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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 | [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) 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. [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 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. | [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

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 ,

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.

2x5/2 ! 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

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.

1

Last Updated on February 10, 2019 by Essay Pro