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\begin{document}






\title{{\large\bf Math 325, Test 2}}
\comment{ \large 
\begin{minipage}[c]{6in}
1. Please cross \fbox{$\times $} the correct answers. \\
2.  This test will be exactly 50 minutes in length.  
When you are told to begin, but not before, glance 
through the entire test and put your name on each
page.  It is YOUR RESPONSIBILITY to make sure your 
test consists of 6 PAGES with 10 PROBLEMS.  The point value for each 
multiple choice problem is 9 points. 
Problems 9 and 10 are worth 14 points each giving a total of 100 points.
Use the back of the test pages  for scratch work.\\
3.   On all partial credit problems, show your work, indicating clearly 
 how you arrived at your answer. 
 The points you receive for problems  9 and 10 will depend on the
extent to which the work you show convinces the grader 
(at the time of grading) that you did all or
part of the problem correctly.  
\end{minipage}}
\answersheetfootline{\large\bf 
\fbox{\rule[-2mm]{0mm}{8mm}Sign your name:\hspace{6cm}}
\hspace{6cm}}
\twocolumn{6} 
\date{{\large Nov. 2, 1992}}
\lastversion=3 \noprofessor \nosection \vertmultchoice=2.5in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm
\boxdepth=0.4cm

\pagenumberstyle=1
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\problem     %%{\bf Fall 92, Test2} 
 The inverse  of the matrix 
$A = \left( \hspace{-2mm} \begin{array}{ccc}1&0&0\\ 0&2&1\\  0&5&3\\
\end{array} \hspace{-2mm} \right) $ is:


\correct $ \left( \hspace{-2mm} \begin{array}{ccc}1&0&0\\ 0&3&-1\\  0&-5&2\\
\end{array} \hspace{-2mm} \right) $
\wrong $ \left( \hspace{-2mm} \begin{array}{ccc}1&0&0\\ 0&-2&5\\  0&1&-3\\
\end{array} \hspace{-2mm} \right) $
\wrong $ \left( \hspace{-2mm} \begin{array}{ccc}-1&0&0\\ 0&-2&5\\  0&1&-3\\
\end{array} \hspace{-2mm} \right) $
\wrong $ \left( \hspace{-2mm} \begin{array}{ccc}-1&0&0\\ 0&3&-1\\  0&-5&2\\
\end{array} \hspace{-2mm} \right) $
\wrong $ \left( \hspace{-2mm} \begin{array}{ccc}2&1&0\\ 5&3&0\\  0&0&1\\
\end{array} \hspace{-2mm} \right) $

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\problem     %%{\bf Fall 92, Test2} 
Consider the following system of 3 equations in the 3 unknowns $x_1,x_2,x_3$.
$$  \begin{array}{ccccccc}x_1&+&x_2&+&x_3 
&=&a\\ x_1&+&2x_2&+&3x_3&=&a\\ x_1&+&7x_2&+&13x_3&=&a\\
\end{array} \hspace{-2mm}  $$ Which of the following statements is true?


\correct One has infinitely many solutions $x  $ for any given value $a$.
\wrong One has no solution $x  $ if $a= 0$.
\wrong One has no solution $x =\left( \hspace{-2mm} \begin{array}{c}x_1\\ x_2\\  x_3\\
\end{array} \hspace{-2mm} \right) $ if $a\neq 0$.
\wrong One has only one solution $x  $ if $a= 0$.
\wrong One has exactly one solution $x  $ if $a\neq 0$.

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 \problem     %%{\bf Fall 92, Test2} 
 Which of the following numbers is an
eigenvalue of 

$A = \left( \hspace{-2mm} 
\begin{array}{cccc} 1&2&3&4 \\ 1&2&3&4\\ 1&2&3&4\\ 1&2&3&4\\
\end{array} \hspace{-2mm} \right)$


\correct $0$
\wrong $1$
\wrong $2$
\wrong $4$
\wrong $8$

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 \problem     %%{\bf Fall 92, Test2} 
 Find the general solution of $x' =Ax$ where $A= \left( \hspace{-2mm}  \begin{array}{cc}
0 & 1 \\ -1 & 0 \\  \end{array} \hspace{-2mm} \right) $. 

