2020 Semester Two (November-December 2020)
Examination Period
Faculty of Science
EXAM CODES: MTH1030
TITLE OF PAPER: Techniques for Modelling
EXAM DURATION: 2 hours 10 mins
Rules
During an exam, you must not have in your possession any item/material that has not been authorised for your exam. This includes
books, notes, paper, electronic device/s, mobile phone, smart watch/device, calculator, pencil case, or writing on any part of your
body. Any authorised items are listed above. Items/materials on your desk, chair, in your clothing or otherwise on your person will
be deemed to be in your possession.
You must not retain, copy, memorise or note down any exam content for personal use or to share with any other person by any
means following your exam.
You must comply with any instructions given to you by an exam supervisor.
As a student, and under Monash University’s Student Academic Integrity procedure, you must undertake your in-semester tasks,
and end-of-semester tasks, including exams, with honesty and integrity. In exams, you must not allow anyone else to do work for
you and you must not do any work for others. You must not contact, or attempt to contact, another person in an attempt to gain
unfair advantage during your exam session. Assessors may take reasonable steps to check that your work displays the expected
standards of academic integrity.
Failure to comply with the above instructions, or attempting to cheat or cheating in an exam may constitute a breach of instructions
under regulation 23 of the Monash University (Academic Board) Regulations or may constitute an act of academic misconduct
under Part 7 of the Monash University (Council) Regulations.
Authorised Materials
CALCULATORS YES NO
DICTIONARIES YES NO
NOTES YES NO
PERMITTED ITEM YES NO
if yes, items permitted are:
Page 1 of 21
Instructions
ALL answers in this exam that are not multiple choice are integers or fractions.
INTEGERS: You must enter integers in their simplest form.
For example, enter the number twenty-three as 23.
Extra spaces before, in the middle or following the integer, or inputs such as 22.999... or 23.000 or 24-1, etc. will ALL result in an
answer to be counted as incorrect.
FRACTIONS: Similarly, you must enter non-integer fractions in lowest terms and in their simplest form possible.
So, enter three halves as 3/2
Extra spaces before, in the middle or following a fraction, or input as 6/4, 1.5, etc. will ALL result in an answer to be counted as
incorrect.
NEGATIVE SIGN: If you want to enter a negative integer or fraction you must enter the negative sign immediately before the number,
e.g. -23 or -3/2. Again, extra spaces in between the minus sign and the number or anywhere else will result in an answer to be
counted as incorrect.
ATTEMPT ALL QUESTIONS: There are no negative marks for incorrect answers.
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Instructions
Information
ALL answers in this exam that are not multiple choice are integers or fractions: You MUST enter
integers in their simplest form.
For example, enter the number twenty-three as 23.
Extra spaces before, in the middle or following the integer, or inputs such as 22.999... or 23.000 or
24-1, etc. will ALL result in an answer to be counted as incorrect.
FRACTIONS: Similarly, you must enter non-integer fractions in lowest terms and in their simplest form
possible.
So, enter three halves as 3/2
Extra spaces before, in the middle or following a fraction, or input as 6/4, 1.5, etc. will ALL result in
an answer to be counted as incorrect.
NEGATIVE SIGN: If you want to enter a negative integer or fraction you must enter the negative sign
immediately before the number, e.g. -23 or -3/2. Again, extra spaces in between the minus sign and
the number or anywhere else will result in an answer to be counted as incorrect.
ATTEMPT ALL QUESTIONS: There are no negative marks for incorrect answers.
Page 3 of 21
Integration by parts and substitution
Question 1
Here are a couple of integrals that can be dealt with using integration by parts (IBP) or integration by substitution (IBS) or a
combination of both. Among the given choices, make the best possible choice resulting in the least number of calculations :)
a) This integral
can be dealt with using
• an IBP with f(t)=t and g'(t)=e^t • an IBP with f(t)=e^t and g'(t)=t • an IBP with f(t)=te^t and g'(t)=1 • an IBS with u=t
• an IBS with u=e^t • an IBS with u=te^t
b) This integral
can be dealt with using
• an IBP with f(x)=1/x and g'(x)=(ln(x))^2 • an IBP with f(x)=(ln(x))^2/x and g'(x)=1 • an IBS with u=ln(x)
• an IBS with u=(ln(x))^2/x
c) This integral
can be dealt with using
• an IBP with f(x)=x^3 and g'(x)=e^(x^2) followed by an IBS • an IBP with f(x)=e^(x^2) and g'(x)=x^3 followed by another IBP
• an IBP with f(x)=x^2e^(x^2) and g'(x)=x • an IBS with u=x^2 followed by an IBP • an IBS with u=x^3 followed by an IBP
d) This integral
can be dealt with using
• an IBS only • an IBP only • a combination of IBS and IBP
e) The Integration by Parts formula results from integrating
• the chain rule • the integrating factor • the product rule • the partial fraction approach • none of the above
f) The formula for an Integration by Substitution results from integrating
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Marks
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• the chain rule • the integrating factor • the product rule • the partial fraction approach • none of the above
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Sequences and series
Question 2
a) For each of the following series, determine whether they are absolutely convergent, conditionally convergent, or divergent.
(i)
• absolutely convergent • conditionally convergent • divergent
(ii)
• absolutely convergent • conditionally convergent • divergent
(iii)
• absolutely convergent • conditionally convergent • divergent
(iv)
• absolutely convergent • conditionally convergent • divergent
b) For each of the following series, determine whether they are absolutely convergent, conditionally convergent, or divergent.
(i)
• absolutely convergent • conditionally convergent • divergent
(ii)
• absolutely convergent • conditionally convergent • divergent
(iii)
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Marks
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• absolutely convergent • conditionally convergent • divergent
(iv)
• absolutely convergent • conditionally convergent • divergent
c) For each of the following series, determine whether they are an alternating series, a p-series, a harmonic series, a geometric
series, or none of the above.
