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MATH2070/2970 Practice Exam
创建日期：2024-05-16 15:15:57
所属分类：数学mathematics
标签： MATH2070
mathematical

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MATH2070/2970 Practice Exam

Quiz Instructions
Quiz Instructions
All exam questions will appear only in this quiz. Do not click submit until you have finished
answering all questions.
Section 1
13 Multiple choice/short answer questions
Section 2
2 Extended answer questions of different value
Section 2 questions should be handwritten on white paper in blue or black pen. You must scan or
photograph your work only after you have submitted the quiz as the use of your mobile phone during
the quiz is prohibited. You must upload a separate document (pdf or docx files only) for each
question's specified assignment dropbox within 25 minutes after the quiz submission. Do not use your
name anywhere in your submission.
Extended Answer Question 1 – submit to the assignment titled 'MATH2070/2970 Practice Exam
Submission Question 1'
You must not handle your mobile device at any point prior to submitting this quiz. The use of a
mobile device during the quiz will be flagged as an integrity breach by ProctorU.
If you experience technical issues during your exam, please refer to the 'Help Centre' in the left-hand
menu.
Permitted resources
Locally saved notes on your computer and paper-based resources (including printed/handwritten
notes and textbooks), handheld calculator, multiple sheets of blank scratch paper and a pen/pencil.
Be mindful that you may not finish the exam within the permitted time period if you frequently search
other resources to help you answer the questions.
During this exam, you will NOT be permitted to open additional browsers or tabs, or to access
Wolfram Alpha, or other online mathematical computing software.

Disclaimer: Note that this practice exam is shorter than the real final exam. In the
real final exam, you will have 20 multiple choice/short answer questions and 3
extended answer questions.
Section 1
Multiple Choice/Short Answer Section (50%)
There are 13 multiple choice questions.
Each question is of equal value.
There are no penalties for wrong answers.
No working will be marked.
2 ptsQuestion 1
The primal is unbounded and the dual is infeasible.
Both the primal and dual are infeasible.
The primal is infeasible and the dual is unbounded.
Both the primal and dual are unbounded.
Both the primal and dual are feasible and finite.
Which of the following is NOT possible for a pair of primal and dual linear
programs?

Information for Questions 2-3
The following is the final simplex tableau for a linear programming problem:
2 ptsQuestion 2
, and
, and
and
and
Which are the basic variables?
2 ptsQuestion 3
The LP problem has multiple solutions, which can be parametrised by , with
. Give the range of .

Information for Questions 4-6
Consider the function .
2 ptsQuestion 4
The function has a critical point at .
In digit form and without any spaces, give the values (could be positive, negative
or zero) of :


2 ptsQuestion 5
negative semidefinite
indefinite
negative definite
positive definite
Give the definiteness of the Hessian matrix at the critical point.

positive semidefinite
2 ptsQuestion 6
local minimum
saddle point
cannot be determined
local maximum
Based on the definiteness of the Hessian matrix, what is the nature of the critical
point?
Information for Questions 7-9
Suppose an agent has the utility function for .
2 ptsQuestion 7
What is the absolute risk aversion of this agent?

2 ptsQuestion 8
What is the relative risk aversion of this agent?
2 ptsQuestion 9
risk-averse
risk-loving
risk-neutral
What is the risk attitude of this agent?
Information for Questions 10-13
An investor is confronted with three investment opportunities as indicated in the
following table:

2 ptsQuestion 10
Investment C
Investment B
Investment A
Which investment will be chosen according to the Principle of Expected Return?
2 ptsQuestion 11
Investment A
Investment B
Investment C
Which investment would an investor with a utility function and
current wealth of $15 choose according to the Principle of Expected Utility?
2 ptsQuestion 12
Which investment would an investor with a utility function and
current wealth of $20 choose according to the Principle of Expected Utility?

Investment C
Investment B
Investment A
2 ptsQuestion 13
6.25
5.557
0.5
0.693
Consider Investment C above and an agent with a utility function
. Assume that the agent's initial wealth is 0.
Based on the Principle of Expected Utility, how much is this agent willing to pay to
receive the expected return of this investment without bearing the risk?
Section 2
Extended Answer Section (50%)
There are 2 extended answer questions, each with multiple parts.
The questions are of different value.
Working must be shown.
These questions must be answered individually on your own
paper, scanned/photographed (pdf or docx), and uploaded into the corresponding
canvas assignment:

Extended Answer Question 1 – submit to the assignment titled
'MATH2070/2970 Practice Exam Submission Question 1'
Extended Answer Question 2 – submit to the assignment titled
'MATH2070/2970 Practice Exam Submission Question 2'
The questions ONLY appear in this quiz. Do not submit this quiz until you have
finished answering ALL extended response questions.
Extended Answer Question 1 (10 marks)
A company can purchase up to 12 units of a raw material at a cost of $0.25/unit.
Four units of the raw material can be processed into one unit of product X at a
processing cost of $3.00 for each unit of X produced, and two units of the raw
material can be processed into one unit of product Y at a processing cost of $1.50
for each unit of Y produced. If units of product X are produced, product X sells
for $ per unit. If units of product Y are produced, product Y sells for $
per unit. The company wishes to maximise its profit.
A. Show that, by producing units of X and units of Y, the profit in dollars of the
company is given by:
Formulate the problem as a nonlinear programming problem subject to
constraints.
B. Convert the problem in part (a) to a minimisation problem and show that the
corresponding Lagrangian function, ignoring the non-negativity conditions, is
where .
C. Give the Karush-Kuhn-Tucker (KKT) conditions for the minimisation problem.
D. Find the optimal solution to the problem. You may either use the Lagrangian
approach or the KKT conditions.

Extended Answer Question 2 (11 marks)
Consider a 2-asset portfolio with parameters:
A. Find and , and hence show that the feasible set consists of the pair
of straight lines
B. Sketch in the -plane the following:
the feasible set;
the locations of the two cases, where all initial wealth is invested in Asset 1
and in Asset 2, respectively;
the minimum risk portfolio .
C. An investor has risk-aversion parameter . Find this investor's optimal
portfolio.
D. If no short-selling of assets is allowed, what is the optimal portfolio for the
investor with risk-aversion parameter ?
E. Which investor will have an optimal portfolio with ?
0 ptsQuestion 14
I confirm that:
I have answered all exam questions including the extended response
questions as instructed. (If not, go back and complete the extended response
questions now.)
I also understand that:
I will first need to submit this quiz, and then I have 25 minutes to (scan), save
(as one pdf or docx file) and upload my extended response question answers
to the Canvas assignment. Submissions of the extended response questions
after this upload time will receive no marks.
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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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