Mathematics

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.

Q2 Calculate the mean, standard deviation and variance for a set of ungrouped data · Completing a tabular approach to processing ungrouped data.
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.
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.

Q5 Use two of the compound angle formulae and verify their results. · Simplify trigonometric terms and calculate complete values using compound formulae.
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.
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.
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.
Q9 Use complex numbers to solve a parallel arrangement of impedances giving the answer in both Cartesian and polar form · See Task.
Q10 Use differential calculus to find the maximum/minimum for an engineering problem. · See Task.
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. · See Task.
Q12 Use numerical integration and integral calculus to analyse the results of a complex engineering problem · See Task.

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 parall

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