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\author{Teschl Gerald}
\title{Sample Test 1}


\begin{document}

\maketitle

\noindent 1) Solve the system
\beas
x + 2y &=& 1\\
6y+ 3x &=& 2
\eeas

a) $x=0,y=0$ b) $x=0,y=1$ c) $x=1,y=1$ d) $x=1-2a$, $y=a$ with $a$ arbitrary e) no
solution\\[6mm]

\noindent 2) Urban Developers has two size lots in its development. One sells for 
\$2500, and the other for \$3000. One month the developer sold 22 lots for a total of of
\$61,500. How many of each did it sell?

a) (9,13) b) (10,12) c) (8,14) d) (12,10) e) (11,11)\\[6mm]


\noindent 3) Which of the following matrices is in reduced echelon form?
\bd
\mb{a)} \left[ \ba{ccc} 1 & 2 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\ea\right| \left. \ba{c} 0 \\ 0 \\ 1\ea\right] \quad
\mb{b)} \left[ \ba{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\ea\right| \left. \ba{c} 3 \\ 0 \\ 2\ea\right] \quad
\mb{c)} \left[ \ba{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\ea\right| \left. \ba{c} 0 \\ 0 \\ 2\ea\right]
\ed
\bd
\mb{d)} \left[ \ba{ccc} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\ea\right| \left. \ba{c} 0 \\ 7 \\ 2\ea\right] \quad 
\mb{e)} \left[ \ba{ccc} 1 & 7 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\ea\right| \left. \ba{c} 0 \\ 0 \\ 1\ea\right] 
\ed
\vspace*{3mm}\\[6mm]

\noindent 4) Let $x,y,z$ be a solution of
\beas
x + 2 y &=& 0\\ -y + z &=& 2 \\ -y + 2z &=& 5.
\eeas
What is $x+y+z$?

a) 1 b) 2  c)  3 d) 4 e) 5\\[6mm]

\noindent 5) What is the reduced row echelon form of the augmented matrix corresponding
to the following system?
\beas
x + 3y - 11 &=& 0\\
3x - 4y + 6 &=& 0\\
2x + 7y + 17 &=& 0
\eeas
\bd
\mb{a)} \left[ \ba{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0\ea\right| \left. \ba{c} 1 \\ 0 \\ 0\ea\right] \quad
\mb{b)} \left[ \ba{cc} 1 & 1 \\ 0 & 1\\ 0 & 0\ea\right| \left. \ba{c} 0 \\ 0 \\ -1\ea\right] \quad
\mb{c)} \left[ \ba{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0\ea\right| \left. \ba{c} 0 \\ 0 \\ 1\ea\right]
\ed
\bd
\mb{d)} \left[ \ba{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0\ea\right| \left. \ba{c} 0 \\ 0 \\
-1\ea\right] \quad 
\mb{e)} \left[ \ba{cc} 1 & 0 \\ 0 & 0 \\ 1 & 0\ea\right| \left. \ba{c} 0 \\ 0 \\ 0\ea\right] 
\ed
\vspace*{3mm}\\[6mm]

\noindent 6) An electric firm assembles and tests 3 kinds of computers, I,II, and III.
Type I,II,III require 4,4,6 hours of assembling and 1,2,3 hours of testing respectively.
The firm has 460 hours for assembling and 200 hours for testing available. How many of
each type should be made such that a total of 100 computers is produced?

a) (30,30,30) b) (30,40,30)  c) (30,40,40) d) (40,40,30) e) (40,40,40)\\[6mm]

\noindent 7) For which values of $x$ and $y$ does the following equality hold?
\bd
\left[ \ba{cc} 1 & x^2-1 \\ 3y & 0\ea\right] = \left[ \ba{cc} 1 & 8 \\ x & y-1\ea\right]
\ed

a) $x=-3,y=1$ b) $x=3,y=1$  c) $x=2,y=0$ d) $x=-3,y=-1$ e) for no values of $x$ and
$y$\\[6mm]

\noindent 8) What is the dimension of the following product
\bd
\left[ \ba{cc} 1 & 2 \\ 3 & 4\ea\right] \left[ \ba{ccc} 1 & 2 & 3\\ 4 & 5 & 6\ea\right] ?
\ed

a) 2 rows, 2 column b) 2 rows, 3 columns c) 2 row, 1 columns\\ d) 2 rows, 3 columns e)
product is not defined\\[6mm]

\noindent 9) Let $a$ and $b$ be numbers such that $ab=1$. Then the inverse of
\bd
\left[ \ba{cc} a & 7 \\ 0 & b\ea\right]
\ed
is
\bd
a) \left[ \ba{cc} b & 7 \\ 0 & a\ea\right] \quad
b) \left[ \ba{cc} a & -7 \\ 0 & b\ea\right] \quad
c) \left[ \ba{cc} b & -7 \\ 0 & a\ea\right] \quad
d) \left[ \ba{cc} a & 7 \\ 0 & b\ea\right] \quad
e) \mb{does not exist}
\ed

