ENG1090
Assignment 3
School of Mathematics
Monash University
Semester 1 2021
ENG1090 FOUNDATION MATHEMATICS
Assignment 3
Assignment instructions
Complete the following questions, upload and submit them in Moodle in a size A4 pdf file (see submission instructions
elow) in week 7 of the semester, no later than Friday April 23, 10 pm Melbourne time (8 pm Malaysia time).
Submission Instructions: In Moodle, click on the following links: Grades, Asst 2, and Add submission. Upload you
assignment and click save changes. At this point, you have the options of editing, removing, or submitting your assignment.
Make sure you submit your assignment and it is not left as ‘draft’, otherwise this will incur in a penalty of 0 marks.
Make sure you use a reliable program or mean to produce pdf files (for instance use the university li
ary photocopy
machine to scan your assignment). Do not include any special characters in your assignment as Moodle might not be
able to process your assignment. If you write your assignment in a tablet, make sure the final pdf file is not editable.
If you take a picture of your assignment and then convert it to a pdf, make sure it is clear, has clear borders, the pape
is not creased, and there is only a slight and well proportioned margin outside the paper.
All steps in your working should be shown, as you must express a mathematical argument clearly in both sentences and
co
ect mathematical notation. (See the ‘Guidelines for writing mathematics’ on Moodle for an indication of what is
equired). Marks are awarded both for many of the explanation steps (not just the final answer), and your mathematics
communication and presentation skills (up to 3 marks). Complete and co
ect solutions to this assignment contribute up
to 4% of the final unit mark for ENG1090.
Special Consideration: If you cannot complete an in-semester assessment due to circumstances beyond your control, in
the first instance you should submit a request for special consideration. Requests for special consideration have to be
submitted no later than two university working days after the due date. For detailed information about how to apply
for special consideration and what supporting documentation you need to submit please check the university’s special
consideration web page.
Short extensions: In addition to special consideration requests which are processed centrally by the University, unit
coordinators may be able to grant an extension of up to two calendar days if you are experiencing short term exceptional
circumstances such as carer responsibilities or a car accident. If you would like to ask for such a short extension, you
have to let your unit coordinator (Clayton: Santiago, Malaysia: Lily) know via e–mail at least 12 hours before the due
date. Any such request must be supported by a justification and, if at all possible, by documentation as in the case of a
special consideration request. Extensions are not available for quizzes. Also, technical issues or workload in other units
are not considered as a reasonable basis for extensions. If you miss the 12–hour cutoff, you can only apply for special
consideration. If you request an extension, even a short one, your unit coordinator may ask you to submit a special
consideration request instead.
In the case of an extension request or a special consideration request, you may only hear back after the deadline. Unit
coordinators will always try to get back to you about any extension request before the due date. However, during particu-
larly busy times you may only hear back from us after two working days. The latest you can submit a special consideration
equest is two university working days after the due date. After you submitted such a request it may then take a few more
days for your request to be processed (5 days is not unusual). At the same time, please keep in mind that assessment
tasks submitted late without approval for an extension or special consideration will incur a 10% penalty per every 24
hour overdue. Assessment tasks submitted more than 7 calendar days late will receive a mark of 0. Therefore to avoid
losing marks in case your request is denied please always: Submit your request for an extension or special consideration
as soon as possible, preferably well before the due date. Even if you have submitted a request for an extension or special
consideration, you should always aim to submit your assessment as soon as possible.
Note that tutors are not authorised to approve late submission of assignments without penalty.
Assignment questions
Question XXXXXXXXXXmarks)
Let S be the last non-zero digit of your student number. Write down your student number. Find the fourth roots
of the complex number 3i−S in exact principal argument form, and plot them on the Argand plane. Make sure
you label each root co
ectly on the diagram.
Question XXXXXXXXXXmarks)
A wave signal travelling along an electrical transmission line (usually called incident wave) will be reflected
ack in the opposite direction when the travelling signal encounters a discontinuity in the characteristic impe-
dance. This can happen in real life if we join dissimilar transmission lines together. Signals travelling alone
the line will be partially reflected at the junction (see illustration below; note that up to this point we have not
asked you any question yet).
Suppose a transmission line with characteristic impedance Z0 is terminated at one end with an impedance of
ZL (recall that impedance is a complex number Z = X+ iY where X is the resistance and Y is the reactance),
then the reflection coefficient Γ is given by the voltage reflection equation
Γ =
ZL − Z0
ZL + Z0
where Γ, ZL = XL + iYL, and Z0 = X0 + iY0 are complex numbers.
(a) What is the reflection coefficient Γ when a transmission line with characteristic impedance Z0 = 3+2i is
terminated by a resistor of S Ohms, where S is the last non-zero digit of your student number? Express
your answer in Cartesian form. (NOTE: an ideal resistor has no reactance, which means the terminal
impedance ZL is purely real, and equals to the resistance).
(b) The argument of the reflection coefficient Γ can be interpreted as the phase shift in the reflected signal
compared to the incident signal. Determine the argument θ of the reflection coefficient Γ, in terms of
XL, YL, X0 and Y0 (do not use the specific values in question 3.2a, since question 3.2b asks for a general
answer). HINT: be careful about the quadrant of Γ, you may need to adjust your answer depending on
the quadrant.
Question XXXXXXXXXXmarks)
A system of pulleys and weights hangs in equili
ium position as shown in the following diagram:
Suppose the pulleys are frictionless, the gravitational force is 40N on weightW1, 50N on weightW2, and 20N
on weight W3.
(a) Determine the tension forces T1 and T2 as vectors in terms of the angles θ and φ.
(b) Determine the angles θ and φ.