\input LaTest.tex
\documentstyle{latest}
\textwidth 6.3in\textheight 8.5in\parindent 0cm
\topmargin -1cm\oddsidemargin 0.7cm

\begin{document}
\baselineskip 0.5cm\everyproblem={\mathsurround=4pt}
\normalafterproblemskip=2.5cm\pageannouncements={usual}
                                         \input exam.perm7
                                         \lastversion=0
                                         \date{{\large October 29,
                                             1997}}
                                         \title{{\large\bf Math
                                             108,  Section 1, 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 7 PAGES with 11 PROBLEMS.
  The point value for each multiple choice problem is 7 points.
  Problems 9 and 10 and 11 are worth 44 points total.
  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, 10 and 11 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}\noprofessor \nosection \vertmultchoice=2.5in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm
\boxdepth=0.4cm\pagenumberstyle=1
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\problem Find the length of the vector $\vec{u}=
\left( \begin{array}{c} 2\\ 4\\ -4\end{array}\right) $

\correct 6
\wrong 36
\wrong 2
\wrong 4
\wrong 10


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\problem
Solve the following differential equation with given initial
condition:
$$y^{\prime}=\frac{1}{2}y-3,\ y(0)=4.$$


\correct $f(t)=6-2e^{\smallfrac{1}{2}t}$
\wrong $f(t)=4-2e^{\smallfrac{1}{2}t}$
\wrong $f(t)=6-\frac{1}{2}e^{\smallfrac{1}{2}t}$
\wrong $f(t)=6-e^{\smallfrac{1}{2}t}$
\wrong $f(t)=5-e^{3t}$

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\problem  Assume $y^{\prime}=t^2$ and  $y(0)=1$. Compute $y(3)$.

\correct 10
\wrong 3
\wrong 9
\wrong 0
\wrong 12

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\problem  Find the equation of the plane through the points
$(8,0,0),(0,4,0)$ and $(0,0,2)$.


\correct $x+2y+4z=8$
\wrong $8x+4y+2z=1$
\wrong $8x+4y+2z=8$
\wrong $\smallfrac{1}{8}x+\smallfrac{1}{4}y+\smallfrac{1}{2}z=0$
\wrong $\smallfrac{1}{8}x+\smallfrac{1}{4}y+\smallfrac{1}{2}z=8$



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\problem  For what value of $k$ does the following system of
equations have {\bf no solution}.
$$
\begin{array}{rcl}
4x-14y&=&-17\\
-6x+ky&=&13
\end{array}
$$


\correct $k=21$
\wrong $k=-21$
\wrong $k=\smallfrac{2}{7}$
\wrong $k=7$
\wrong $k=-7$





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\problem Solve the differential equation $y'=e^{t-y}$

\correct $y=\ln (e^t+C)$
\wrong $y= e^t+C$
\wrong $y= e^{t-y}+C$
\wrong $y= e^{t-y}+e^t+C$
\wrong $y=\ln (e^t+t+C)$


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\problem Determine which of the following functions is a solution
of the differential equation $y''-5y'+6y=0$.


\correct $e^{3t}$
\wrong $e^{4t}$
\wrong $e^{5t}$
\wrong $e^{6t}$
\wrong $e^{-5t}$

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\problem
Find the equation of the plane which passes through the
point $P=(1,1,1)$ and which is perpendicular to the line
$\left( \begin{array}{c}x\\ y\\ z\end{array}\right) =
\left( \begin{array}{c}5\\ -6\\ 2\end{array}\right)+\alpha
\left( \begin{array}{c}3\\ -7\\ 5\end{array}\right)$


\correct $6x-14y+10z=2$
\wrong  $x+y+z=1$
\wrong $5x-6y+2z=1$
\wrong $3x-7y+5z=0$
\wrong $5x-6y+2z=0$




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\problem
(5 pts) Find the constant solutions for the differential
equation $y\prime =y^2-9$.

\vspace*{4cm}

(10 pts) In the following graph draw the constant solutions and
sketch the solutions having initial condition $y(0)=1$ and $y(0)=4$.


%\vspace*{17cm}


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\problem (15 pts) A person opens an Individual Retirement account
(IRA) with the initial amount of \$40,000.  Then \$12,000 per
year are deposited in this IRA in a uniform and continuous
manner.  Assume that the interest rate is 7.5\% compounded
continuously.\medskip


(5 pts) Model this problem as a Calculus problem by finding a
{\bf differential equation} and an {\bf initial condition}
describing the amount of money, $M(t)$, in the IRA at any time
$t$.

\vspace*{4cm}

(5 pts) Solve the obtained differential equation, i.e.  find
$M(t)$ at any time $t$.

\vspace*{8cm}

(5 pts) Compute the balance in the IRA after 20 years. (If you do
not have a calculator you can approximate $e^{1.5}$ with 4.5).

\vspace*{5cm}

%\vspace*{17cm}

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\problem

(14 pts)
Find the complete solution of the following system of
linear equations. If no solution exists state this.

$
\begin{array}{ccccccccc}
2x&-&2y&+&z&=&4\\
4x&-&4y&+&4z&=&16\\
-x&+&y&-&z&=&-4
\end{array}
$





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\endtest
