ENM1600 Engineering Mathematics, S1–2020

Enactment 2

Value: 10%. Due Era: 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 congruous tour are resolute by the controlthcoming method of rectirectidirect equations.

7I1 – 3I2 – I3 = -48 -I2 + 4I3 = 15 -4I1 + 8I2 = 13

(a) Write the method in matrix coercionm Ax = b; (2 marks)

(b) Solve the method of rectirectidirect equations using Cramer’s Rule. (18 marks)

Restrain your exculpation by substituting into the method of rectirectidirect equations.

QUESTION 2 (20 marks)

The steady-state concentrations of chemicals in three vats C1, C2 and C3, measured in milligrams per firm metre (mg/ m3), are resolute by the controlthcoming method of rectirectidirect equations.

-3C1 + 4C2 + 2C3 = 24 -9C1 – 5C2 + 8C3 = 18 6C1 – 9C2 + C3 = -17

(a) Write the method of rectirectidirect equations in augmented matrix coercionm. (3 marks)

(b) Portraiture Gaussian exportation on the augmented matrix to meet the three concentrations, and

write the grade influence portraitured present to each innovating grade. (17 marks)

Restrain your exculpation by substituting into the method of rectirectidirect equations.

QUESTION 3 (12 marks)

1

Let f(x) = 2x – 1 and g(x) = sin4x.

(a) Meet the composite operation f(g(x)) and yield its inclosure. (6 marks)

(b) Meet the composite operation g(f(x)) and yield its inclosure. (6 marks)

QUESTION 4 (16 marks)

The controlthcoming reasoning operations were base in analyzing the motion of a spring-damper method. Express these reasoning operations as severial elements:

8s2 + 26s + 15; (8 marks)

s – 20 + 7s2

(b) s3 + 6s2 + 10s. (8 marks)

QUESTION 5 (16 marks)

On a comicality precinct ride the aspect of the mien at age t = 0 (in minutes) is yieldn by the parametric operation

x=4cos5pt+~10+2cos4pt~cospt

y=4sin5pt+ ~10 + 2cos4pt~ sinpt

where twain x and y are measured from the axis of recurrence and are yieldn in metres.

(a) Demonstration the squared space from the axis of recurrence, i.e. d2 = x2 + y2, can be rewritten as

d2 = 20 cos2 4pt + 120 cos 4pt + 116. (7 marks)

(b) Using (a) meet complete ages, t = 0, (in direct coercionm i.e. in stipulations of elements) when the space d is

4 m i.e. d=4ord2 = 16. (7 marks)

(c) What is the principal age that the space d of the mien is 4 m?

Yield your exculpation directly (as a element) and almost as a decimal. (2 marks)

QUESTION 6 (16 marks)

The expedite, V (km/h), of a car at age t (seconds) is yieldn by

where 0 = t = 100. V = f(t) = ~ pt ~

70 – 10 cos 100

~ pt ~

2 + cos 100

(a) We shortness to understand when the car procure arrive-at a actual expedite. Solve coercion the age t in stipulations of the

expedite V , and hence meet the coercionmula coercion the inverse operation f_1(t). (6 marks)

(b) Write down the inclosure of f_1(t). (2 marks)

(c) Concoct f(t) from t = 0 to t = 100 and then concoct f_1(t) on the selfselfcorresponding axes notwithstanding solely coercion the values

of t in the inclosure of the inverse operation base in (b). (6 marks)

(d) Describe the geometric similarity that you can conceive when you think twain f(t) and f_1(t). (2 marks)

Note: You are encouraged to portraiture MATLAB or Scilab coercion your concocts.

Note that the MATLAB instructs coercion require and cos_1 t are cos(t) and acos(t). Principal explain values of t from 0 to 100 coercion your concoct of f.

t1=[0:0.01:100];

gives values from 0 to 100, in steps of 0.01. Then meet the V-values coercion these t-values using

y1=(70-10.*cos(pi*t1/100))./(2+cos(pi*t1/100)); Then concoct f using

plot(t1,y1).

Portraiture the instruct

axis resembling

to procure resembling layers on the axes.

To infer the concoct of f-1, explain innovating t’s coercion its inclosure. That is with

t2=[…]

Then calcupast the y-values.

/t + 1

Coercion entreaty coercion the operation 7 sin-1 + 5 we would portraiture the instruct

t – 2

y2 = 7*asin((t2+1)./(t2-2)) + 5;

We can then concoct twain operations on the selfselfcorresponding graph using the instruct concoct(t1,y1,t2,y2).

Note the sin-1 x and cos-1 x operations are evaluated in MATLAB using asin(x) and acos(x) respectively.

Include a concoct mode to characterize the graphs, if you effort.

To infer an inferitional concoct of y = t to demonstration the geodesy, portraiture the selfselfcorresponding process: explain the t’s you shortness coercion the concoct principal (t3=[…]), calcupast the y’s, then concoct by including t3,y3 succeeding y2 to the earlier instruct

plot(t1,y1,t2,y2,t3,y3).

End of Enactment 2 (100 marks Total)

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