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MATHEMATICAL TRIPOS Part IA
Friday 28th May 2004 1.30 to 4.
PAPER 2
Before you begin read these instructions carefully.
Each question in Section II carries twice the number of marks 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.
Additional credit will be awarded for substantially complete answers.
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 B and F according to the code letter affixed to each question. Include in the same bundle questions from Sections 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.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
SECTION I
1B Differential Equations By writing y(x) = mx where m is a constant, solve the differential equation dy dx
x − 2 y 2 x + y
and find the possible values of m.
Describe the isoclines of this differential equation and sketch the flow vectors. Use these to sketch at least two characteristically different solution curves.
Now, by making the substitution y(x) = xu(x) or otherwise, find the solution of the differential equation which satisfies y(0) = 1.
2B Differential Equations
Find two linearly independent solutions of the differential equation d^2 y dx^2
dy dx
Find also the solution of d^2 y dx^2
dy dx
that satisfies
y = 0,
dy dx
= 0 at x = 0.
3F Probability Define the covariance, cov(X, Y ), of two random variables X and Y.
Prove, or give a counterexample to, each of the following statements.
(a) For any random variables X, Y, Z cov(X + Y, Z) = cov(X, Z) + cov(Y, Z).
(b) If X and Y are identically distributed, not necessarily independent, random variables then cov(X + Y, X − Y ) = 0.
4F Probability
The random variable X has probability density function
f (x) =
cx(1 − x) if 0 6 x 6 1 0 otherwise. Determine c, and the mean and variance of X.
Paper 2
7B Differential Equations
Show how a second-order differential equation ¨x = f (x, x˙) may be transformed into a pair of coupled first-order equations. Explain what is meant by a critical point on the phase diagram for a pair of first-order equations. Hence find the critical points of the following equations. Describe their stability type, sketching their behaviour near the critical points on a phase diagram.
(i) ¨x + cos x = 0 (ii) ¨x + x(x^2 + λx + 1) = 0, for λ = 1, 5 / 2.
Sketch the phase portraits of these equations marking clearly the direction of flow.
8B Differential Equations Construct the general solution of the system of equations
x˙ + 4x + 3y = 0 y˙ + 4y − 3 x = 0
in the form (^) ( x(t) y(t)
= x =
∑^2
j=
aj x(j)eλj^ t
and find the eigenvectors x(j)^ and eigenvalues λj.
Explain what is meant by resonance in a forced system of linear differential equations.
Consider the forced system
x˙ + 4x + 3y =
∑^2
j=
pj eλj^ t
y ˙ + 4y − 3 x =
∑^2
j=
qj eλj^ t^.
Find conditions on pj and qj (j = 1, 2) such that there is no resonant response to the forcing.
Paper 2
9F Probability
Let X be a positive-integer valued random variable. Define its probability generating function pX. Show that if X and Y are independent positive-integer valued random variables, then pX+Y = pX pY.
A non-standard pair of dice is a pair of six-sided unbiased dice whose faces are numbered with strictly positive integers in a non-standard way (for example, (2, 2 , 2 , 3 , 5 , 7) and (1, 1 , 5 , 6 , 7 , 8)). Show that there exists a non-standard pair of dice A and B such that when thrown
P {total shown by A and B is n} = P {total shown by pair of ordinary dice is n}
for all 2 6 n 6 12.
[Hint: (x + x^2 + x^3 + x^4 + x^5 + x^6 ) = x(1 + x)(1 + x^2 + x^4 ) = x(1 + x + x^2 )(1 + x^3 ).]
10F Probability
Define the conditional probability P (A | B) of the event A given the event B. A bag contains four coins, each of which when tossed is equally likely to land on either of its two faces. One of the coins shows a head on each of its two sides, while each of the other three coins shows a head on only one side. A coin is chosen at random, and tossed three times in succession. If heads turn up each time, what is the probability that if the coin is tossed once more it will turn up heads again? Describe the sample space you use and explain carefully your calculations.
11F Probability The random variables X 1 and X 2 are independent, and each has an exponential distribution with parameter λ. Find the joint density function of
Y 1 = X 1 + X 2 , Y 2 = X 1 /X 2 ,
and show that Y 1 and Y 2 are independent. What is the density of Y 2?
Paper 2 [TURN OVER
