DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 1833

FINAL EXAMINATION
April 1998
TIME: 3 HOURS
MARKS: 60

INSTRUCTIONS:

(i) Answer all 15 questions.
(ii) Show details of your solutions.
(iii)Calculators are permitted.
(iv)4 points for each question. Total points 60.

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

1. A manufacturer has a fixed cost of $4000. It costs him $4000 to manufacture 1000 units. He sells them (1000 units) for $6000. Find the cost function, revenue function and break-even point.
2. Let ${\displaystyle A = \left[
\begin{array}{rrr}
1 & 2 & -4\\
3 & 0 & 2
\end{array...
...;\;C = \left[ \begin{array}{cc}
1 & 2\\
3 & 1\\
0 & 1
\end{array} \right]\;.}$
Find the value of (2A)(3B-4C).
3. Find the inverse of
\begin{displaymath}A = \left[ \begin{array}{rrr}
1 & 0 & 1\\
2 & 1 & 0\\
0 & 1 & -1
\end{array} \right]\;. \end{displaymath}
4. Using Gauss-Jordan elimination method, solve
\begin{displaymath}\left. \begin{array}{lcrclcr}
x & - & y & + & z & = & 4\\
x ...
...& - & z & = & 2\\
x & + & y & & & = & 3
\end{array} \right. . \end{displaymath}
5. Maximize 150x + 120y such that
\begin{displaymath}x + y \leq 150,\;\;\;2x+y \leq 200,\;\;\;x \geq 0,\;\;\;y \geq 0.
\end{displaymath}
6. An owner of a gasoline station can have at most 64,000 litres of gasoline in his tanks. He estimates that he can sell at least 20,000 litres of regular gas and at least 8,000 litres of premium gas. He also knows that he cannot sell more than 3 times as much regular gas as premium gas. If regular gas sells at 55 cents per litre and premium gas at 60 cents per litre, find his maximum return. EXPRESS this as a linear programming problem. DO NOT SOLVE IT!!
7. Let $U = \{1,2,3,4,5,6,7\}$ be the universal set. Let $A =
\{1,2,3\},\;\;B = \{2,3,4,5\}$ and $C = \{3,4,5,6\}$. Find $A \cup (B
\cap C)'$.
8. 100 charitable organizations receive aid from at least one of the 3 groups A, B or C. 40 from A, 55 from B and 65 from C, 15 from A and B, 30 from B and C and 20 from A and C. How many receive aid from all 3 groups?
9. From a group of 10 women and 6 men a committee of 5 is to be formed with the condition that there should be more women than men in the committee. How many different committees can be formed?
10. A box contains 3 red, 5 white and 2 blue balls. Two balls are picked at random one after another without replacement. What is the probability that one is blue and the other is red?
11. In section A there are 5 boys and 7 girls. In section B there are 16 boys and 20 girls. A die is rolled. If 1 or 6 comes up a student is selected at random from section A. If any other number comes up a student is selected at random from section B. If the student selected is a girl, what is the probability that she is from section B?
12. At end end of 7 years, I need $100,000. How much should I deposit now in a bank that pays a nominal rate of 9% compounded monthly?
13. Bank I has a nominal rate of 6% compounded quarterly. Bank II has a nominal rate of 6.10% compounded semi-annually. Which one is a better rate?
14. Suppose $50 is deposited at the end of each quarter for 10 years, the nominal rate being 8% compounded quarterly. What is the total at the end of 10 years? How much is the interest earned?
15. I borrow $50,000 from a bank whose nominal rate is 7% compounded yearly. The debt is amortized over a period of 10 years and each payment is made at the end of each year.
(a) How much should each payment be?
(b) What is the outstanding balance at the end of 7 years?