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Assignment attached
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Q1 |
Perform the two basic operations of multiplication and division to a complex number in both rectangular and polar form, to demonstrate the different techniques. |
· Dividing complex numbers in rectangular and polar forms. · Converting complex numbers between polar and rectangular forms and vice versa. |
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Q2 |
Calculate the mean, standard deviation and variance for a set of ungrouped data |
· Completing a tabular approach to processing ungrouped data. |
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Q3 |
Calculate the mean, standard deviation and variance for a set of grouped data |
· Completing a tabular approach to processing grouped data having selected an appropriate group size. |
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Q4 |
Sketch the graph of a sinusoidal trig function and use it to explain and describe amplitude, period and frequency. |
· Calculate various features and coordinates of a waveform and sketch a plot accordingly. · Explain basic elements of a waveform. |
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Q5 |
Use two of the compound angle formulae and verify their results. |
· Simplify trigonometric terms and calculate complete values using compound formulae. |
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Q6 |
Find the differential coefficient for three different functions to demonstrate the use of function of a function and the product and quotient rules |
· Use the chain, product and quotient rule to solve given differentiation tasks. |
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Q7 |
Use integral calculus to solve two simple engineering problems involving the definite and indefinite integral. |
· Complete 3 tasks; one to practise integration with no definite integrals, the second to use definite integrals, the third to plot a graph and identify the area that relates to the definite integrals with a calculated answer for the area within such. |
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Q8 |
Use the laws of logarithms to reduce an engineering law of the type y = axn to a straight line form, then using logarithmic graph paper, plot the graph and obtain the values for the constants a and n. |
· See Task. |
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Q9 |
Use complex numbers to solve a parallel arrangement of impedances giving the answer in both Cartesian and polar form |
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Q10 |
Use differential calculus to find the maximum/minimum for an engineering problem. |
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Q11 |
Using a graphical technique determine the single wave resulting from a combination of two waves of the same frequency and then verify the result using trig formulae. |
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Q12 |
Use numerical integration and integral calculus to analyse the results of a complex engineering problem |
Level of Detail in Solutions: Need to show work leading to final answer
Need
Question 1
(a) Find:
(4 + i2)
(1 + i3)
Use the rules for multiplication and division of complex numbers in rectangular form.
(b) Convert the answer in rectangular form to polar form
(c) Repeat Q1a by first converting the complex numbers to polar form and then using the rules for multiplication and division of complex numbers in polar form.
(d) Convert the answer in polar form to rectangular form.
Question 2
The following data within the working area consists of measurements of resistor values from a production line.
For the sample as a set of ungrouped data, calculate (using at least 2 decimal places):
1. arithmetic mean
1. standard deviation
1. variance
The following data consists of measurements of resistor values from a production line:
51.4
54.1
53.7
55.4
53.1
53.5
54.0
56.0
53.0
55.3
55.0
52.8
55.9
52.8
50.5
54.2
56.2
55.6
52.7
56.1
52.1
54.2
50.2
54.7
56.2
55.6
52.7
52.1
56.1
54.2
50.2
54.7
55.1
54.8
56.5
55.8
55.3
54.5
57.0
56.0
53.9
57.3
55.3
54.4
49.6
54.1
51.6
53.2
54.6
56.4
53.9
50.9
54.0
51.8
56.1
53.2
54.6
56.4
53.9
50.9
54.0
51.8
56.1
‘n’
No.
Value (X)
Mean
(X – Mean)
(X-Mean)2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
= Total X
Mean = Total X / ‘n’(max) =
‘n’ max – 1 (Y) =
Total (X-Mean)2 ……. let’s call this (Z)
Variance
Variance = (Total(X-Mean)2) / (‘n’ max -1)
= Z / Y =
Standard Deviation
Standard Deviation = square-root of Variance
=
Question 3
Using the data in Q2, we now need to arrange the data into groups so that we can “tally” the data accordingly.
No.
Data Range
Gap
Freq. (F)
Mid-Point (X)
(F x X)
Gap x F
(Bar Area)
(X – Mean)
(X-Mean)2
(X-Mean)2 x F
1
2
3
4
5
6
7
8
9
10
11
Total F…Let’s call this (Y)
Total (F x X)
Mean = (Total (F x X)) / (Y)
Total (X-Mean)2 x F ……. let’s call this (Z)
Variance
Variance = (Total(X-Mean)2 x f) / (Total F)
= Z / Y
Standard Deviation
Standard Deviation = square-root of Variance
=
Question 4
1. For a sinusoidal trigonometric function, explain what is meant by:
· amplitude
· periodic time
· frequency
Use diagrams or paragraphs if you like.