\noindent 10)  Suppose you have a {\em Leontief production model} consisting of an electric
company and a steel company. To produce a certain amount of electricity the electric
company needs 7\% of its own electricity and 4\% of the value of its own electricity
worth of steel. The steel company uses 12\% of its own steel, and 40\% of the value of
its own steel worth of electricity. Find the value of the produced electricity and the
value of the steel products if the customers demand 6 million dollars worth of electricity
and 4 million dollars worth of steel products. Compute the internal demand.

a) $\D \left[ \ba{c} 0.8 \\ 1.8 \ea\right]$ b) $\D \left[ \ba{c} 4.4 \\ 1.8 \ea\right]$ c)
$\D \left[ \ba{c} 5.6 \\ 2.4 \ea\right]$ d) $\D \left[ \ba{c} 0.8 \\ 4.0 \ea\right]$ e)
$\D \left[ \ba{c} 2.4 \\ 5.6 \ea\right]$\\[6mm]

\noindent 11) Which of the following points $(x,y)$ is a feasible solution to the system
of inequalities $2x + 4y \ge 2$ and $x-y <0$?

a) (2,2),\hspace{3mm}b) (0,$\frac{1}{10}$),\hspace{3mm}c)
($\frac{1}{2}$,$\frac{1}{5}$),\hspace{3mm} d)  (2,0),\hspace{3mm}e)
(1,$\frac{3}{2}$)\\[6mm]

\noindent 12) An investor wants to invest in tax exempt bonds with a 7.5\% return and
taxable bonds with a 9\% return so that the total return will be at least \$8100. Which of
the following inequality expresses this information?

a) $7.5 x  + 9 y \ge 8100$ b) $0.075 x  + 0.09 y > 8100$ c) $0.075 x  + 0.09 y \le 8100$
d) $7.5 x  + 9 y > 8100$ e) $0.075 x  + 0.09 y \ge 8100$\\[6mm] 

\noindent 13) Which of the following points is a corner of the following system of
inequalities?
\bd
5x+2y \le 6, \quad  x+ 2y \le 2, \quad x \ge 0, \quad y \ge 0.
\ed

a) (1,1) b) ($\frac{1}{2}$,$\frac{1}{2}$) c) (1,0) d) (1,$\frac{1}{2}$) e)
(0,$\frac{1}{2}$)\\[6mm]

\noindent 14) Find the maximum value of $z$ given the objective function $z = 12x +y$ subject to
\bd
5x+2y \le 6, \quad  x+ 2y \le 2, \quad x \ge 0, \quad y \ge 0.
\ed

a) 0 b) 3 c) $\frac{25}{2}$ d) $\frac{72}{5}$ e) 15\\[6mm]

\noindent 15) A sewing machine operator may sew coats or trousers. The trousers require two minutes sewing time and the operator receives \$0.50 each. A coat requires eight minutes sewing time, and the operator receives \$1.50 each. The operator must sew at least four coats per hour. How many coats and how many trousers should the operator sew each hour to maximize hourly income?

a) 18 trousers, 3 coats b) 14 trousers, 4 coats c) 18 trousers, 4 coats d) 12 trousers, 5
coats e) 10 trousers, 6 coats  

\begin{center}
\Large Calculator problems
\end{center}

\noindent 16) Compute the reduced echelon form of the following matrix:
\bd
\left[ \ba{ccc} 0 & 1 & -3 \\ 2 & 4 & 6 \\ 3 & 5 & 2\ea\right| \left. \ba{c} 2 \\ -4 \\ 2\ea\right] 
\ed

\noindent 17) Solve if possible the system
\beas
0.3x + 0.7y + 0.4z -0.8 &=& 0\\
0.2x + 0.9y + 0.2z -0.1 &=& 0\\
0.5x + 0.2y + 0.9z -0.6 &=& 0
\eeas
\vspace{5mm}

\noindent 18) Compute
\bd
\left[ \ba{cc} 112 & 27 \\ 18 & 53\ea\right] \left[ \ba{cc} 1.22 & 3.45\\ 7.18 & 5.01 \\ 6.33 & 2.78 \ea\right]
\ed
\vspace{5mm}

\noindent 19) Compute the inverse of the following matrix
\bd
\left[ \ba{ccc} 0 & 4 & -2 \\ 1 & 7 & 3 \\ 1 & 3 & 5\ea\right]
\ed
\vspace{5mm}

\noindent 20) Solve if possible the system
\beas
0.1x + 0.2y &=& 0.7\\
0.4y - 0.4z &=& 1.6\\
0.3x + 0.5y + 0.1z &=& 1.9
\eeas
\vspace{5mm}


\end{document}