\correct
 $c_{1}\left( \hspace{-2mm}  
\begin{array}{c}\sin\; t\\ \cos\; t\\\end{array} \hspace{-2mm} \right)$
+$c_{2} \left( \hspace{-2mm}  
\begin{array}{c} -\cos\; t \\   \sin\; t \\ \end{array} \hspace{-2mm} \right)$

\wrong
 $c_{1}e^{t} \left( \hspace{-2mm}  
\begin{array}{c} 1 \\ 0\\ \end{array} \hspace{-2mm} \right)$ +
$c_{2}e^{-t}\left( \hspace{-2mm}  
\begin{array}{c}-1\\ 0 \\ \end{array} \hspace{-2mm} \right)$

\wrong
 $c_{1} \left( \hspace{-2mm}  
\begin{array}{c} \cos\; t \\ \cos\; t \\ \end{array} \hspace{-2mm} \right)$ +
$c_{2}\left( \hspace{-2mm}  
\begin{array}{c}\sin\; t\\ \sin\; t \\ \end{array} \hspace{-2mm} \right)$

\wrong
 $c_{1} \left( \hspace{-2mm} 
 \begin{array}{c} e^{t} \cos\; t \\ -e^{t}\sin\; t \\ \end{array}
\hspace{-2mm} \right)$ + $c_{2}\left( \hspace{-2mm} 
 \begin{array}{clcr}e^{t}\cos\; t\\ e^{t} \sin\;t \\ \end{array}
\hspace{-2mm} \right)$

\wrong
 $c_{1}e^{t} \left( \hspace{-2mm}  
\begin{array}{c} 1 \\ 1 \\ \end{array} \hspace{-2mm} \right)$ +
$c_{2}e^{-t}\left( \hspace{-2mm}  
\begin{array}{c}0\\ 1\end{array} \hspace{-2mm} \right)$


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\problem    %%{\bf Fall 92, Test2}
You are given a matrix
$A = \left( \hspace{-2mm} \begin{array}{cc}2 & 4\\ -1 & 6\\ 
\end{array} \hspace{-2mm} \right) $ which has a double eigenvalue at $r=4$
with a corresponding eigenvector $\left( \hspace{-2mm}
\begin{array}{c} 2 \\ 1 \\ \end{array} \hspace{-2mm} \right) $.
Therefore the system $x'=Ax$ has the  solution
$\left( \hspace{-2mm}
\begin{array}{c} 2 \\ 1 \\ \end{array} \hspace{-2mm} \right) e^{4t}$.
Find the general solution for this system.


\correct $x(t) = C_{1} \left( \hspace{-2mm} \begin{array}{c} 2 \\ 1 \\ \end{array} 
\hspace{-2mm} \right) e^{4t} + C_{2} \left( \hspace{-2mm}
\begin{array}{c} 2t-1 \\ t \\ \end{array} \hspace{-2mm} \right) e^{4t}$

\wrong $x(t) = C_{1} \left( \hspace{-2mm} \begin{array}{c} 2 \\ 1 \\ \end{array} 
\hspace{-2mm} \right) e^{4t} + C_{2}t \left( \hspace{-2mm}
\begin{array}{c} 2 \\ 1 \\ \end{array} \hspace{-2mm} \right) e^{4t}$

\wrong $x(t) = C_{1} \left( \hspace{-2mm} \begin{array}{c} 2 \\ 1 \\ \end{array} 
\hspace{-2mm} \right) e^{4t} + C_{2} \left( \hspace{-2mm}
\begin{array}{c} 2t \\ t-4 \\ \end{array} \hspace{-2mm} \right) e^{4t}$

\wrong $x(t) = C_{1} \left( \hspace{-2mm} \begin{array}{c} 2 \\ 1 \\ \end{array} 
\hspace{-2mm} \right) e^{4t} + C_{2} t\left( \hspace{-2mm}
\begin{array}{c} 1 \\ 1\\ \end{array} \hspace{-2mm} \right) e^{4t}$

\wrong $x(t) = C_{1} \left( \hspace{-2mm} \begin{array}{c} 2 \\ 1 \\ \end{array} 
\hspace{-2mm} \right) e^{4t} + C_{2} \left( \hspace{-2mm}
\begin{array}{c} 1 \\ t \\ \end{array} \hspace{-2mm} \right) e^{4t}$