(b) An alternating current voltage is given by:
V = 310 Sin (285t + 0.65)
1. sketch the waveform, marking on all main values
1. state whether the waveform is leading or lagging
1. state the phase angle in degrees
1. state the amplitude
1. calculate the periodic time
1. calculate the frequency
Note – for an accurate plot of the waveform you will need to carry out the following steps:
Using the values in the equation work out the Periodic Time. Divide this into 4 quarters:
0 x t
0.25 x t
0.5 x t
0.75 x t
1 x t
Use these values to work out the amplitude for each value of t. Then sketch the plot.
Question 5
Using compound angle formulae, simplify:
(a) Sin (θ – 90o)
(b) Cos (θ + 270o)
In each case, verify your answer by substituting θ = 30o
Question 6
Differentiate:
1. y = (3×2 – 2x)7
1. y = 6×3 .sin4x
1. y = 5 e6x
x – 8
Question 7
(a) Integrate the following:
(i) (4Cos 3θ+ Sin 6θ) dθ
(ii) (2 + Cos 0.83θ) dθ
(b) Evaluate:
(i)
(ii)
(c)
(i) Plot the curve y = 3×2 + 6 between x = 1 and 4
(ii) Find the area under the curve between x = 1 and 4 using integral calculus
Question 8
Create a document titled Laws of Logarithms
Use the laws of logarithms to reduce an engineering law of the type y = axn to a straight line form, then using logarithmic graph paper, plot the graph and obtain the values for the constants a and n.
The following set of results was obtained during an experiment:
X: 2 2.5 3 3.5 4
Y: 8 6.4 5.3 4.6 4
The relationship between the two quantities is of the form y = axb
Complete the law:
(i) using the laws of logarithms to reduce the law to a straight line form and determine the gradient and intercept.
(ii) graphically, using logarithmic graph paper
Question 9
Use complex numbers to solve a parallel arrangement of impedances giving the answer in both Cartesian and polar form
Two impedances Z1 and Z2 are given by the complex numbers:
Z1 = 2 + j10
Z2 = j14
Find the equivalent impedance Z if:
(i) Z = Z1 + Z2 when Z1 and Z2 are in series
(ii) 1 = 1 + 1 when Z1 and Z2 are in parallel
Z Z1 Z2
Question 10
Use complex numbers to solve a parallel arrangement of impedances giving the answer in both Cartesian and polar form Use differential calculus to find the maximum/minimum for an engineering problem.
A sheet of metal is 340 mm x 225 mm and has four equal squares cut out at the corners so that the sides and edges can be turned up to form an open topped box shape.
Calculate:
(i) The lengths of the sides of the cut out squares for the volume of the box to be as big as possible.
(ii) The maximum volume of the box. Z Z1 Z2
Question 11
Using a graphical technique determine the single wave resulting from a combination of two waves of the same frequency and then verify the result using trig formulae.
A sheet of metal is 340 mm x 225 mm and has four equal squares cut out at the corners so that the sides and edges can be turned up to form an open topped box shape.
Calculate:
1. Using a vector or phasor diagram add the following waveforms together and present your answer in the form Vr = V sin(ωt+α).
Show all working including phasor diagrams.
Add the following together:
V1=10sin(ωt)
V2=20sin(ωt+)
1. Verify your result by using trigonometric formulae.
Question 12
In calculating the capacity of absorption towers in Chemical Engineering it is necessary to evaluate certain definite integrals by approximate methods. Evaluate the following definite integrals by use of Simpson’s rule and the trapezoid rule using the given data (xi is an empirical function of x, yi is an empirical function of y):
1
)
x
–
x
(
=
s
Deviation
dard
tan
S
1
–
n
)
x
–
x
(
s
Variance
2
2
2
–
å
å
=
n
1
)
x
–
x
(
=
s
Deviation
dard
tan
S
1
–
n
)
x
–
x
(
s
Variance
2
2
2
–
å
å
=
n
Variance s
x
–
x
)
n
S
dard Devia
tion s
=
x
–
x
)
2
2
2
=
å
å
(
tan
(
n
(
)
ò
+
+
4
1
2
7
6
2
dx
x
x
(
)
dx
e
x
ò
6
2
125
.
0
4
2
p
