Assignment title Coursework 2

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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 [12]

2) Label the following complex numbers on an Argand diagram: [6]

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 [12]

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 [12]

5) Sketch the locus of z when; [6]

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 [12]

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

2

5

2

x

x

x

[10]

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 . [10]

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. [12]

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. [16]

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. [3]

(ii) it will take between 50 and 125 minutes to deal with a reported leak. [4]

(iii)the mean time to deal with a random sample of 90 reported leaks is

less then 70 minutes. [5]

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? [4]

(ii) The regulatory capacity will be exceeded? [6]

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. [20]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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