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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.
