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NATURAL SCIENCES TRIPOS Part IA
Wednesday 12 June 2002 9 to 12
MATHEMATICS (2)
Before you begin read these instructions carefully:
You may submit answers to no more than six questions. All questions carry the same number of marks.
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.
Questions marked with an asterisk (*) require a knowledge of B course material.
At the end of the examination:
For each question you have attempted, attach a blue cover sheet to your answer and write the question number and letter (for example, 3B) in the ‘section’ box on the cover sheet.
List all the questions you attempted on the yellow master cover sheet.
Every cover sheet must bear your candidate number and your desk number.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
1A
(a) Prove that if the vectors a, b and c are non-coplanar, then a, a × b and a × c are non-coplanar. (^) [8]
(b) Solve the vector equation a × r + λr = c
for r, where λ 6 = 0. (^) [12]
2A*
(a) Calculate the derivative with respect to a of
∫ (^) a^2
0
sin ax x
dx.
[8]
(b) Evaluate the integral ∫ (^) ∞
0
cos αx e−βxdx (β > 0)
by writing cos αx = Re(eiαx), or otherwise. (^) [4]
Hence evaluate (^) ∫ (^) ∞
0
x cos αx e−βxdx (β > 0)
using the method of differentiation with respect to a parameter. (^) [8]
3B Express the Cartesian coordinates x, y, z in terms of spherical polar coordinates r, θ, φ. Write down the standard volume element in spherical polar coordinates. (^) [4]
(a) Fluid is contained within a sphere of radius a and centre the origin. The density of the fluid is ρ = μ(2 + (z/r)) where μ is constant. Calculate the total mass of fluid. [6]
(b) A distribution of electric charge has charge density (i.e., charge per unit volume) ρ = λxy with λ a constant. It occupies the region of space with r 6 a and x, y, z > 0. Calculate the total charge. (^) [10]
Paper 2
6C* A vector field is defined by
A = ( y(z^2 −1), x(1−z^2 ), 0 )
(using Cartesian coordinates and components). Compute B = ∇ × A and, from this answer, compute ∇ · B. (^) [4]
Calculate explicitly ∫
C
A · dx ,
D
B · dS ,
H
B · dS ,
where the curve C is the circle of unit radius in the xy plane with centre the origin, the surface D is the disc of unit radius in the xy plane with centre the origin, and the surface H is the hemisphere of unit radius x^2 + y^2 + z^2 = 1 with z > 0. (^) [12]
Explain how your results illustrate (i) Stokes’ Theorem; (ii) the Divergence Theorem. [4]
7D
(a) Find the most general solution of
d^2 y dx^2
dy dx
subject to dy/dx = 0 when x = 0. (^) [10]
(b) Using the substitution x = cos θ, find the general solution of
sin θ
d^2 y dθ^2
− cos θ
dy dθ
[10]
8D
(a) Show that if a matrix A is symmetric and orthogonal, then A−^1 = A. Let B be an orthogonal, anti-symmetric matrix. Is AB necessarily (i) anti-symmetric? (ii) orthogonal? (^) [7]
(b) Determine the eigenvalues and normalised eigenvectors of the matrix
Verify that the eigenvectors are orthogonal. (^) [13]
Paper 2
9E Write down the Taylor expansion of f (x) about the point x = a. (^) [2]
Find, by any method, the first three non-zero terms in the Taylor expansions about x = 0 of the following functions:
(i) (^1) −xe−x ; (^) [5] (ii) tan(x + π/4) ; (^) [5] (iii) ln sec x. (^) [8]
10E* State the comparison and ratio tests for the convergence of a series. (^) [5]
Determine for which real values of α the following series are convergent:
(i)
n=
cos αn n^2 ;^ [3] (ii)
n=
2 α α^2 +
)n ; [4] (iii)
n=1 n
(^2) e−αn (^) ; [3] (iv)
n=1[(n
(^4) + α (^4) ) 1 / (^2) − n (^2) ]. [5]
11F Consider the change of variables
x = e−s^ sin t , y = e−s^ cos t such that u(x, y) = v(s, t).
(a) Use the chain rule to express ∂v/∂s and ∂v/∂t in terms of x, y, ∂u/∂x and ∂u/∂y. [6]
(b) Find, similarly, an expression for ∂^2 v/∂t^2. (^) [6]
(c) Hence transform the equation
y^2
∂^2 u ∂x^2
− 2 xy
∂^2 u ∂x∂y
∂^2 u ∂y^2
into a partial differential equation for v. (^) [8]
12F
(a) Consider a function f (x, y). Sketch the contours of constant f in the vicinity of (i) a maximum; (ii) a minimum; (iii) a saddle point. (^) [5]
(b) Find all stationary points of the function
f (x, y) = x^4 − y^4 − x^2 + 4y^2
and classify each as a maximum, minimum or saddle point. Draw a diagram showing the positions of all stationary points in the xy plane and sketch contours on which f (x, y) is constant. (^) [15]
Paper 2
