DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 1833

FINAL EXAMINATION
December 1997
TIME: 3 HOURS
TOTAL POINTS = 60

INSTRUCTIONS:

(a) Answer all questions in the space provided. If you need extra space, use the reverse side of the page and clearly label the question number.
(b) This exam has 7 pages, including a blank page at the end for rough work (DO NOT DETACH).
(c) Show ALL intermediate calculations.
(d) Calculators are permitted.
(e) 4 points for each questions. Total points 60.
 

FORMULAS:
\begin{eqnarray*}S & = & P(1 + i)^n\\
S_n & = & Rs_{n \rceil i} = R \left[ \fra...
... & = & Ra_{n \rceil i} = R \left[ \frac{1-(1+i)^{-n}}{i} \right]
\end{eqnarray*}

1. For a certain commodity the demand equation is q = 15,000 - 150p and the supply equation is q = -200 + 100p where the price p is in dollars and the supply q is in thousands of units. Find the equilibrium price and quantity.
2. Let ${\displaystyle A = \left[
\begin{array}{rrr}
2 & 3 & -1\\
5 & -2 & 6
\end{arra...
...C = \left[ \begin{array}{rr}
3 & -4\\
1 & 0\\
3 & 4
\end{array} \right]\;\;.}$
Find 2A + 3B and (2A)(3C).
3. Find the inverse, if it exists, of ${\displaystyle A = \left[
\begin{array}{rrr}
1 & 3 & 0\\
2 & 2 & 3\\
0 & -1 & 1
\end{array} \right]\;.}$
4.Using the Gauss-Jordan elimination method, solve:
\begin{displaymath}\begin{array}{rcrcrcr}
-x & + & 3y & & & = & 1\\
& & 3y & + & z & = & 6\\
2x & - & y & + & 2z & = & 9
\end{array} . \end{displaymath}
5.Find the maximum and minimum values of z if z = 2x + 3y and x and y are subject to the constraints: $2x - y \leq 20,\;\;x + y
\leq 220,\;\;x \geq 20$ and $y \geq 80$.
6.Jack bought 7 acres of land on which he wishes to grow wheat, barley and corn. The cost of planting one acre is $120 for wheat, $80 for barley and $90 for corn. Jack has only $720 available for planting. The anticipated profit per acre is $1,000 for wheat, $800 for barley and $400 for corn. How many acres of each should he plant to maximize his profit? 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 $A = \{1,2,3\},\;\;B = \{3,4,5,6\}$ and $C = \{2,3,4\}$.
Find $A' \cap (B \cup C)'$.
8. In how many ways can 15 persons be assigned to 3 departments if 5 must be assigned to department A, 7 to department B and 3 to department C?
9.Montreal and Toronto are playing a best of 5 tournament. (The first team that wins 3 games is the series winner.) Games 1 and 2 are at Montreal, games 3 and 4 (if necessary) are at Toronto and game 5 (if necessary) is at Toronto. Both teams are equally matched and the only advantage is the home ice. For each game the probability of the home team winning is .6. What is the probability that Montreal will win the series even though it lost the first home game?
10. Out of the 100 students enrolled in an evening college 73 are taking an English course and 27 are taking a Math course and 18 are taking both.
(a) What is the probability that a student, chosen at random, is taking the English course given that he is taking the Math course?
(b) What is the probability that a student, chosen at random, is taking neither?
11. A novice golfer must hit a certain shot. He has six clubs in his bag, only one of which is the right club. The probability that he hits a good shot is .23 if he uses the right club, but only .1 if he uses a wrong club. Suppose he picks a club at random and hits a good shot, what is the probability that he selected the right club?
12.The value of a building which cost its owner $200,000 is declining at an annual rate of 10% of that year's value. What is it worth at the end of 5 years?
13. An amount $P deposited now in a bank (that gives a nominal rate of j% p.a., compounded monthly, has a value of $1061.68 at the end of 1 year and has a value of $1127.16 at the end of 2 years. Find P.
14. Julie deposits $300 at the end of each month. The rate is 6% p.a., compounded monthly. How much is in her account immediately after 40 deposits? How much is earned interest?
15.Ann bought a new car for $12,500. She is required to make a 20% down payment. She borrows the rest to be amortized over a 5 year period at 9% p.a., compounded monthly.
(a) What is her monthly payment?

(b) If Ann decides to sell the car after 3 years, how much should she sell it for to pay the loan?