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\begin{center} 
{\Large TEST 2}

Math 105 - 5
\end{center}

Show all your work and make your work legible.
Each problem is worth 10 points (except number 1).

\begin{enumerate}

\item  Name \rule{5in}{.3pt}

\item  Solve using your favorite method:
\[
\begin{array}{rrrrrrr}
x & + & y & + & z & = & 1 \\
2x & - & 3y & + & z & = & 2 \\
\end{array}
\]
\vspace{3in}

\item  You are working late one night plotting the graph of a parabola and you plot the points: $(1,2), (2,4), (-1,-4)$.  Its late and you don't finish the graph.  When you get back to work the next day, your dog has eaten the paper and you can't remember what the original function was.  Find the function.  (Remember a parabola comes from a quadratic function $y=ax^2+bx+c$)
\vspace{3in}

\item  Find the partial fraction decomposition for:
\[
\frac{x-2}{x^2(x-1)}
\]
\vspace{2in}
%\newpage

\item  Graph the system of inequalities and find the maximum and minimum of 
$z=-2x-5y$ subject to the constraints:
\[
\begin{array}{ccccc}
& & x & \geq & 0 \\
& & y & \geq & 0 \\
2x & + & y & \leq & 6 \\
\end{array}
\]

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\item  Find the following if possible.  If not possible, say so.

\[
A=\left[ \begin{array}{rrr}
3 & 2 & 2 \\
2 & 2 & 2 \\
-4 & 4 & 3 \\
\end{array} \right]\hspace{.5in}
B= \left[ \begin{array}{rr}
2 & -1\\
3 & -2 \\
-5 & 7
\end{array} \right]
\]

\begin{enumerate}
\item  $AB$
\vspace{1.5in}
\item  $BA$
\vspace{1.5in}
\item  $A^{-1}$
\vspace{1.5in}
\item  $B^{-1}$
\vspace{1.5in}
\item  $\mbox{det}(A)$
\vspace{1.5in}
\end{enumerate}
%\newpage

\item  Find the inverse of the matrix $A$.  No decimals!  If there are fractions, write them as fractions.
\[
A=\left[ \begin{array}{rr}
9 & 7 \\
-4 & 5 \\
\end{array} \right]
\]
\vspace{2in}

\item   Using a determinant, find the equation of a line passing through the vertex of the parabola: $y=3(x+2)^2-5$ and the point $(1,1)$
\vspace{2.5in}

\item  Solve for $x$:
\[
\left|
\begin{array}{rr}
x & x-2 \\
1 & x-1 \\
\end{array} \right|
=5
\]
\newpage

\item  For each of the following sequences, determine if the sequence is arithmetic, geometric or neither.  If arithmetic, find the common difference.  If geometric, find the common ratio.  If neither, just say so.
\begin{enumerate}
\item  
\[
 1,\frac{1}{2}, \frac{1}{6}, \frac{1}{24} ,\frac{1}{120}, . . .
\]
\vspace{.5in}

\item  
\[
1, \frac{5}{3}, \frac{7}{3}, 3, \frac{11}{3}, . . .
\]
\vspace{.5in}

\item
\[
8, 12, 18, 27, \frac{81}{2}, . . .
\]
\vspace{1in}

\end{enumerate}

\item  Find the sums (no decimals, if there are fractions, use fractions):
\begin{enumerate}
\item 
\[
\sum_{k=1}^{10}3k+2
\]
\vspace{2in}

\item
\[
\sum_{n=2}^{5} \left( \frac{1}{2} \right) ^n
\]

\end{enumerate}

\end{enumerate}

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