Qn: # Mark Score Total 100

Transcription

1 DEPARTMENT OF MATHEMATICS University of Toronto at Mississauga MAT 33Y, Test October 20, 2003 Time 6.0pm.8.00 pm Fill in the following information in INK! Last Name:. Given Name:. Student #:. Tutor s Name J.I. J.T. S.C. N.H. W.N. E.W. S.L. Tutorial No. (Please Circle your Tutor s initials and enter the tutorial number) Instructions Calculators are allowed. No other aids. There are 5 problems in total. You may solve them in any order. Read your answers carefully. Write down all your work carefully and in an organized manner. There are 2 pages to this test. Make sure you have all of them. Qn: # Mark Score Total 00

2 Page 2 Question ( = 20 Marks) Q.(a) If an initial investment of $3000 grows to $8,000 in ten years, find the nominal rate of interest compounded monthly, that was earned by the investment. Q.(b) How much money must be invested now at an interest rate of 7.25% compounded quarterly to have $0,000 in two years?

3 Page 3 Q.(c) Suppose that Teresa can invest $3,000 in a business that guarantees her the following cash flows: $6000 at the end of 2 years, $5000 at the end of 4 years, and $4000 at the end of 6 years. Assuming an interest rate of 6% compounded monthly, find the net present value of the cash flows. Is the investment profitable?

4 Page 4 Question 2 ( = 20 marks) Q.2 (a) Consider the following annuity: $2000 due at the end of each year for two years, and $3000 due thereafter at the end of each year for three years. At an interest rate of 4% compounded annually, what is the present value of the annuity? Q.2 (b) A $5000 loan is to be repaid over three years by equal payments due at the end of every quarter. If interest is at the rate of 20% compounded quarterly, determine the quarterly payment.

5 Page 5 Q.2(c) Suppose a diagnostic machine will yield a net of $000 per quarter for 5 years, after which the machine can be sold for $000. How much should a firm pay for the machine if it wants to earn 7.5% annually on its investment and also set up a sinking fund to replace the purchase price? For the fund, assume quarterly payments and a rate of 5.5% compounded quarterly.

6 Page 6 Question3 (2+8 = 20 marks) A survey is conducted in a community to estimate the outcome of an upcoming election; the results of the survey indicated the following preferences for each of the candidates A B C Males Females Calculate: Q3(i) If an individual is selected at random from that community, what is the probability that He is a male She is a female, and will vote for B Either she is a female, or the individual will vote for B. The individual is a female, knowing the individual voted for B.

7 Page 7 Q3(ii). Assume candidate C will receive a subsidy of $00,000 with probability 5%, and nothing with probability 95%. Calculate its expectation (mean) and standard deviation.

8 Page 8 Question 4 (0+0 =20 marks) Q.4(a) After a merger between two companies A and B, a new company C is created. The board of the new company will consist of 0 members, 5 from each of the original companies. If both A and B had 7 members in each board, how many different board compositions can we have for C? Q.4(b) The new company C will have an executive committee consisting of 4 members from its own board. What is the probability that former company A will have no representation in the executive committee?

9 Page 9 Question 5(8+4+2 = 4 marks) Q.5(a) A theatre company sold out all 300 seats on opening night. The admission prices are $50 for adults, $30 for students, and $20 for children under 4 years of age. Total sales were $0,00. It was agreed by management that the number of adults, students and children attending would not be affected if the prices were raised to $70 for adults, $40 for students, and $30 for children. With the increased prices the total revenue would increase by $3,800.Express this information in a 3 3 matrix equation and solve using row reduction to find the number of adults, students and children attending.

10 Page 0 Q.5(b) (i) Solve the matrix equation [ ] = y x for the scalars x,y. (ii) Use the result in (i) to evaluate [ ] [ ] Τ y x x y

11 Page Q.5(c) Solve the linear system, show all row operations performed. x 2y + 3z = x + 3y = 0 2x 5y + 5z = 7

12 Page 2 List of Formulae S compound amount or future value P Principal, r periodic rate, n number of periods Ordinary Annuity Annuity Due ( ( + r) n ) A = R (present value) r n ( + r) S = R (future value) r n+ ( + r) A = R + (present value) r ( + r) S = R r n+ (future value)