(i)
• alternating series • p-series • harmonic series • geometric series • none of the above
(ii)
• alternating series • p-series • harmonic series • geometric series • none of the above
(iii)
• alternating series • p-series • harmonic series • geometric series • none of the above
d) Choose the correct value for the sum of the following series.
(i)
• 3/4 • 5/4 • 4/5 • 2 • 4/3
(ii)
• 1/3 • 12/5 • 3 • 5/3 • 4/3
Page 7 of 21
Power series
Question 3
Let
a) What is the radius of convergence R of the power series?
Answer:
R =
b) What is the interval of convergence of the power series?
Answer:
• (-R,R) • [-R,R) • (-R,R] • [-R,R] • None of the above
c) Suppose a function f admits the following Maclaurin series near 0.
What is f'(1)?
f'(1) =
d) Suppose f is a function with f(0) = 0, and f' admits the following Maclaurin series near 0.
What is f(1)?
f(1) =
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Marks
Page 8 of 21
Differential equations
Question 4
a) One of the following differential equations is separable. Determine which it is.

• (i) • (ii) • (iii) • (iv)
b) Suppose we wish to solve the following differential equation using the method of the integrating factor.
What should we choose as our integrating factor?
• 1/x^2 • 1/x • e^x • ln(x) • x
c) The differential equation
has a general solution of the form
where a, b, c, and d are constants. Let us choose a and b so that a>0 and b<0. What are a and b?
Answers:
a =
b =
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Marks
Page 9 of 21
Vectors
Question 5
Fill in the correct numbers or choose the correct answer.
Let u=(1,0,1), v=(0,2,0) and w=(2,1,-2) be vectors in R . Then:
a) (2v) · (3u) =
b) The x-coordinate of (2v) × (3u) is
c) u × (-2v) · w =
d) The angle between v and w is
• 0 • larger than pi/2 • less than pi/2 • pi/2 • pi/4 • pi/6 • pi/3
e) The square of the area of the parallelogram spanned by u and v is
f) The volume of the parallelepiped spanned by u, v and w is
g) u, v, w (in this order) form
• a left-handed system • a right-handed system • neither a left-handed nor a right-handed system
8
Marks
3
Page 10 of 21
Points, lines, and planes
Question 6
a) A plane is given by the following equation 2x-y+3z=2. This plane has a normal vector with x-coordinate -4. What is its z-
coordinate?
Answer:
b) There is a point with x-coordinate 1 and z-coordinate 1 on this plane. What is its y-coordinate?
Answer:
c) Find the y-coordinate of the point of intersection of the line with equation (1,12,3) + t(1,2,1) and this plane.
Answer:
d) L is a line with equation (1,3,1)+t(2,1,d). If L is contained in the plane, what is d?
Answer:
e) The closest point on the line L to the point (2,15,3) is given by the parameter t. Then t =
Answer:
f) What is the distance between the two lines with equations (1,0,-1)+t(1,2,0) and (-1,-1,4)+s(3,-2,0)?
Answer:
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Marks
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Reduced row echelon form
Question 7
a) The augmented matrix of a linear system is given by
The number of equations is
The number of variables is
b) Here are reduced row echelon forms of three linear systems. How many solutions do these systems have?
Answer:
• 0 solutions • exactly 1 solution • exactly 2 solutions • infinitely many solutions • this question makes no sense
Answer:
• 0 solutions • exactly 1 solution • exactly 2 solutions • infinitely many solutions • this question makes no sense
Answer:
• 0 solutions • exactly 1 solution • exactly 2 solutions • infinitely many solutions • this question makes no sense
c) Let's say you are dealing with a linear system with 1000 unknowns and 1000 equations with randomly generated coefficients.
What is most likely?
Answer:
Most likely this system has
• 0 solutions • exactly 1 solution • exactly 2 solutions • infinitely many solutions • this question makes no sense
d) The reduced row echelon form of an augmented matrix is
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How many free variables does this reduced row echelon form have?
Answer:
The unknowns in the linear system are x, y, z, w, from left to right. There is a solution to the linear system in which z = -2 and w =
-1. In this solutions what are x and y?
Answers:
x =
y =
Page 13 of 21
Determinants
Question 8
Calculate the following determinants?
a)
Answer:
b)
Answer:
c)
Answer:
d)
Answer:
e)
Answer:
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Marks
Page 14 of 21
Linear transformations
Question 9
The reflection across the plane with equation 2x-2y+z=0 is a linear transformation of R with matrix
Then:
a) The determinant of the matrix A is
b) The matrix has two eigenvalues, one is 1, the other one is
c) The dimension of the eigenspace with respect to 1 is
d) If an eigenvector of A has x-coordinate -4, y-coordinate 4 and negative z-coordinate, then its z-coordinate is
e) If an eigenvector of A has x-coordinate 1 and y-coordinate 2, then its z-coordinate is
f) What are the entries a, b and c?
Answers:
a =
b =
c =
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Marks
3
Page 15 of 21
Find the inverse
Question 10
Let
a) The determinant of A is equal to
• 0 • a negative number • a positive number • nonsense: the matrix does not have a determinant
b) Let k be the smallest positive integer such that A = O. Then k is
• 1 • 2 • 3 • 4 • 5 • nonsense: k does not exist
c) Recall the geometric series (1–x) = 1 + x + x + x + ... for |x| < 1. This can be used to compute the inverse of B = I –A (Hint:
formally 1 = I, and let x = A). The first row of B is (a, b, c). Then
a =
b =
c =
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Marks
k
–1 2 3
–1
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Some useful facts
Information
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Information
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Information
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Information
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