### Maths for Energy Systems

December 8, 2017
###### principles of strategic management
December 8, 2017

Assignment title Coursework 2
Assignment value
Instructions for Assignment
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General guidance on Assignment ANSWER ALL QUESTIONS
Guidance for breakdown of marks Marks will be awarded for:
a) Mathematical correctness.
b) Appropriate mathematical method.
c) Accuracy and appropriate use of
decimal places and significant
figures.
d) Clarity of presentation.
Method of Feedback
Comments on BART sheet No
Individual electronic feedback Yes
Group electronic feedback No
Other feedback method details Verbal feedback after the deadlineComplex 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) SStatistics 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 subsystems. C7 and C8 are seperate power systems (R=0.95). C6 is a multifibre 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
inputCoursework 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 ,
yy0 = 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.
2×5/2 ! 3y5/3
2×5/2y2/3 dx + 3y35x/33/!2y25x/35/2 dy = 0 .
4! (a) Find the general solutions of the di↵erential equations
8>>>>>>><>>>>>>>:
d2y
dx2 ! 4y = 0 ,
d2y
dx2 ! 6
dy
dx + 13y = 0 ,
d2y
dx2 + 6
dy
dx + 9y = 0 .
(b) Solve the initial value problem
d2y
dx2 ! 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
dx2 + 9y = 2 cos 3x + 3 sin 3x ,
d2y
dx2 !
dy
dx ! 2y = 3x + 4 ,
d2y
dx2 + 9y = 2x2e3x + 5 .
(b) Solve the initial value problem d2y
dx2 + 9y = sin 2x; y (0) = 1, y0 (0) = 0.

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