\magnification=\magstep1
\baselineskip=12pt
\parskip=6pt
\def\ds{\displaystyle}
\input amstex
\line{April 7, 1997\hfil Math 262 Exam 2\hfil Name\ {\hbox to 2truein{\hrulefill}}}
\bigskip

\item{(10) \bf 1.}
The vectors $(1,2,1),\ (3,4,5),\ (2,0,k)$ are linearly {\bf dependent} if
\medskip
\hskip4truein{A.\ \ $k=1$}

\hskip4truein{B.\ \ $k=6$}

\hskip4truein{C.\ \ $k\ne6$}

\hskip4truein{D.\ \ $k=0$}

\hskip4truein{E.\ \ $k\ne0$}


\vfill

\item{(10) \bf 2.} If $T\colon P_3 @>>> P_3$ is a linear transformation such
that $T(x^2-1)=x^2+x-3,\ T(3x)=6x$ and $T(2x+1)=4x+4$, then $T(x^2)$ is

\hskip4truein{A.\ \ $x^2$}

\hskip4truein{B.\ \ $x^2+x-2$}

\hskip4truein{C.\ \ $x^2+x-1$}

\hskip4truein{D.\ \ $x^2+x$}

\hskip4truein{E.\ \ $x^2+x+1$}

\vfill\eject

\item{(10) \bf 3.}
Use {\it Cramer's Rule\/} to solve the system below for the unknown
functions $u_1(x)$ and $u_2(x)$.
$$\aligned
u_1\sin x+ u_2\cos x & = 0 \\
u_1\cos x- u_2\sin x & = e^x
\endaligned$$

\vfill

\item{(10) \bf 4.}
What is the correct {\it form\/} of $y_p$ to use when
finding a particular solution to the equation
$y'' + y = x \cos x$
using the method of undetermined coefficients?
\newline
{\bf Do not compute the coefficients}.  Just write
down the {\bf FORM} of the particular solution.  (For example,
if the right hand side were $x^2$, the correct form of $y_p$ would
be $Ax^2+Bx+C$.)

\vskip.8in

\item{(20) \bf 5.}Let
$$
A=\bmatrix\format\r&\quad\r&\quad\r\\ 1&1&-2\\ 0&1&a\\ 2&4&-3\endbmatrix
$$
\itemitem{\bf a)}for what value(s) of $a$ is $\det A\not= 0$.
\itemitem{\bf b)}Find all $a$ such that the equation $Ax=0$ has a nontrivial
solution.
\vfill\eject

\item{(20) \bf 6.} Find the general solution $y(x)$ to the differential
equation
$$
y''+3y'+2y=10\sin x.
$$


\vfill\eject
\item{(20) \bf 7.}Let $T\colon R^4 @>>> R^3$ be defined by $Tx=Ax$ where
$$
A=\bmatrix\format\r&\quad\r&\quad\r&\quad\r\\ 1&1&-1&-3\\ 0&1&1&-4\\ 2&2&-2&-6\endbmatrix.
$$
Find a basis for ker$(T)$.  What is the dimension of ker$(T)$?

\enddocument