R^3 for λ = 1 but not for λ 6 = 1.
[Hint: decompose
−λ 1 λ
(^) in terms of the eigenbasis of the matrix in (1).]
(c) For λ = −1, find the solution of (2) which goes through the point (0, 1 , 0) at t = 0.
Paper 2, Section II 8A Differential Equations (a) State how the nature of a critical (or stationary) point of a function f (x) with x ∈ Rn^ can be determined by consideration of the eigenvalues of the Hessian matrix H of f (x), assuming H is non-singular.
(b) Let f (x, y) = xy(1 − x − y). Find all the critical points of the function f (x, y) and determine their nature. Determine the zero contour of f (x, y) and sketch a contour plot showing the behaviour of the contours in the neighbourhood of the critical points.
(c) Now let g(x, y) = x^3 y^2 (1 − x − y). Show that (0, 1) is a critical point of g(x, y) for which the Hessian matrix of g is singular. Find an approximation for g(x, y) to lowest non-trivial order in the neighbourhood of the point (0, 1). Does g have a maximum or a minimum at (0, 1)? Justify your answer.
Part IA, 2010 List of Questions
Paper 4, Section II 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, 2010 List of Questions
Paper 4, Section II 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^.
Paper 4, Section II 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.]
Part IA, 2010 List of Questions [TURN OVER
Paper 3, Section II 7D Groups Let G be a group, X a set on which G acts transitively, B the stabilizer of a point x ∈ X.
Show that if g ∈ G stabilizes the point y ∈ X , then there exists an h ∈ G with hgh−^1 ∈ B.
Let G = SL 2 (C), acting on C∪{∞} by M¨obius transformations. Compute B = G∞, the stabilizer of ∞. Given g =
a b c d
∈ G
compute the set of fixed points
x ∈ C ∪ {∞}
∣ gx^ =^ x
Show that every element of G is conjugate to an element of B.
Paper 3, Section II 8D Groups Let G be a finite group, X the set of proper subgroups of G. Show that conjugation defines an action of G on X.
Let B be a proper subgroup of G. Show that the orbit of G on X containing B has size at most the index |G : B|. Show that there exists a g ∈ G which is not conjugate to an element of B.
Part IA, 2010 List of Questions [TURN OVER
Paper 4, 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).
Paper 4, Section I 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.
Part IA, 2010 List of Questions
Paper 4, Section II 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^.
Paper 4, Section II 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, 2010 List of Questions
Paper 2, Section I 3F Probability Jensen’s inequality states that for a convex function f and a random variable X with a finite mean, Ef (X) > f
EX
(a) Suppose that f (x) = xm^ where m is a positive integer, and X is a random variable taking values x 1 ,... , xN > 0 with equal probabilities, and where the sum x 1 +... + xN = 1. Deduce from Jensen’s inequality that
∑^ N
i=
f (xi) > N f
N
(b) N horses take part in m races. The results of different races are independent. The probability for horse i to win any given race is pi > 0, with p 1 +... + pN = 1. Let Q be the probability that a single horse wins all m races. Express Q as a polynomial of degree m in the variables p 1 ,.. ., pN. By using (1) or otherwise, prove that Q > N 1 −m.
Paper 2, Section I 4F Probability Let X and Y be two non-constant random variables with finite variances. The correlation coefficient ρ(X, Y ) is defined by
ρ(X, Y ) =
E
[
(X − EX)(Y − EY )
]
Var X
Var Y
(a) Using the Cauchy–Schwarz inequality or otherwise, prove that
− 1 6 ρ(X, Y ) 6 1.
(b) What can be said about the relationship between X and Y when either (i) ρ(X, Y ) = 0 or (ii) |ρ(X, Y )| = 1. [Proofs are not required.] (c) Take 0 6 r 6 1 and let X, X′^ be independent random variables taking values ±1 with probabilities 1/2. Set
Y =
X, with probability r, X′, with probability 1 − r.
Find ρ(X, Y ).
Part IA, 2010 List of Questions [TURN OVER
Paper 2, Section II 11F Probability In a branching process every individual has probability pk of producing exactly k offspring, k = 0 , 1 ,.. ., and the individuals of each generation produce offspring independently of each other and of individuals in preceding generations. Let Xn represent the size of the nth generation. Assume that X 0 = 1 and p 0 > 0 and let Fn(s) be the generating function of Xn. Thus
F 1 (s) = EsX^1 =
∑^ ∞
k=
pksk, |s| 6 1.
(a) Prove that Fn+1(s) = Fn(F 1 (s)).
(b) State a result in terms of F 1 (s) about the probability of eventual extinction. [No proofs are required.] (c) Suppose the probability that an individual leaves k descendants in the next generation is pk = 1/ 2 k+1, for k > 0. Show from the result you state in (b) that extinction is certain. Prove further that in this case
Fn(s) = n − (n − 1)s (n + 1) − ns , n > 1 ,
and deduce the probability that the nth generation is empty.
Part IA, 2010 List of Questions [TURN OVER
Paper 2, Section II 12F Probability Let X 1 , X 2 be bivariate normal random variables, with the joint probability density function fX 1 ,X 2 (x 1 , x 2 ) =
2 πσ 1 σ 2
1 − ρ^2
exp
[
ϕ(x 1 , x 2 ) 2(1 − ρ^2 )
]
where ϕ(x 1 , x 2 ) =
x 1 − μ 1 σ 1
− 2 ρ
x 1 − μ 1 σ 1
x 2 − μ 2 σ 2
x 2 − μ 2 σ 2
and x 1 , x 2 ∈ R.
(a) Deduce that the marginal probability density function
fX 1 (x 1 ) =
2 πσ 1
exp
[
(x 1 − μ 1 )^2 2 σ^21
]
(b) Write down the moment-generating function of X 2 in terms of μ 2 and σ 2. [No proofs are required.]
(c) By considering the ratio fX 1 ,X 2 (x 1 , x 2 )
fX 2 (x 2 ) prove that, conditional on X 2 = x 2 , the distribution of X 1 is normal, with mean and variance μ 1 + ρσ 1 (x 2 − μ 2 )
σ 2 and σ 12 (1 − ρ^2 ), respectively.
Part IA, 2010 List of Questions