
\documentstyle[11pt]{article}
\title{MATH 110 Final Exam}
\author{}
\date{March 15, 1990}
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Name \hrulefill\ \hspace{.5in} Student Number \hrulefill\
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{\bf Show your work! No credit will be given if no work is shown.
Partial credit will be given only if the corresponding work is shown.}
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1.(10 pts) A test is composed of 4 multiple choice problems and 4
questions that can be answered true or false.  Each multiple choice
problem has 3 choices.  How many different response sheets are possible
if only one choice is marked for each problem?
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2.(10 pts) From a group of 8 people, in how many ways can we choose a
chairperson, a vice-chairperson, and a treasurer, assuming one person
cannot hold more than one position? In how many ways can we choose
a committeee of 3 people?   
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3.(10 pts) A committee of 4 people is to be chosen from a group of
4 men and 5 women. What is the probability that the committee consists
of 2 men and 2 women? 
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4.(10 pts) 1000 tickets for a concert were sold at \$10 each. Tickets
are to be drawn at random and  prizes are awarded as follows:
one prize of \$500, two prizes of \$100, and five prizes of \$50. What
is the expected value to you if you buy one ticket?

5.(10 pts) A spinner is numbered from 1 through 10, and each number is
as likely to occur as any other. Find the probabilities of
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(a) the dial will stop at a number less than 3 or larger than 7;
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(b) the dial will stop at an odd number or a number divisible by 3.
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6.(10 pts) A survey of 100 people found that 60 people own cats,
70 people own dogs and 45 people own both cats and dogs.  If a
person is selected at random, what is the probability that the person
owns neither cats nor dogs?
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7.(10 pts) A box of 10 parts consists of 8 good ones and 2 defective
ones. A sample of 2 is chosen from these 10 parts. What is the
probability that one or more parts in the sample are defective?
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8.(10 pts) A basketball team is to play two games in a tournament.
The probability of winning the first game is .60. If the first game
is won, the probability of winning the second game is .50. if the first
game is lost, the probability of winning the second game is .70. What
is the probability that the first game was won if the second game is
lost?

9.(10pts) Solve the linear system of equations by augmented matrix
using Gauss-Jordan elimination:
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\hspace{8mm}$x_1 - 3x_2 - 2x_3 = 9$

\hspace{8mm}$x_2 - 3x_3 = 7$

\hspace{8mm}$x_1 + x_3 = -6$
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10.(10pts) Given 

\[{\em A} = \left[ \begin{array}{rrr}
   1 & -3 & 0 \\
   0 & 3 & 1 \\
   2 & -1 & 2
 \end{array} \right] \]

Find ${\em A}^{-1}$.
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11.(10 pts) If \$5,000 is to be invested, part at 12\% and part at 8\%,
how much should be invested at each rate so that the annual return will
be \$520?
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12.(10 pts) Given

\[{\em A}= \left[ \begin{array}{rrr}
   -1 & -4 & 3 \\
   2  &  0 & 1 
 \end{array} \right] \hspace{3mm}and \hspace{3mm} 
{\em B}= \left[ \begin{array}{rr}
   2 & -3 \\
   1 &  2 \\
   0 & -1 
 \end{array} \right] \]

Find {\em AB}. 
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13.(10 pts) Given the following linear programing problem:

\[ \mbox{ Maximize } P=x_{1} - x_{2} + 3x_{3} \]
\[ \mbox{ Subject to } \begin{array}[t]{lllll}
  x_{1}& +x_{2} & & \leq & 20 \\
  x_{1} & & +x_{3}& = & 5 \\
 & x_{2} & +x_{3} & \geq & 10 \\
 x_{1}, & x_{2}, & x_{3} & \geq 0  \end{array} \]
introduce slack variables, surplus variables and artificial variables
and form the modified problem for big M method. Then write the
preliminary simplex tableau and label the columns. {\bf Don't perform
any operations}
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14.(10 pts) A catering company makes three kinds of jello salads:
orange, strawberry, and lemon.  It supplies two outlet stores, Outlet
A and Outlet B.  The company can make up to 14 trays of orange,
20 trays of strawberry, and 12 trays of lemon per day.  Outlet A 
needs at least 26 trays of jello per day and
Outlet B needs at least 10 trays of jello per day.  The transportation
cost for shipping one tray from the company to the outlets are given in
the chart below:
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\begin{tabular}{|l|c|c|}
\hline
Type of jello  &  Outlet A & Outlet B \\
\hline
Orange &  \$2  &  \$3 \\
Strawberry & \$3 & \$2 \\
Lemon & \$4 & \$4 \\ 
\hline
\end{tabular} 
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Formulate a linear programming problem that will determine a shipping
schedule that will minimize the cost of transporting the jello salads
to fill the needs of two outlets. {\bf Do not solve the linear
programing problem.} 
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15.(10 pts) Solve the following linear programming problem using the
simplex method:
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Maximize $P=x_1-x_2$

Subject to $x_1+x_2 \leq 6$

\hspace{19mm}$3x_1+10x_2 \leq 30$

\hspace{19mm}$x_1, x_2 \geq 0$
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16.(10pts) Solve the following linear programming problem by applying 
the simplex method to the dual problem:
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Minimize $C=x_{1}+4x_{2}$

Subject to $x_{1}+7x_{2} \geq 2$

\hspace{19mm}$2x_{1}+x_{2} \geq 5$

\hspace{19mm}$4x_{1}+8x_{2} \geq 1$

\hspace{19mm}$x_{1}, x_{2}, x_{3} \geq 0$
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17.(10 pts) Suppose you deposit \$500 into a bank at 12\%. How much 
would you have in the account after one year if
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(a)the account is compounded quarterly;

(b)the account is compounded monthly.
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18.(10 pts) What is the future value of an ordinary annuity at the end
of 3 years if \$400 is deposited each year into an account earning 12\%
compounded annually?
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19.(10 pts) A new boat was sold. The buyer paid \$500 down and accepted
a  financing plan under which he would pay \$300  at the end of each 
month in the next 48 months. 
The rate of the financing plan is 9\% compounded monthly.
What's the selling price of the boat? 
How much interest would the buyer pay?
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20.(10pts) You agree to pay off a \$8,000 loan in 24 monthly payments.
The interest rate is 0.75\% per month on the unpaid balance.  What is
the unpaid balance after 6 monthly payments have been made?
(Hint: The unpaid balance is the ``present value'' for the remaining
payments. You must
find the monthly PMT first.)



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