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\noindent{\bf Name:}\hskip 2 in {\bf Professor:} \bigskip
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\centerline{\bf {Mathematics 325}}
\centerline{\bf Fall Semester 1999}
\centerline{\bf Test 2}
\centerline{\bf {October 12, 2000}}

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\noindent
1. This Examination contains twelve problems worth a total of 100 points.
(There are 9 problems in Part A worth a total of 45 points. There are 3
problems in Part B worth a total of 45 points as well. You begin with 10
points). The test will be exactly
75 minutes in length.

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2. For each multiple choice question, please mark an $X$ on the correct
answer on the answer sheet.
{\bf Do not circle it.}

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3. On problem 10, 11 and 12, show your work, indicating clearly in the
space provided how you arrive at your
answer; otherwise no credit will be given.

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4. Calculators, books and notes are not allowed.

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5. A table of Laplace transforms is supplied at the end of the booklet.

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6. Hand in the entire test.

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\noindent{\bf Sign the pledge:} \lq \lq {On my honor, I have neither given nor
received unauthorized aid on this Exam}":

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\hskip .7 in\vrule height .10pt depth 0 pt width 2 in \bigskip

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\centerline{\bf GOOD LUCK}

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\noindent
1. Find the Laplace transform of
$f(t)=\delta(t-\pi) \cos t$.\\
(a) $-e^{\pi s}$\ \ \
(b) $ - e^{-\pi s}$\ \ \
(c) $-1$\ \ \ (d) $e^{-\pi s}$\ \ \ (e) $e^{-\pi s}\cos s $.\\
2. Let $F(s)=\frac{1}{s^{3}(s^{2}+4)}$
Find the inverse Laplace transform of $F$.\\
(a) $t^2\sin 2t$\ \
(b) $\frac 12 t^2\sin 2t $\ \
(c) $\frac 14 t^2\sin 2t $\\
(d) $\frac 12\int _0^t(t-\tau)^2 \sin 2\tau d\tau $\ \
(e) $\frac 14\int _0^t(t-\tau)^2 \sin 2\tau d\tau $\\
3. Find the inverse of the matrix
$\left(\begin{array}{ll}
2 & 4\\1 & 3
\end{array}\right)$\\
(a) $\left(\begin{array}{ll} -1 & 3\\
2 & -4\end{array} \right)$
(b) $\left(\begin{array}{ll} \frac {3}{2}& -2\\
-\frac {1}{2}&1\end{array} \right)$
(c) $\left(\begin{array}{ll}4&3\\
2&1\end{array}\right)$
(d) $\left(\begin{array}{ll}2 & 1\\
\frac{3}{2} & \frac{1}{2}\end{array}\right)$
(e) $\left(\begin{array}{ll} -2&1\\
\frac {3}{2}&-\frac {1}{2}\end{array} \right)$
\noindent
4. Which of the following statements is incorrect?\\
(a) The system $\begin{array}{lll} x_{1} + x_{3} & = & 0 \\
x_{1}+x_{2} & = & 0\\
x_{2}+x_{3} & = & 0\end{array}$
has only the trivial solution.\\
(b) The vectors $\textbf{x}^{(1)}(t) =\left(\begin{array}{l}
t^{2}\\t \end{array}\right)$
and
$\textbf{x}^{(2)}(t)=\left(\begin{array}{l}t\\1\end{array}\right)$ are linearly
dependent on $0 < t < 1$.\\
(c) The vectors $\textbf{x}^{(1)}
=\left(\begin{array}{l}1\\2\end{array}\right)$ and
$\textbf{x}^{(2)} =\left(\begin{array}{l}2\\0\end{array}\right)$ are linearly
independent.\\
(d) The vectors $\textbf{x}^{(1)} =
\left(\begin{array}{l}1\\1\\0 \end{array}\right)$,
$\textbf{x}^{(2)} =\left(\begin{array}{l} 0\\1\\1\end{array}\right)$,
$\textbf{x}^{(3)} =\left(\begin{array}{l} 1\\0\\1\end{array}\right)$
are linearly independent.\\
(e) The two vectors $\textbf{x}^{(1)}(t) =
\left(\begin{array}{l}\sin t\\\cos t\end{array}\right)$ and
$\textbf{x}^{(2)}(t)=\left(\begin{array}{l}-\cos t\\ \ \
\sin t\end{array}\right)$ are linearly independent on $-\infty <t<\infty$.\\ 5.
Using elementary row operations, find the number $k$ such that the system
$\begin{array}{lll}x_{1}-2x_{2}+x_{3}&=&k\\
x_{1}-x_{2}+2x_{3}&=&1\\
2x_{1}-3x_{2}+3x_{3}&=&0\end{array}$
has a solution.\\
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(a) $k=0$\ \ (b) $k=2$\ \ (c) $k=1$\ \ (d) $k=3$\ \ (e) $k=-1$\\
6. Suppose that $\textbf{x}^{(1)} =
\left(\begin{array}{l}4\\2\end{array}\right)$. Which
$\textbf{x}^{(2)}$
below will make $\textbf{x}^{(1)}$, $\textbf{x}^{(2)}\}$ linearly
independent?\\
(a) $\textbf{x}^{(2)} = \left(\begin{array}{l}-2\\ -1\end{array}\right)$\ \
(b) $\textbf{x}^{(2)} = \left(\begin{array}{l}2\\1\end{array}\right)$\ \
(c) $\textbf{x}^{(2)}= \left(\begin{array}{l}0\\0\end{array}\right)$\\
(d) $\textbf{x}^{(2)} = \left(\begin{array}{l}\ \ 6\\-4\end{array}\right)$\ \
(e) $\textbf{x}^{(2)} = \left(\begin{array}{l}8\\4\end{array}\right)$\\
7. Compute the eigenvalues of the matrix $\left(\begin{array}{lll}
4&100&99\\0&\ \ 3&\ 2\\0&\ \ 2&\ 3 \end{array}\right).$\\
(a) $100, 99, 4$\ \
(b) $4, 5, 1$\ \
(c) $4, 3, 2$\ \
(d) $3+2i, 3-2i, 4$\ \
(e) $3, 2, 3i$

