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Exam information
Course code
and title MATH2001 Calculus and Linear Algebra II
Semester Summer Examinations 2022
Exam type Online, non-invigilated, end-of-semester examination
Exam
technology File upload to Blackboard Assignment
Exam date and
time
Your examination will begin at the time specified in your personal
examination timetable. If you commence your examination after this
time, the end for your examination does NOT change.
The total time for your examination from the scheduled starting time
will be:
2 hours 10 minutes (including 10 minutes reading time during which
you should read the exam paper and plan your responses to the
questions).
A 15-minute submission period is available for submitting your
examination after the allowed time shown above. If your examination
is submitted after this period, late penalties will be applied unless you
can demonstrate that there were problems with the system and/or
process that were beyond your control.
Exam window
You must commence your exam at the time listed in your
personalised timetable. You have from the start date/time to the end
date/time listed in which you must complete your exam.
Permitted
materials
This is an Open Book examination
Any calculator permitted – unrestricted
Your personal study notes and all course materials available on the
course Blackboard page are permitted
Recommended
materials
Ensure the following materials are available during the exam:
bilingual dictionary; phone/camera/scanner
Instructions
You will need to download the question paper included within the
Blackboard Test. Once you have completed the exam, upload the
completed exam answers file to the Blackboard assignment
submission link. You may submit multiple times, but only the last
uploaded file will be graded.
You can print the question paper and write on that paper or write your
answers on blank paper (clearly label your solutions so that it is clear
which problem it is a solution to) or annotate an electronic file on a
suitable device.
Who to contact Given the nature of this examination, responding to student queries
and/or relaying corrections to exam content during the exam may not
be feasible.
If you have any concerns or queries about a particular question or
need to make any assumptions to answer the question, state these at
the start of your solution to that question. You may also include
queries you may have made with respect to a particular question,
should you have been able to ‘raise your hand’ in an examination-
type setting.
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You are permitted to refer to the allowed resources for this
exam, but you cannot cut-and-paste material other than your
own work as answers.
You are not permitted to consult any other person – whether
directly, online, or through any other means – about any
aspect of this examination during the period that it is available.
If it is found that you have given or sought outside assistance
with this examination, then that will be deemed to be cheating.
Less than 5 minutes – 5% penalty
From 5 minutes to less than 15 minutes – 20% penalty
More than 15 minutes – 100% penalty
If you experience any interruptions to your examination, please
collect evidence of the interruption (e.g. photographs, screenshots or
emails).
If you experience any issues during the examination, contact ONLY
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Chat: support.my.uq.edu.au/app/chat/chat_launch_lib
Phone: +61 7 3335 7047
Email: examsupport@library.uq.edu.au
You should also ask for an email documenting the advice provided so
you can provide this as evidence for a late submission.
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incomplete
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In the event of a late submission, you will be required to submit
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corrupt file, missing pages, poor quality scan). We strongly
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(even if you submit on time).
If you submit your paper after the due time, then you should send
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Important exam
condition
information
You are responsible for managing your multi-factor authentication in
this examination. Please check the guidance on How do I MFA before
an online exam?
Academic integrity is a core value of the UQ community and as such
the highest standards of academic integrity apply to all examinations,
whether undertaken in-person or online.
This means:
If you submit your online exam after the end of your specified reading
time, duration, and 15 minutes submission time, the following
penalties will be applied to your final examination score for late
submission:
These penalties will be applied to all online exams unless there is
sufficient evidence of problems with the system and/or process
that were beyond your control.
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Undertaking this online exam deems your commitment to UQ’s
academic integrity pledge as summarised in the following declaration:
“I certify that I have completed this examination in an honest, fair and
trustworthy manner, that my submitted answers are entirely my own
work, and that I have neither given nor received any unauthorised
assistance on this examination”.
Summer Examinations, 2022 MATH2001
Page 1 of 15


This exam paper must not be removed from the venue

Venue ____________________
Seat Number __________
Student Number |__|__|__|__|__|__|__|__|
Family Name ____________________
First Name ____________________


School of Mathematics & Physics
Summer Examinations, 2022
MATH2001 Calculus and Linear Algebra II
This paper is for St Lucia Campus and St Lucia Campus (External) students.

Examination Duration: 120 minutes

Planning Time: 10 minutes

Exam Conditions:

•Learn.UQ non-invigilated
•File upload to Blackboard Assignment (Blackboard test is used to
release the question paper)
•This is an Open Book examination
•Any calculator permitted - unrestricted
•During Planning Time - Students are encouraged to review and plan
responses to the exam questions
•This examination paper will be released to the Library

Materials Permitted in the Exam Venue:
(No electronic aids are permitted e.g. laptops, phones)

your personal study notes and all course materials available on the
course Blackboard page are permitted

Materials to be supplied to Students:
Additional exam materials (e.g. answer booklets, rough paper) will
be provided upon request.

None

Instructions to Students:
If you believe there is missing or incorrect information impacting
your ability to answer any question, please state this when writing
your answer.


