\documentclass[11pt]{article} \usepackage{amsmath,amssymb,epsfig} \voffset=-1.4in \hoffset=-1.2in \textwidth=6.8in \textheight=10.5in \begin{document} \thispagestyle{empty} \centerline{\textbf{\huge FINAL EXAM MAT 119}} \vspace*{.3in} \textbf{MAT119 - Spring 2003 RUEDEMANN \hspace*{1.0in} NAME \hrulefill} \vspace*{.1in} \vspace*{.3in} \noindent \begin{center} \textit{(Please show all work for credit. Answers without work will not receive credit. Please box your final answers.)} \end{center} \vspace*{0.2in} \begin{enumerate} \item (15 points) Solve \\ $ 2x+3y = 6 $ \\ $ x-y=8 $\\ by (a) The substitution method and (b) the elimination method. You must show work for credit. \vspace*{1.0in} \begin{flushright} \hspace*{4.0in} $(a)$\hrulefill \end{flushright} \vspace*{1.0in} \begin{flushright} \hspace*{4.0in} $(b)$\hrulefill \end{flushright} \item (15 points) The reduced row-echelon form of the augmented matrices for different systems of linear equations is given. Classify them by number of solutions.\\ (a) $ \begin{bmatrix} 1 & 0 & 1 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix} $ \hspace*{0.5in} (b) $ \begin{bmatrix} 1 & 0 & 1 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 7 \end{bmatrix} $ \hspace*{0.5in} (c) $ \begin{bmatrix} 1 & 0 & 0 & 8 \\ 0 & 1 & 0 & 9 \\ 0 & 0 & 1 & 10 \end{bmatrix} $ \begin{flushright} \hspace*{3.0in} (a)Number of Solutions = \hrulefill \end{flushright} \begin{flushright} \hspace*{3.0in} (b) Number of Solutions = \hrulefill \end{flushright} \begin{flushright} \hspace*{3.0in} (c)Number of Solutions = \hrulefill \end{flushright} \newpage \begin{flushright} \hspace*{1.5in} Name:\hrulefill \end{flushright} \item (25 points) Let A = $ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $ and B = $ \begin{bmatrix} -1 & 0 \\ 1 & 5 \end{bmatrix} $ Find the following: \vspace*{0.5in} \begin{flushright} \hspace*{4.0in} (a) A + B = \hrulefill \end{flushright} \vspace*{0.5in} \begin{flushright} \hspace*{4.0in} (b) A - 2B = \hrulefill \end{flushright} \vspace*{0.5in} \begin{flushright} \hspace*{4.0in} (c) AB = \hrulefill \end{flushright} \vspace*{0.5in} \begin{flushright} \hspace*{4.0in} (d) BA = \hrulefill \end{flushright} \vspace*{0.5in} \begin{flushright} \hspace*{4.0in} (e) $A^{-1}$ = \hrulefill \end{flushright} \vspace*{0.2in} \vspace*{0.2in} \newpage \item (25 points) Use the simplex method to maximize $ P = 3x_{1} + 2x_{2} $ subject to\\ $ x_{1} + x_{2} \leq 5 $ \\ $ 2x_{1} + x_{2} \leq 6 $ \\ $ x_{1} \geq 0 , \; \; \; x_{2} \geq 0 $ \\ (You must use the simplex method and you must show work for credit.) \vspace*{7.0in} \begin{flushright} \hspace*{1.2in} Answer: \hrulefill \end{flushright} \newpage \item (10 points) Which is the better investment rate, 8.0\% compounded monthly or 8.2\% compounded annually? Explain why. \vspace*{1.5in} \begin{flushright} \hspace*{1.2in} Answer: \hrulefill \end{flushright} \item (20 points) A couple wishes to purchase a house for \$140,000 with a down payment of 20\%. They amortize the balance at 8\% for 30 years. (a) What is their monthly payment? (b) What is the total interest paid? (c)What is their equity after 20 years? \vspace*{4in} \begin{flushright} \hspace*{3in} (a) monthly payment = \hrulefill \end{flushright} \vspace*{0.2in} \begin{flushright} \hspace*{3in} (b) total interest = \hrulefill \end{flushright} \vspace*{0.2in} \begin{flushright} \hspace*{3in} (c) equity after 20 years = \hrulefill \end{flushright} \vspace*{0.2in} \newpage \item (10 points) Frank borrows \$1000 from Nancy and promises to pay back \$1200 in 6 months. What simple rate of interest will Frank pay? \vspace*{1.2in} \begin{flushright} \hspace*{3in} simple interest rate = \hrulefill \end{flushright} \item (10 points) There are 20 freshmen, 12 sophomores, and 7 juniors in a class. How many ways are there to select 5 freshmen, 3 sophomores and 1 junior to be on a committee? \vspace*{1.2in} \begin{flushright} \hspace*{3in} \# committees = \hrulefill \end{flushright} \item (10 points) How many 7-digit telephone numbers can be formed if adjacent digits cannot be the same? \vspace*{1.2in} \begin{flushright} \hspace*{3in} \# phone numbers = \hrulefill \end{flushright} \item (10 points) How many distinct "words" can be formed using all the letters in the word committee? What is the probability that a given word starts with an "e"? \vspace*{0.5in} \begin{flushright} \hspace*{2.5in} (a) \# words = \hrulefill \end{flushright} \vspace*{0.5in} \begin{flushright} \hspace*{2.50in} (b) probability of beginning with an "e": = \hrulefill \end{flushright} \newpage \item (20 points) You toss a pair of dice? Let E and F be the events\\ \vspace*{0.2in} E: Sum is 6 \hspace*{1in} F: Doubles are thrown\\ Find the following: \vspace*{0.5in} \begin{flushright} \hspace*{2.50in}(a) $P(E)$ = \hrulefill \end{flushright} \vspace*{0.5in} \begin{flushright} \hspace*{2.50in} (b) $P(F) $ = \hrulefill \end{flushright} \vspace*{0.5in} \begin{flushright} \hspace*{2.5in} (c) $P(E|F)$ = \hrulefill \end{flushright} \vspace*{0.5in} \begin{flushright} \hspace*{2.50in} (d) Are $E$ and $F$ independent? Circle one: Yes or No \end{flushright} \item (15 points) Cars are being produced by two plants. Plant I produces 40\% of the car output, plant II produces the remaining 60\% of the cars. One percent of the cars produced by plant I are defective, while 2\% of those from plant II are defective. A car is chosen at random from the annual output and is found to be defective. What is the probability it came from plant I? \vspace*{2.2in} \begin{flushright} \hspace*{3in} Answer: \hrulefill \end{flushright} \newpage \item (15 points) A coin is unfair with heads more likely to occur than tails. Specifically, $P(H)=4/5$ while $P(T)=1/5$ You flip the coin 10 times. \vspace*{0.5in} (a) What is the probability of getting at least one head? \vspace*{0.5in} \begin{flushright} \hspace*{2.5in} (a) \hrulefill \end{flushright} \vspace*{0.5in} (b) What is the probability of getting exactly 6 heads? \vspace*{0.5in} \begin{flushright} \hspace*{2.5in} (b) \hrulefill \end{flushright} \vspace*{0.5in} (c) What is the expected number of heads if you toss the coin 100 times? \vspace*{0.5in} \begin{flushright} \hspace*{2.5in} (c) \hrulefill \end{flushright} \end{enumerate} \end{document}