\input calc
\def\problemspace{.25in}
\let\ds=\displaystyle

%% TITLE
{\bf Math 226: Calculus IV}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam I}\quad{\sl February 14, 1991}\hfill Score:\hbox to 2.5truein{\hrulefill}

\bigskip
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this answer sheet. There are 16 multiple
choice questions worth 6 points each. You start with 4~points.

%% PROBLEM %1
Let 
$$\A = \bmatrix{2&0\cr1&-6\cr3&1\cr}\qquad
\B =\bmatrix{-1\cr1\cr5\cr}$$
Find $\A - 2\B$.
%% CORRECT
{{\sl Not defined}}
%% WRONG
{$\bmatrix{4&0\cr-1&-6\cr-7&1\cr}$}
%% WRONG
{$\bmatrix{2&2\cr1&-8\cr3&-9\cr}$}
%% WRONG
{$\bmatrix{4\cr-1\cr-7\cr}$}
%% WRONG
{$\bmatrix{2\cr-8\cr9\cr}$}

%% PROBLEM %2
Calculate the matrix product
$$\bmatrix{-1&0&2\cr3&1&1\cr}\bmatrix{2&-1\cr0&-1\cr4&0\cr}$$

%% CORRECT
{$\bmatrix{6&1\cr10&-4\cr}$}
%% WRONG
{$\bmatrix{-2&0\cr0&-1\cr}$}
%% WRONG
{$\bmatrix{6&1&0\cr1&-1&0\cr}$}
%% WRONG
{$\bmatrix{-2&0\cr0&-1\cr12&0\cr}$}
%% WRONG
{{\sl Not defined}}

%% PROBLEM %3
Which of the following matrices are in row-echelon form?
$$
\A = \bmatrix{
1&-2&1&0\cr
0&0&0&0\cr
0&0&0&1\cr
}
\qquad
\B = \bmatrix{
0&0&1&2\cr
1&-3&1&0\cr}
\qquad
\C = \bmatrix{
0&1&0&-1\cr
0&0&1&2\cr
0&0&0&1\cr
0&0&0&0\cr}
$$
$$
\D = \bmatrix{
0&1\cr0&2\cr}
\qquad
\E = \bmatrix{
1&0&0\cr
0&0&0\cr
0&0&0\cr}
$$
%% CORRECT
{$\C$ and $\E$ only}
%% WRONG
{$\E$ only}
%% WRONG
{$\D$, $\C$, and $\E$ only}
%% WRONG
{$\B$, $\C$, $\D$, and $\E$ only}
%% WRONG
{$\A$, $\B$, $\C$, $\D$, and $\E$}

%% PROBLEM %4
Translate the following system of equations into an augmented matrix.
$$\eqalign{
x_1 - x_2 + x_4 &= 5\cr
2x_2 +x_3&=-3\cr
x_1 - x_3 + 2x_4 &= 1\cr
x_2 - x_4 &= 0\cr}$$
%% CORRECT
{$\bmatrix{
1&-1&0&1&:&5\cr
0&2&1&0&:&-3\cr
1&0&-1&2&:&1\cr
0&1&0&-1&:&0\cr}$}
%% WRONG
{$\bmatrix{
1&-1&1&:&5\cr
2&1&0&:&-3\cr
1&-1&2&:&1\cr
1&-1&0&:&0\cr}$}
%% WRONG
{$\bmatrix{
1&-1&1\cr
0&2&1\cr
1&-1&2\cr
0&1&-1\cr}$}
%% WRONG
{$\bmatrix{
1&-1&0&1\cr
0&2&1&0\cr
1&0&-1&2\cr
0&1&0&-1\cr}$}
%% WRONG
{$\bmatrix{
1&-1&1&0&:&5\cr
0&2&1&0&:&-3\cr
1&-1&2&0&:&1\cr
0&1&-1&1&:&0\cr}$}

%% PROBLEM %5
Which of the following is the general solution of the system of
equations associated to the augmented matrix
$$\bmatrix{
1&1&-1&0&4&:&0\cr
0&0&1&5&3&:&1\cr
0&0&0&1&-2&:&2\cr}$$
%% CORRECT
{\hsize=1.5truein\vtop{$$\eqalign{
x_1 &= -9 - x_2 - 17x_5\cr
x_3 &= -9 -13x_5\cr
x_4 &= 2 + 2x_5\cr}$$}}
%% WRONG
{\hsize=1.5truein\vtop{$$\eqalign{
x_1 &=  - x_2 - 4x_5\cr
x_3 &= 1 -3x_5\cr
x_4 &= 2 + 2x_5\cr}$$}}
%% WRONG
{\hsize=1.5truein\vtop{$$\eqalign{
x_1 &=0\cr
x_3 &= 1\cr
x_4 &= 2\cr}$$}}
%% WRONG
{\hsize=1.5truein\vtop{$$\eqalign{
x_1 &=  - x_2 +x_3 - 4x_5\cr
x_3 &=  1 - 5x_4 + 3x_5\cr
x_4 &= 2 - 2x_5\cr}$$}}
%% WRONG
{{\sl System is inconsistent}}


