DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 1833

FINAL EXAMINATION
APRIL 1997

TIME: 3 hours


FORMULAS:


PART I: MULTIPLE CHOICE QUESTIONS

    The solution of each of the following 15 problems is one of four choices (A, B, C, D) listed at the end of the problem. Select the letter corresponding to the right answer and put it in the boxes below. Each question is worth 2 marks. Do your rough calculations in the booklet provided (do not hand it in).

    QUESTION123456789101112131415
    ANSWER               

    1. The equation of the line through (-1,2) with slope -3/4 is

      (A) 4x + 3y - 2 = 0 (B) 3x + 4y - 5 = 0 (C) 3x - 4y + 11 = 0 (D) None of these

    2. A product costs $4 per unit to manufacture with a fixed cost of $12,000 per month. If the product sells for $10 per unit, then the monthly break-even unit is

      (A) 3,000 units (B) 1,200 units (C) 2,000 units (D) None of these

    3. If a bank deposit paying 5% interest compounded continuously grows to $3,100 in 1 year, then the initial deposit was

      (A) $2,938.40 (B) $2,948.81 (C) $2,952.38 (D) None of these

    4. Corresponding to an annual nominal rate of interest of 12% compounded every 3 months, the equivalent effective rate of interest per year is

      (A) 12.55% (B) 16% (C) 12.49% (D) None of these

    5. If $1,000 is invested at 10% compounded semi-annually then the accumulated amount of the investment after 8 years is

      (A) $1,477.45 (B) $2,143.58 (C) $4,594.67 (D) $2,182.87

    6. If + = , then y equals

      (A) 2 (B) 4 (C) 6 (D) None of these

    7. If is the augmented matrix of a linear system of equations in x, y and z, then the number of solutions is

      (A) None (B) One (C) Two (D) Infinite

    8. If is the inverse of the matrix then the entries (c,d) are

      (A) (-1,-1/3) (B) (1,2) (C) (-1,-2) (D) (0,-1/3)

    9. A survey consists of 20 questions with 4 possible answers for each question. If every question is answered then the maximum possible number of different responses is

      (A) 16000 (B) 4845 (C) 80 (D) None of these

    10. A committee of 4 is to be formed from a group of 8 men and 6 women. If the committee must consist of two men and two women, the number of ways it can be formed is

      (A) 672 (B) 8! · 6! (C) 420 (D) None of these

    11. A bowl contains 5 red and 10 black marbles. Three marbles are selected at random. The probability that at least one red is selected is

      (A) 0.74 (B) 0.92 (C) 0.98 (D) None of these

    12. The number of different permutations of the letters in FREDERICTON is

      (A) 11! (B) (C) 7! (D) 7! · 4

    13. If an experiment consists of tossing a coin and rolling a die, then the total number of outcomes is

      (A) 8 (B) 16 (C) 12 (D) None of these

    14. If and are independent events with and then equals

      (A) 0.082 (B) 0.28 (C) 1.1 (D) 0.42

    15. If in the given sample space of 20 equally-likely outcomes, events A, B and D are such that

      then the probability is

      (A) 3/10 (B) 3/12 (C) 3/20 (D) 3/7


    PART II: FULL SOLUTIONS
    SHOW ALL CALCULATIONS

    Solve the following FIVE problems, and enter the details of your solutions in the spaces provided below the statements of the problem. Each question is worth 8 marks.

    1. A couple have just purchased a $70,000 house and have made a down-payment of $15,000. They can amortize the balance for 25 years at 9% per annum compounded monthly.

      (a) What are the monthly payments?
      (b) What is the total interest payment?
      (c) What is the present value of the remaining payments after 20 years?

    2. (a) Use the Gauss elimination or Gauss-Jordan method to find, if any, all the solutions of the system:

        x - 5z = 1, y + 4z = 0, x + 3y = -6.

      (b) Write the system of equations x + y - z = 1, 2x + y + z = -1, x + z = 2 in the form AX = B. If A-1 is

      find the solution of the equations by matrix multiplication.

    3. A firm manufactures bumper bolts and fender bolts for cars. The bumper bolt machine can produce at most 120 boxes of bumper bolts per day. The fender bolt machine can produce at most 130 boxes of fender bolts per day. The packaging department can only package a total of 230 boxes of bolts per day. The firm makes $50 profit on each box of fender bolt and $40 profit on each box of bumper bolts. How many boxes of each type of bolts should be produced each day to maximize the profit?

    4. During June, ABC Motors sold

        75cars with air conditioning
        95cars with power steering
        100cars with automatic transmission
        20cars with all three options
        10cars with none of these options
        10cars with only air conditioning
        50cars with both automatic transmission and power steering
        60cars with both automatic transmission and air conditioning.

      (a) How many cars were sold in June?
      (b) How many cars had only power steering?

    5. (a) Machines I, II, III in a factory produce 55%, 30% and 15% of total production, respectively. The percentage of defective output of these machines is 1%, 2% and 3%, respectively. An item is chosen at random and it is defective. What is the probability that it came from I?
      (b) In the rolling of a pair of fair dice, determine the probability that either the sum of the numbers rolled is odd or two equal numbers are rolled.