For Examiner Use Only

Question

Mark





























































Total _________


Summer Examinations, 2022 MATH2001
1. (10 marks) Solve the initial value problem
cos(x+ y)dx+
(
3y2 + 4y + cos(x+ y)
)
dy = 0, y(0) = π.
Show all working.
Page 2 of 15
Summer Examinations, 2022 MATH2001
2. (Total of 10 marks) Consider the non-homogeneous ODE
t2y′′ − 2y = 3t2 − 1, t > 0.
(a) (5 marks) Show that y1 = t
2 and y2 = t
−1 are solutions to the corresponding homogeneous
ODE t2y′′ − 2y = 0.
(b) (5 marks) Find the general solution of the non-homogeneous ODE.
Show all working.
Page 3 of 15
Summer Examinations, 2022 MATH2001
3. (10 marks) Show that y1 = t
1/2 and y2 = t
−1 form a fundamental set of solutions of
2t2y′′ + 3ty′ − y = 0, t > 0.
Show all working.
Page 4 of 15
Summer Examinations, 2022 MATH2001
4. (10 marks) Let M2,2 have the standard inner product
⟨u,v⟩ = Tr(uTv).
Find the cosine of the angle between the vectors
u =
(
0 2
3 1
)
and v =
( −1 0
3 2
)
.
Show all working.
Page 5 of 15
Summer Examinations, 2022 MATH2001
5. (10 marks) Find the least squares straight line fit to the data
(1, 0), (1, 2) and (3, 7).
Show all working.
Page 6 of 15
Summer Examinations, 2022 MATH2001
6. (10 marks) Consider the Euclidean inner product of the vector space R3. Apply the Gram-
Schmidt process to transform the basis
{u1 = (1, 1, 1),u2 = (1, 0, 1),u3 = (1, 1, 3)}
into an orthonormal basis. Show all working.
Page 7 of 15
Summer Examinations, 2022 MATH2001
7. (Total of 10 marks) Consider the expression
I =
∫ 1√
2
0
∫ y
0
xy dx dy +
∫ 1
1√
2
∫ √1−y2
0
xy dx dy.
(a) (5 marks) Express I as a single iterated integral in terms of polar coordinates.
(b) (5 marks) Determine the value of I.
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Summer Examinations, 2022 MATH2001
8. (10 marks)
Let C be the path in the xy-plane comprising the strait line from (0,0) to (5,3) followed by the
straight line from (5,3) to (2,0). Evaluate the integral∫
C
x4 dx+ e−y
2
dy.
Page 9 of 15
Summer Examinations, 2022 MATH2001
9. (10 marks) Consider a hemisphere in the upper-half of R3 of radius R with base centred at the
origin of the xy-plane. The hemisphere has density ρ(x, y, z) = σz
√
x2 + y2 + z2 where σ is a
positive constant. Determine the moment of inertia of the hemisphere about the z-axis.
(Hint: use spherical coordinates (r, θ, ϕ).)
Page 10 of 15
Summer Examinations, 2022 MATH2001
10. (10 marks)
Calculate the net outward flux of the vector field F (x, y, z) = (y2 + zey, x2 + xz2ex, 2z2) across
the surface of the solid bounded by the hemispheres z =
√
9− x2 − y2 and z = √1− x2 − y2
and the plane z = 0.
Page 11 of 15
Summer Examinations, 2022 MATH2001
11. (10 marks)
Let F(x, y, z) = (x2exz, y3zez, xy). Use Stokes’ Theorem to evaluate
∮
C
F · dr, where C is the
curve of intersection of the sphere x2 + y2 + z2 = 5 with z > 0 and the cylinder x2 + y2 = 1
oriented counterclockwise as viewed from above.
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Summer Examinations, 2022 MATH2001
12. (10 marks)
Let
I =
x
D
ln(9x2 + 4y2) dA
where D is the annulus region bounded by 9x2 + 4y2 = 1 and 9x2 + 4y2 = 4.
Express I in terms of the new coordinates (ρ, θ) given by
x =
1
3
e−ρ cos(θ)
y =
1
2
e−ρ sin(θ).
Do not evaluate the integral. (Hint: use the Jacobian.)
Working space next page
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Summer Examinations, 2022 MATH2001
Working space only
END OF EXAMINATION
Page 14 of 15
Summer Examinations, 2022 MATH2001
Formula sheet
ˆ Variation of parameters
W = y1y
′
2 − y′1y2
u = −
∫
y2r
W
dx v =
∫
y1r
W
dx
ˆ Hyperbolic and trigonometric functions
sinhx =
ex − e−x
2
coshx =
ex + e−x
2
tanhx =
sinhx
coshx
sin 2x = 2 sin x cosx cos 2x = cos2 x− sin2 x
ˆ Spherical coordinates
x = r cos θ sinϕ y = r sin θ sinϕ
z = r cosϕ r =
√
x2 + y2 + z2
ˆ Centre of mass/moments of inertia (3d)
x¯ =
t
xρ dV
t
ρ dV
y¯ =
t
yρ dV
t
ρ dV
z¯ =
t
zρ dV
t
ρ dV
Ix =
y
(y2 + z2)ρ dV Iy =
y
(x2 + z2)ρ dV
Iz =
y
(x2 + y2)ρ dV
ˆ Green’s theorem
x
D
(
∂F2
∂x
− ∂F1
∂y
)
dx dy =
∮
C
(F1 dx+ F2 dy)
ˆ Divergence theorem
{
S
F · n dS =
y
V
div(F) dV
ˆ Stokes’ theorem
x
S
(curlF) · n dA =
∮
C
F · dr
ˆ Orthogonal projection
ProjU(v) = ⟨v, e1⟩e1 + . . .+ ⟨v, ek⟩ek
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