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NATURAL SCIENCES TRIPOS Part IA
Wednesday 13 June 2007 9 to 12
MATHEMATICS (2)
Before you begin read these instructions carefully:
The paper has two sections, A and B. Section A comprises short questions and carries 20 marks in total. Section B contains ten questions, each carrying 20 marks.
You may submit answers to all of section A, and to no more than five questions from section B.
The approximate number of marks allocated to a part of a question is indicated in the right hand margin.
Write on one side of the paper only and begin each answer on a separate sheet. (For this purpose, your section A attempts should be considered as one single answer.)
Questions marked with an asterisk (*) require a knowledge of B course material.
At the end of the examination:
Tie up all of your section A answer in a single bundle, with a completed blue cover sheet.
Each section B question has a number and a letter (for example, 2Y).
Section B answers must be tied up in separate bundles, marked R, S, T, X, Y or Z according to the letter affixed to each question. Do not join the bundles together. For each bundle, a blue cover sheet must be completed and attached to each bundle, with the appropriate letter written in the section box.
A separate yellow master cover sheet listing all the questions attempted must also be completed. (Your section A answer may be recorded just as A: there is no need to list each individual short question.)
Every cover sheet must bear your examination number and desk number.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS 6 blue cover sheets and treasury tags None Yellow master cover sheet Script paper
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
SECTION A
- Let a = (1, − 2 , 1) and b = (1, 0 , 1). Find a vector perpendicular to both a and b. (^) [1]
- Evaluate the integrals
(a) I =
∫ (^) π
0
sin 2x sin 4x dx
[1]
(b)
J =
∫ (^) π/ 2
0
sin x cos x dx.
[1]
- The function f (x, y) = 2x − x^2 y + y
is given.
(a) Verify that the point (x, y) = (1, 1) is a stationary point of f (x, y). (^) [2]
(b) Find the other stationary point. (^) [1]
- A particle is confined by a potential well given by the function
v(x, y, z) = 3x^2 + 3y^2 + z^2.
(a) Find the components of the force vector f = −∇v on the particle. (^) [1]
(b) Find ∇ · f. (^) [1]
(c) Find ∇ ∧ f. (^) [1]
- The column vectors a =
and b =
are given. Evaluate
(a) aTb (^) [1]
(b) abT. (^) [1]
- Give the modulus and the argument (either in degrees or radians) of (1+i)(
3+i). (^) [2]
- Give the first two non-zero terms of the Taylor expansion at x = 0 of
ln (1 + x).
[2]
Paper 2
SECTION B
1Y
Find in polar coordinates (r, θ) the equation for the circle S which has radius 1 and is centred at x = 1, y = 0. (^) [3]
Find in polar coordinates the equation for the tangent T to the circle S at the point (2, 0). (^) [3]
A curve C (known as the Cissoid of Diocles) is defined as follows. Draw a straight line from the origin O which intersects the circle S (again) at point Q, and intersects the tangent T at point R. The point P on the line is defined so that OP = QR. As the point Q moves around the circle, the point P traces out a curve C. Find the polar equation for the curve C. (^) [6]
Hence or otherwise show that the Cartesian equation for C is
y^2 (2 − x) = x^3.
[4]
Sketch the circle S, the tangent T and the curve C. (^) [4]
2Y
Use the substitution t = tan(x/2) to show that
∫ (^) π/ 2
0
dx 2 + sin x
0
dt t^2 + t + 1
[10]
Hence, or otherwise, show that ∫ (^) π/ 2
0
dx 2 + sin x
π 3
[10]
Paper 2
3X
(a) The probability of an experiment that involves counting events having the result N = n (where n is a non-negative integer) is
Pn = Aρn,
where ρ (0 < ρ < 1) is given. Find the normalising constant A. (^) [3]
Calculate the probability that N > n. (^) [2]
Calculate the probability that N > n, conditional on N > m (n > m). (^) [2]
(b) The probability density function for a continuous random variable X is
f (x) = Bρx^ ≡ Be−λx, (λ = ln(ρ−^1 ))
where x takes values between 0 and ∞. Find the normalising constant B. (^) [3]
Calculate the probability that X > x, conditional on X > y (x > y). (^) [4]
Deduce the probability density function for X, conditional on X > y. (^) [2]
Calculate the variance of X, conditional on X > y. (^) [4]
4Z
(a) Find the general solution of the differential equation
d^2 y dx^2
dy dx
[8]
(b) Find the solution of d^2 y dx^2
dy dx
given that y = dy/dx = 0 at x = 0. (^) [8]
Sketch the solution for x > 0. [4]
Paper 2 [TURN OVER
6Z
(a) A force field F is given in Cartesian coordinates by
F = (2xy + z, x^2 + 2y, x).
Find ∇ ∧ F. [3]
(b) Find a suitable potential ψ such that F = −∇ψ. (^) [3]
(c) Evaluate
F · dx along the straight line connecting the origin to the point (1, 1 , 1) and verify that your result is consistent with the change in potential ψ. (^) [4]
(d) The surface S of an ellipsoid is defined parametrically by x = (b sin θ cos φ, b sin θ sin φ, a cos θ), where 0 6 θ 6 π and 0 6 φ 6 2 π. By using
dS =
∂x ∂θ
∂x ∂φ
dθdφ ,
evaluate directly the integral (^) ∫
S
G · dS
over the surface of the ellipsoid, where
G = (xz^2 , xy^2 , z^3 ).
[You should not use the divergence theorem.] (^) [10]
Paper 2 [TURN OVER
7S
(a) Below are statements about square matrices A, B and C all having the same dimension N × N. Moreover A and B are invertible.
(i) det
B−^1 AB
= det (A) (^) [2]
(ii) Tr (ABC) = Tr (BAC) (^) [2]
(iii) A−^1 + B−^1 = A−^1 (A + B) B−^1. (^) [2]
Indicate which of these statements is true and which is false. If a statement is true, give a brief proof of the relation.
(b) Given
C =
Show that C 2 = C−^1. Hence, or otherwise, compute C 16. (^) [4]
(c) The matrix M is defined by
M =
μ 1 0 1 0 1 0 1 μ
where μ is a real parameter. (i) What condition must μ satisfy for the inverse M−^1 of M to exist? (^) [2]
(ii) Express M−^1 as a function of the parameter μ. (^) [4]
(d) The variables x, y and z satisfy the following set of simultaneous linear equations
μx + y = 1 x + z = 2 y + μz = 1
where μ is a real parameter. (i) Find the values of x, y and z for all nonzero values of μ. (^) [2]
(ii) Determine the solutions of these equations when μ = 0. What is their locus in Cartesian space (x, y, z)? (^) [2]
Paper 2
10T*
Fluid is flowing between two parallel plates situated at y = 0 and y = 1. The driving pressure is turned off at time t = 0 and the fluid velocity u(y, t) subsequently satisfies the partial differential equation
∂u ∂t
= ν
∂^2 u ∂y^2
where ν is a positive constant.
Suppose that u can be expressed as the product of two functions,
u(y, t) = Y (y)T (t).
Show that d^2 Y dy^2
= aY,
where a is an arbitrary constant, and find a corresponding ordinary differential equation for T (t). (^) [8]
The boundary conditions are u(0, t) = u(1, t) = 0 for all t > 0. If initially u(y, 0) = sin(πy), find the solution for u(y, t) in the interval 0 6 y 6 1. (^) [8]
State the principle of superposition for linear differential equations. Bearing this in mind, write down the solution for the initial condition u(y, 0) = sin(πy) + 1 10 sin(3πy).^ [4]
END OF PAPER
Paper 2
