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(a) In solving a typical eigenvalue problem, what is given and what is sought?

19 / 01 / 2019 Maths

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Question 1.

Consider the double integral [5 marks]

I =

Z

3

−2

dx

4−x2 Z

0

f(x, y) dy +

Z

0

3

dx

2−

4−x2 Z

0

f(x, y) dy.

(a) Sketch the region of integration (by hand or using computer);

(b) Re-write the integral by reversing the order of integration.

Question 2.

Find the area bounded by the following curves, [5 marks]

x

2 + y

2 = 12, xp

6 = y

2

(x ≥ y

2

/

p

6).

Sketch the area.

1

Question 3.

Find the total electric charge Q of the charged plate bounded by the curves [5 marks]

x = 1/4, y = 0, y2 = 16x (y ≥ 0)

if the charge density is described by the formula ρ(x, y) = 16x + 9y

2/2.

Sketch the plate.

Question 4.

Find the work done by the force [5 marks]

F(x, y, z) = −x

2

y

3

i + 4 j + x k

on moving a charged electric particle along the path given by the equation

r(t) = 2 costi + 2 sin tj + 4 k,

where π/4 ≤ t ≤ 7π/4.

Question 5.

The motion of the spring system with friction is described by the differential [5 marks]

equation

m

d

2

y

dt2

+ c

dy

dt + ky = 0 ,

where y is the displacement, m the mass, c the friction coefficient, and k the

spring constant. Consider the case c

2 = 4mk (called critical).

(a) Find the general solution y(t).

(b) Find and sketch the solution satisfying the initial conditions y(0) = 2,

y

0

(0) = −7.

2

Question 6.

Find the general solution of the differential equation [5 marks]

d

2

y

dx2

− 2

dy

dx + 4y = 2 + 3x + sin x .

Question 7.

Consider the vectors [5 marks]

a = (0, 3, −2, 1, 4); b = (5, 2, 1, 0, −1); c = (7, −3, 6, 21, 0).

(a) Find the length of the vector v = 2a − b;

(b) Are any two of the given vectors parallel or orthogonal? Indicate which,

if any.

Question 8.

Show that the vectors (1, 2, −1, 4),(0, 1, 0, −1),(1, 3, −1, 1),(−2, −4, 2, −1) [5 marks]

are linearly dependent.

Question 9.

Find (a) all eigenvalues of the following matrix and (b) the eigenvectors [5 marks]

corresponding to the smallest eigenvalue.

A =

5 8/3 −2/3

2 2/3 4/3

−4 −4/3 −8/3

 .

Show your working.

3

Question 10.

Write a paragraph answering each of the following questions. [5 marks]

(a) In solving a typical eigenvalue problem, what is given and what is

sought?

(b) What is diagonalisation of a matrix? Describe how diagonalisation

helps to solve systems of linear differential equations.


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