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Complex Numbers Section
1) Using the cartesian form of the complex number only, if u = 4 + 3j and

 v = 6 – j determine the following: a) u + v b) u – v c) uv d) vu e) u / v f) v / u  2) Label the following complex numbers on an Argand diagram: 

a) z1 = 3 – 2j b) z2 = -1 – 3j c) z3 = -1 + 3j d) z4 = 2j e) z5 = -3
3) Convert each of them into the polar form of the complex number, and
perfom the following multiplications:
a) z1 z2 b) z2 z3 c) z3 z4  4) Using the Exponential form of the complex number z = 4 – 3j find:
a) z2 b) z -3 c) z1/2 d) z -1/3  5) Sketch the locus of z when;  a) b)
Integration Section
1) Integrate the following expressions with respect to x :
a) dx
x x
x
I
³

3 9 1
2 3
2 b) dx
x x
x
I
³

3 9 1
2 3
2  2) Find the mean value of the following expression for 5 ≤ x ≤ 6 :
2
5
2

x
x
x
 3) A rotating system can be modelled with the equation:
s 3sin(t) 4cos(t)
Find the RMS value of the function, s between t = 0 to t = π/2 .  4) The equation of a volume of solid of revolution rotated about the y-axis is
given by:
V xy dx
b³a
2S
If y ln(2x 4) use integration by parts and substitution to calculate the
volume between
x = 1 and x = 3.  5
z 2 j 3 arg( z) S
Statistics and Probability Section
1) The measured current I and for values of applied voltage V in a circuit are
related by the law
I = aVn where a and n are constants. From the
values given below use method of least squares to determine the values
of
a and n that best fit the set of recorded values.  V 8 12 15 20 28 36
I 61.7 83.4 98.7 122.4 157.5 190.5
2) A gas supplier maintains a team of engineers who are available to deal
with leaks reported by customers. Most leaks can be dealt with quickly but
some require a long time. The time (excluding travelling time) taken to
deal with a reported leak is found to have a mean of 65 minutes and a
standard deviation of 60 minutes. Assuming that the times follow a
normal distribution, estimate the probability that:
(i) it will take more than 185 minutes to deal with a reported leak.
 (ii) it will take between 50 and 125 minutes to deal with a reported leak.  (iii)the mean time to deal with a random sample of 90 reported leaks is

 less then 70 minutes.  3) An average of 36 vehicles an hour pass along a ‘weak’ bridge each taking 20 seconds to travel along it. The bridge has a regulatory capacity of two vehicles. Using Poisson’s probability distribution formula:

!
( )
ex
f x
x P
P
where µ is the mean value, and x is the number of vehicles, and given
free movement of vehicles, what is the probability that at a given instant;

 (i) The bridge is not being used?  (ii) The regulatory capacity will be exceeded?  4) The process in Figure 1 below is made up of a number of distinct sub systems. C7 and C8 are seperate power systems (R=0.95). C6 is a multi fibre signal wire comprising 6 fully redundant fibres who average 1 flaw per fibre but must not have any for successful signal transmission. C3, C4 and C5 are identical components (R=0.85) and at least two must work for the process to maintain throughput. C2 is a thermal process whose unreliability increases as temperature increases with a probability distribution of N(45, 10). C1 is a quality checking human who works too hard and whose reliability drops by 3% for each hour of their shift above 4. If the temperature reaches 350, 2 hours into the shift and 400 after 6 hours. Compare the system reliability at these two stages. 

Figure 1
(End of Coursework 2 Estimated time required: 10 hours,
Total marks: 200 to include 60 for the Differential Equations section)

 C1 C2

C3

 C4

C5
C6
C7
C8
output
input

## Coursework on Differential Equations for CSM1040-Maths for Energy Systems

Hamid Alemi Ardakani, Department of Mathematics, University of Exeter, Penryn Campus, Cornwall TR10 9EZ, UK
1! (a) Solve the initial value problem
dy
dx
+ y = 2 sin x , y (0) = 1 .
(b) Find the general solution of
xy0 = 3y + x4 cos x .
Prime denotes derivative with respect to x.
(c) Find the particular solution of the di↵erential equation in (b) for which
y (2) = 0.
2
! (a) Solve the di↵erential equations
dy
dx
=
1 +
px
1 + py ,
yy
0 = x !y2 + 1.
(b) Find the explicit particular solution of dy
dx
= 6 exp (2x ! y) for which y (0) = 0.
3
! Verify that the following di↵erential equation is exact; then solve it.
2
x5/2 ! 3y5/3
2x5/2y2/3 dx + 3y35x/33/!2y25x/35/2 dy = 0 .
4! (a) Find the general solutions of the di↵erential equations
8>>>>>>><>>>>>>>:
d2y
dx
2 ! 4y = 0 ,
d
2y
dx
2 ! 6
dy
dx
+ 13y = 0 ,
d
2y
dx
2 + 6
dy
dx
+ 9y = 0 .
(b) Solve the initial value problem
d2y
dx
2 ! 6
dy
dx
+ 25y = 0 , y (0) = 3 , y0 (0) = 1 .
5! (a) Find a particular solution yp of the given inhomogeneous di↵erential equations.
8>>>>>>><>>>>>>>:
d2y
dx
2 + 9y = 2 cos 3x + 3 sin 3x ,
d
2y
dx
2 !
dy
dx
! 2y = 3x + 4 ,
d
2y
dx
2 + 9y = 2x2e3x + 5 .
(b) Solve the initial value problem d2y
dx
2 + 9y = sin 2x; y (0) = 1, y0 (0) = 0.
1

Last Updated on February 10, 2019 by EssayPro