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\noindent
8. Suppose $A$ is a $2 \times 2$ real matrix with a complex
eigenvalue
$r = 3 + 2i$, and $\left(\begin{array}{l}\ \ 1\\2 + i\end{array} \right)$ is
the corresponding eigenvector.  Which of the following is the general
solution to the first order system $\textbf{x}' = A\textbf{x}$?\\
(a) $c_{1}e^{3t}\left(\begin{array}{l}
2\cos 2t - \sin 2t\\ \ \ \ \cos 2t
\end{array}\right) + c_{2}e^{3t}\left(\begin{array}{l}2\sin 2t + \cos 2t\\
\ \ \ \ \sin 2t\end{array}\right)$\\
(b) $c_{1}e^{2t} \left(\begin{array}{l}\ \ \ \ \cos 3t \\2\cos 3t - \sin 3t
\end{array}\right) + c_{2}e^{2t}\left(\begin{array}{l}\ \ \ \ \sin 3t\\
2\sin 3t + \cos 3t\end{array}\right)$\\
(c) $c_{1}e^{3t}\left(\begin{array}{l}\ \ \ \ \cos 2t\\2\cos 2t-\sin 2t
\end{array}\right)+c_{2}e^{3t}\left(\begin{array}{l}\ \ \ \ \sin 2t\\2\sin
2t+\cos 2t
\end{array}\right)$\\
(d) $c_{1}e^{-3t}\left(\begin{array}{l}\ \ \ \ \cos 2t\\2\cos 2t-\sin 2t
\end{array}\right) + c_{2}e^{-3t}\left(\begin{array}{l}\ \ \ \ \sin 2t\\
2\sin 2t + \cos 2t\end{array}\right)$\\
(e) $c_{1}e^{-2t}\left(\begin{array}{l}\ \ \ \ \cos 3t\\2\cos 3t - \sin 3t
\end{array}\right)+c_{2}e^{-2t}\left(\begin{array}{l}\ \ \ \ \sin 3t\\
2\sin 3t+\cos 3t
\end{array}\right)$\\
9. Suppose $A$ is a $3\times 3$ matrix with a repeated eigenvalue
$r_{1} = r_{2} = 5$, and
$\left(\begin{array}{l}1\\0\\0\end{array}\right)$ and $\left(\begin{array}{l}
0\\0\\1\end{array}\right)$ are the corresponding eigenvectors.
Then which of the following are solutions of
$\textbf{x}'=A\textbf{x}$?\\
(a) $ c_1 e^{5t} \left(\begin{array}{l}1\\0\\0\end{array}\right)
+ c_{2}e^{5t}\left(\begin{array}{l}0\\0\\1\end{array}\right)$, for all
numbers $c_{1}$ and $c_{2}$\\
(b) $c_{1}e^{5t}\left(\begin{array}{l}1\\0\\0\end{array}\right)
+ c_{2}te^{5t}\left(\begin{array}{l}0\\0\\1\end{array}\right)$, for all
numbers $c_{1}$ and $c_{2}$\\
(c) $c_{1}e^{5t}[t\left(\begin{array}{l}1\\0\\0\end{array}\right) +
\left(\begin{array}{l}0\\0\\1\end{array}\right)] + c_{2}e^{5t}
\left(\begin{array}{l}0\\0\\1\end{array}\right)$, for all  numbers $c_{1}$ and
$c_{2}$\\
(d) $ c_{1}e^{5t} \left(\begin{array}{l} 1\\0\\0\end{array}\right) + c_{2}
e^{5t}[t\left(\begin{array}{l} 0\\0 \\ 1\end{array} \right) +
\left(\begin{array}{l}1\\0\\ 0\end{array}\right)]$, for all  numbers $c_{1}$
and $c_{2}$\\
(e) $c_{1}e^{5t} [t\left(\begin{array}{l}0\\0\\
1\end{array}\right) +
\left(\begin{array}{l}1\\0\\0\end{array}\right)] +
c_{2}e^{5t}\left(\begin{array}{l}0\\0\\1\end{array}
\right)$, for all  numbers $c_1$ and $c_2$\\
10.	Find the solution of the initial value problem\\
$y''+4y'+5y=\delta(t-1),\ y(0)=0,\ y'(0)=1$.\\
11. (a) Find the general solution of
$ \textbf{x}' = \left(\begin{array}{ll}\
\ 2&1\\-2&5\end{array}\right)
\textbf{x}$.\\
(b) Find the solution that satisfies the initial condition
$\textbf{x}(0) = \left(\begin{array}{l}2\\3\end{array}\right)$.\\
12. Find the general solution of
$\textbf{x}' = \left(\begin{array}{ll}4&0\\1&4\end{array}\right)\textbf{x}$.
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