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  \problem     %%{\bf Spring 92 and Fall92, Test2} 
 If $x(t)$ is the solution of the initial value problem
$$x' = \left( \hspace{-2mm} \begin{array}{cc}0 & 1\\ -1 & 0\\ 
\end{array} \hspace{-2mm} \right)x, \hspace{7mm}
x(0)=\left( \hspace{-2mm} \begin{array}{c}-1\\ 1\\ 
\end{array} \hspace{-2mm} \right),$$ then $x(\smallfrac{\pi}{2})$ is:

\correct $\left( \hspace{-2mm} \begin{array}{c} 1\\ 1\\ 
\end{array} \hspace{-2mm} \right)$
\wrong $\left( \hspace{-2mm} \begin{array}{c} 0\\ 1\\ 
\end{array} \hspace{-2mm} \right)$
\wrong $\left( \hspace{-2mm} \begin{array}{c} -1\\ 0\\ 
\end{array} \hspace{-2mm} \right)$
\wrong $\left( \hspace{-2mm} \begin{array}{c} 1\\ -1\\ 
\end{array} \hspace{-2mm} \right)$
\wrong $\left( \hspace{-2mm} \begin{array}{c}0\\ 0\\ 
\end{array} \hspace{-2mm} \right)$


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 \problem   %%{\bf Fall 92, Test2}
 Find the unique solution of the initial value problem
$x' = \left( \hspace{-2mm} \begin{array}{cc}0 & 1\\ 1 & 0\\ 
\end{array} \hspace{-2mm} \right) x + \left( \hspace{-2mm}
\begin{array}{c}e^{2t}\\ e^{2t}\\ \end{array} \hspace{-2mm} \right)
\hspace{5mm}x(0)=\left( \hspace{-2mm}
\begin{array}{c}2\\ 0\\ \end{array} \hspace{-2mm} \right).$


\correct $\left( \hspace{-2mm} \begin{array}{c}e^{2t}+e^{-t}\\ e^{2t}-e^{-t}\\ 
\end{array} \hspace{-2mm} \right)$

\wrong $\left( \hspace{-2mm} \begin{array}{c}2\cosh t\\   \sinh t\\ \end{array} 
\hspace{-2mm} \right)$

\wrong $\left( \hspace{-2mm} \begin{array}{c}1\\ -1\\ \end{array} 
\hspace{-2mm} \right)e^{t} + \left( \hspace{-2mm}
\begin{array}{c}1\\ 1\\ \end{array} \hspace{-2mm} \right) e^{2t}$

\wrong $\left( \hspace{-2mm} \begin{array}{c}2\\ 0\\ \end{array} 
\hspace{-2mm} \right) e^{2t}$


\wrong $\left( \hspace{-2mm} \begin{array}{c}2\cos t\\ 2 \sin t\\ 
\end{array} \hspace{-2mm} \right)$
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\problem   %%{\bf Fall 92, Test2}
 Consider the system
$x' = Ax$, where $A$ has eigenvalues -3 and 1 with corresponding eigenvectors
$\left( \hspace{-2mm} \begin{array}{c}1\\ -1\\ 
\end{array} \hspace{-2mm} \right)$ and
$\left( \hspace{-2mm} \begin{array}{c}1\\ 2\\ 
\end{array} \hspace{-2mm} \right)$. If $\Phi (t)$ is the fundamental matrix
with $\Phi (0)=I$ then the determinant of $\Phi (t)$ equals:


\correct $e^{-2t}$
\wrong $e^{t}$
\wrong $e^{-3t}$
\wrong $e^{2t}$
\wrong $e^{3t}$
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%%%%%%%%%%%%%%%%%%%%  Partial Credit        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\problem   %%{\bf Fall 92, Test 2}
 (14 pts.) 
Find the general solution of the system
$x' = \left( \hspace{-2mm} \begin{array}{ccc}3&5&7\\ 0&2&-1\\  0&1&2\\
\end{array} \hspace{-2mm} \right) x.$

\vspace*{15cm}

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\problem   %%{\bf Fall 92, Test 2}
 (14 pts.) 
 Consider the system $x' = \left( \hspace{-2mm} \begin{array}{cc}0 & 
1\\ 0 & 0\\ \end{array}
\hspace{-2mm} \right)x.$
 Find the fundamental matrix $\Phi(t)$ satisfying $\Phi(0) = \left( \hspace{-2mm}
\begin{array}{cc}1 & 0\\ 0 & 1\\ \end{array} \hspace{-2mm} \right).$


\vspace*{15cm}

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