08/11/2022, 22:43 Quiz: MATH2070/2970 Practice Exam https://canvas.sydney.edu.au/courses/45030/quizzes/195369/take 1/11 MATH2070/2970 Practice Exam Started: Nov 8 at 22:43 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’ Extended Answer Question 2 – submit to the assignment titled ‘MATH2070/2970 Practice Exam Submission Question 2’ 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. 08/11/2022, 22:43 Quiz: MATH2070/2970 Practice Exam https://canvas.sydney.edu.au/courses/45030/quizzes/195369/take 2/11 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 08/11/2022, 22:43 Quiz: MATH2070/2970 Practice Exam https://canvas.sydney.edu.au/courses/45030/quizzes/195369/take 3/11 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 . 08/11/2022, 22:43 Quiz: MATH2070/2970 Practice Exam https://canvas.sydney.edu.au/courses/45030/quizzes/195369/take 4/11 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. 08/11/2022, 22:43 Quiz: MATH2070/2970 Practice Exam https://canvas.sydney.edu.au/courses/45030/quizzes/195369/take 5/11 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 08/11/2022, 22:43 Quiz: MATH2070/2970 Practice Exam https://canvas.sydney.edu.au/courses/45030/quizzes/195369/take 6/11 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: 08/11/2022, 22:43 Quiz: MATH2070/2970 Practice Exam https://canvas.sydney.edu.au/courses/45030/quizzes/195369/take 7/11 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 08/11/2022, 22:43 Quiz: MATH2070/2970 Practice Exam https://canvas.sydney.edu.au/courses/45030/quizzes/195369/take 8/11 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: 08/11/2022, 22:43 Quiz: MATH2070/2970 Practice Exam https://canvas.sydney.edu.au/courses/45030/quizzes/195369/take 9/11 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. 08/11/2022, 22:43 Quiz: MATH2070/2970 Practice Exam https://canvas.sydney.edu.au/courses/45030/quizzes/195369/take 10/11 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. 08/11/2022, 22:43 Quiz: MATH2070/2970 Practice Exam https://canvas.sydney.edu.au/courses/45030/quizzes/195369/take 11/11 Not saved Yes You must upload a separate document for each question’s specified assignment dropbox within 25 minutes after the quiz submission. 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’ You must not handle your mobile device at any point prior to submitting this quiz as it will be flagged as an integrity breach by ProctorU Submit Quiz
