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ENM1600 Engineering Mathematics, S1–2020
Assignment 2
Value: 10%. Due Date: Tuesday 28 April 2020.
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QUESTION 1 (20 marks)
The currents I1, I2 and I3, measured in Amperes, in the three branches of a parallel circuit are determined by the following system of linear equations.
7I1 - 3I2 - I3 = -48 -I2 + 4I3 = 15 -4I1 + 8I2 = 13
(a) Write the system in matrix form Ax = b; (2 marks)
(b) Solve the system of linear equations using Cramer’s Rule. (18 marks)
Check your answer by substituting into the system of linear equations.
QUESTION 2 (20 marks)
The steady-state concentrations of chemicals in three vats C1, C2 and C3, measured in milligrams per cubic metre (mg/ m3), are determined by the following system of linear equations.
-3C1 + 4C2 + 2C3 = 24 -9C1 - 5C2 + 8C3 = 18 6C1 - 9C2 + C3 = -17
(a) Write the system of linear equations in augmented matrix form. (3 marks)
(b) Use Gaussian elimination on the augmented matrix to find the three concentrations, and
write the row operation used next to each new row. (17 marks)
Check your answer by substituting into the system of linear equations.
QUESTION 3 (12 marks)
Let f(x) = 2x - 1 and g(x) = sin4x.
(a) Find the composite function f(g(x)) and give its domain. (6 marks)
(b) Find the composite function g(f(x)) and give its domain. (6 marks)
QUESTION 4 (16 marks)
The following rational functions were found in analyzing the motion of a spring-damper system. Express these rational functions as partial fractions:
8s2 + 26s + 15; (8 marks)
s - 20 + 7s2
(b) s3 + 6s2 + 10s. (8 marks)

QUESTION 5 (16 marks)
On a fun park ride the position of the carriage at time t = 0 (in minutes) is given by the parametric function
y=4sin5pt+ ~10 + 2cos4pt~ sinpt
where both x and y are measured from the axis of rotation and are given in metres.
(a) Show the squared distance from the axis of rotation, i.e. d2 = x2 + y2, can be rewritten as
d2 = 20 cos2 4pt + 120 cos 4pt + 116. (7 marks)
(b) Using (a) find all times, t = 0, (in exact form i.e. in terms of fractions) when the distance d is
4 m i.e. d=4ord2 = 16. (7 marks)
(c) What is the first time that the distance d of the carriage is 4 m?
Give your answer exactly (as a fraction) and approximately as a decimal. (2 marks)
QUESTION 6 (16 marks)
The speed, V (km/h), of a car at time t (seconds) is given by
where 0 = t = 100. V = f(t) = ~ pt ~
70 - 10 cos 100
~ pt ~
2 + cos 100
(a) We want to know when the car will reach a certain speed. Solve for the time t in terms of the
speed V , and hence find the formula for the inverse function f_1(t). (6 marks)
(b) Write down the domain of f_1(t). (2 marks)
(c) Plot f(t) from t = 0 to t = 100 and then plot f_1(t) on the same axes but only for the values
of t in the domain of the inverse function found in (b). (6 marks)
(d) Describe the geometric relationship that you can see when you consider both f(t) and f_1(t). (2 marks)
Note: You are encouraged to use MATLAB or Scilab for your plots.
Note that the MATLAB commands for cost and cos_1 t are cos(t) and acos(t). First define values of t from 0 to 100 for your plot of f.
gives values from 0 to 100, in steps of 0.01. Then find the V-values for these t-values using
y1=(70-10.*cos(pi*t1/100))./(2+cos(pi*t1/100)); Then plot f using

Use the command
axis equal
to get equal scales on the axes.
To add the plot of f-1, define new t’s for its domain. That is with
Then calculate the y-values.
/t + 1
For instance for the function 7 sin-1 + 5 we would use the command
t - 2
y2 = 7*asin((t2+1)./(t2-2)) + 5;
We can then plot both functions on the same graph using the command plot(t1,y1,t2,y2).
Note the sin-1 x and cos-1 x functions are evaluated in MATLAB using asin(x) and acos(x) respectively.
Include a plot style to distinguish the graphs, if you wish.
To add an additional plot of y = t to show the geometry, use the same process: define the t’s you want for the plot first (t3=[...]), calculate the y’s, then plot by including t3,y3 after y2 to the previous command
End of Assignment 2 (100 marks Total)

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