DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 1833

FINAL EXAMINATION
APRIL 1999
TIME: 3 HOURS

Calculators are permitted.
Show all intermediate calculations.

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}

MARKS
(5) 1. Let ${\displaystyle A = \left[ \begin{array}{rrr}
-1 & 4 & 0\\
1 & 2 & -3
\end{array} \right]}$ and ${\displaystyle B = \left[
\begin{array}{rrr}
2 & 1 & -2\\
1 & 0 & 4
\end{array} \right]}$. Find, if possible,
(a) A + 2B, (b) AB, (c) ABt.
(5)2. Use matrices and Gauss-Jordan elimination to solve
\begin{displaymath}\left\{ \begin{array}{rcrcrcrcr}
2x & + & 4y & - & 4z & - & ...
...
- x & - & 2y & + & 4z & - & 10w & = & 2
\end{array} \right. . \end{displaymath}
(5)3. Find the inverse of
\begin{displaymath}A = \left[ \begin{array}{rrr}
-2 & 5 & 1\\
1 & -3 & -1\\
-1 & 2 & 1
\end{array} \right] . \end{displaymath}
(4)4. A company makes electronic hockey and soccer games. Each hockey game requires 2 hours of assembly and 2 hours of testing. Each soccer game requires 3 hours of assembly and 1 hour of testing. Each day there are 42 hours available for assembly and 26 hours available for testing. How many of each game should be produced each day to maximize total daily output? Express this as a linear programming problem. DO NOT SOLVE IT!
(5)5. Maximize P(x,y) = x + 2y
subject to:
\begin{displaymath}\left\{ \begin{array}{ccl}
y & \leq & -x + 100\\
y & \geq & \frac{1}{3}\;x + 20\\
y & \leq x
\end{array} \right. .\end{displaymath}
(3)6. A radio station polled 190 students about music. It found that 114 liked rock, 50 liked country and 41 liked classical. Moreover 14 liked rock and country, 15 liked rock and classical, 11 liked classical and country and 5 liked all three. How many students
(a) like country but not rock?
(b) like exactly one of the types of music?
(c) do not like any of the types of music?
(3)7. A shipment of 100 diskettes has 7 defectives. How many samples of 3 have exactly one defective in them?
(3)8. A man, a woman and their 3 children stand in a row for family pictures. How many different pictures can be taken that have the parents standing together.
(4)9. Three students are deciding which of 3 calculus sections to take (A, B or C). What is the probability that they all choose
(a) the same section?
(b) different sections?
(2)10. A coin is tossed 3 times. What is the probability that the outcome is HHH given that at least 2 heads occur?
(5)11. It is estimated that 10% of Olympic athletes use steroids. A test is 93% effective in correctly detecting steroids in users. It yields false positives in 2% of non-users. What is the probability that someone who tests positive is a user?
(3)12. How much should a 21-year-old invest in a GIC at 6% compounded monthly to yield $10,000 at age 25?
(2)13. What is the effective annual interest rate if the nominal rate is 12% compounded monthly?
(3)14. How much money should be deposited now into an account yielding 6% compounded quarterly in order to receive $3,000 at the end of each quarter for 2 years?
(3)15. A city has a debt of $1,000,000 falling due in 15 years. How much should it deposit at the end of each half-year into an account that yields 4% compounded semi-annually to pay off the debt?
(5)16. Calculate the monthly payments on an $80,000 mortgage amortized over 25 years at an interest rate of 9% compounded monthly. After 3 years how much is owing on the principal?

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