DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 1833

FINAL EXAMINATION
December 1998
TIME: 3 HOURS
MARKS: 60

INSTRUCTIONS:

(i) Answer all 15 questions.
(ii) Show details of your solutions.
(iii) Calculators are permitted.
(iv) Four (4) points for each question.

FORMULAS:
\begin{displaymath}S = P(1 + i)^n\;; \hspace*{0.5cm} S_n = R \left[ \frac{(1+i)^...
...ace*{0.5cm} A_n = R \left[ \frac{1 - (1+i)^{-n}}{i}
\right]\;.
\end{displaymath}

1. A video company is planning to produce an instructional video. The producer estimates that it will cost $84,000 to shoot the video and $15 per unit. The wholesale price of the tape is $50 per unit.
(a) What is the cost and revenue function?
(b) Find the break-even point.
2. Let ${\displaystyle A = \left[ \begin{array}{rrr}
1 & 2 & 3\\
2 & -1 & 0
\end{array...
... C =
\left[ \begin{array}{rr}
1 & 1\\
-2 & -1\\
1 & 0
\end{array} \right]\;.}$
Find the value of 3AC + 2BC.
3. Using the inverse method, solve x + y = 1, y + z = -1 and x + z = 2.
4. Using Gauss-Jordan elimination method, solve
3x + 6y - 9z = 15  
2x + 4y - 6z = 10 .
-2x - 3y + 4z = -6   
5. Minimize 30x + 10y subject to
\begin{displaymath}10x + 2y \geq 84,\;\;8x + 4y \geq 120,\;\;0 \leq x \leq 50,\;\;0 \leq
y \leq 50.
\end{displaymath}
6. Because of new regulations, a chemical plant introduced a process to supplement an older process. The older process emitted 20 grams of sulphur dioxide and 40 grams of particulate matter for each gallon of chemical produced, whereas the new process emits 5 grams of sulphur dioxide and 20 grams of particulate matter for each gallon produced. The company makes a profit of 60 cents per gallon and 20 cents per gallon on the old and new processes, respectively. If the government allows the plant to emit no more than 10,000 grams of sulphur dioxide and 30,000 grams of particulate matter, how many gallons of the chemical should be produced by each process to maximize the profit? Express this as a linear programming problem. DO NOT SOLVE IT!
7. Let U = {1,2,3,4,5,6,7,8} be the universal set.
Let A = {1,2,6,7} and B = {2,4,6,8}. Show that $(A \cup B)' =
A' \cap B'$.
8. A group of 100 people touring Europe includes 42 people who can speak English, 55 who speak French and 17 who speak neither language. How many in the group speak both the languages?
9. From a group of 8 players, a basketball team of 5 is to be selected. How many starting teams are possible if either Mike or Ken (but not both) must be in the starting team?
10. A town council has 11 members, 6 men and 5 women. If a 3 person committee is selected at random, what is the probability that women make up the majority?
11. In a given country, records show that 45% are Liberals, 35% are Tories and 20% are Independents. In an election, 70% of the Liberals, 40% of the Tories and 80% of the Independents voted for abolition of death penalty. If a citizen selected at random is in favour of abolition, what is the probability that the voter is a Tory?
12. A loan of $4,000 was repaid at the end of 8 months with an amount of $4,230. What is the annual rate (Simple Interest)?
13. Parents wishing to have enough money for their childs education 17 years from now decide to buy a $30,000 bond. If the interest rate is 10% compounded annually, what should they pay now?
14. I need $25,000 at the end of 5 years. What amount should I deposit at the end of each month into an account paying 12% compounded monthly?
15. A family purchased a home 10 years ago for $80,000. Their down payment was 20%. They signed a 30 year mortgage at 9% compounded monthly. What is their unpaid balance now?