%% PROBLEM %6
Transform the following matrix into row-echelon form without exchanging any
rows.
$$\bmatrix{
1&2&0&3\cr
2&5&1&6\cr
1&2&0&4\cr
3&7&1&10\cr}$$
%% CORRECT
{$\bmatrix{
1&2&0&3\cr
0&1&1&0\cr
0&0&0&1\cr
0&0&0&0\cr}$}
%% WRONG
{$\bmatrix{
1&2&0&3\cr
0&1&1&0\cr
0&0&1&1\cr
0&0&0&1\cr}$}
%% WRONG
{$\bmatrix{
1&2&0&3\cr
0&0&1&1\cr
0&0&0&1\cr
0&0&0&0\cr}$}
%% WRONG
{$\bmatrix{
1&2&0&3\cr
0&1&0&1\cr
0&0&0&1\cr
0&0&0&0\cr}$}
%% WRONG
{$\bmatrix{
1&0&0&0\cr
0&1&0&0\cr
0&0&1&0\cr
0&0&0&0\cr}$}

%% PROBLEM %7
Find the inverse of the following matrix, if it exists.
$$\bmatrix{
1&2&3\cr
0&1&2\cr
1&0&1\cr}$$
%% CORRECT
{${1\over2}\bmatrix{
1&-2&1\cr
2&-2&-2\cr
-1&2&1\cr}$}
%% WRONG
{${1\over2}\bmatrix{
1&1&-3\cr
0&-2&0\cr
1&0&1\cr}$}
%% WRONG
{$\bmatrix{
1&-2&3\cr
0&1&-2\cr
0&0&1\cr}$}
%% WRONG
{${1\over2}\bmatrix{
-3&1&1\cr
1&1&-2\cr
0&-1&1\cr}$}
%% WRONG
{{\sl Does not exist}}

%% PROBLEM %8
Calculate the determinant of the matrix
$$\bmatrix{
1&-1&1\cr
-2&1&-2\cr
1&-3&1}$$
%% CORRECT
{$0$}
%% WRONG
{$6$}
%% WRONG
{$12$}
%% WRONG
{$-6$}
%% WRONG
{$-12$}

%% PROBLEM %9
Use expansion by cofactors to calculate the determinant of the matrix
$$\bmatrix{
2&-1&3&0\cr
6&1&-1&5\cr
0&3&0&0\cr
4&0&0&2\cr}$$
%% CORRECT
{$-60$}
%% WRONG
{$30$}
%% WRONG
{$-15$}
%% WRONG
{$120$}
%% WRONG
{$0$}

%% PROBLEM %10
Calculate the (2,3)-cofactor, $A_{(2,3)}$, of the matrix
$$\A = \bmatrix{
5&7&1\cr
2&6&2\cr
1&4&20\cr}$$
%% CORRECT
{$-13$}
%% WRONG
{$-8$}
%% WRONG
{$16$}
%% WRONG
{$4$}
%% WRONG
{$0$}

%% PROBLEM %11
Which of the following statements are always true:

\medskip
{\leavevmode\parindent = 20truept
\item{(A)}\enspace If $\X_1$, $\X_2$, $\X_3$, $\X_4$
are linearly dependent, then there are {\bf no} scalars $c_1$, $c_2$, $c_3$,
and $c_4$, not all zero, such that $c_1\X_1 + c_2\X_2 + c_3\X_3 + c_4\X_4 =
0$.

\medskip
\item{(B)}\enspace If $\X_1$, $\X_2$, $\X_3$, and $\X_4$
are linearly independent, then $\X_1$ is {\bf not} a linear combination of
$\X_2$, $\X_3$, and $\X_4$.

\medskip
\item{(C)}\enspace If $\X_1$, $\X_2$, and $\X_3$ are 2-dimensional vectors,
then they are linearly dependent.

\medskip
\item{(D)}\enspace If $\X_1$, $\X_2$,and  $\X_3$
are linearly independent, then $\X_1$, $\X_2$, $\X_3$, and $\X_4$ are
linearly independent.

}
%% CORRECT
{B and C only}
%% WRONG
{A, C, and D only}
%% WRONG
{B, C, and D only}
%% WRONG
{A, and D only}
%% WRONG
{A, B, C, and D}

%% PROBLEM %12
Find all solutions to the equation $\A\X = \B$ where
$$\A = \bmatrix{1&-1&-1\cr-1&1&-1\cr-1&-1&1\cr},\quad
\X = \bmatrix{x_1\cr x_2\cr x_3\cr}, \quad \B =
\bmatrix{-2\cr4\cr2\cr}$$
%% CORRECT
{$\X = \bmatrix{-3\cr0\cr-1\cr}$}
%% WRONG
{$\X = \bmatrix{-2\cr0\cr-1\cr}$}
%% WRONG
{$\X = \bmatrix{-2\cr4\cr2\cr}$}
%% WRONG
{$\X = \bmatrix{-3\cr1\cr-2\cr}$}
%% WRONG
{{\sl No solution exists}}

%% PROBLEM %13
Determine a maximal subset of linearly
independent vectors from the vectors
$$\X_1 = [0,1,0,0]\quad \X_2 =[1,1,1,1]\quad \X_3 =
[0,1,1,1]\quad \X_4 = [1,1,0,0]$$
%% CORRECT
{$\X_1$, $\X_2$, $\X_3$}
%% WRONG
{$\X_1$}
%% WRONG
{$\X_1$, $\X_2$}
%% WRONG
{$\X_3$, $\X_4$}
%% WRONG
{$\X_1$, $\X_2$, $\X_3$, $\X_4$}

%% PROBLEM %14
Use the Gram-Schmidt process to produce an
orthonormal set of vectors from the following vectors:
$\X_1 =[0,-1,0,1]$, $\X_2 = [0,1,1,-1]$, $\X_3 = [1,1,-1,0]$.
%% CORRECT
{\hsize=1.5truein\vtop{$$\eqalign{
\Q_1 &= {1\over\sqrt{2}}[0,-1,0,1]\cr
\Q_2 &= [0,0,1,0]\cr
\Q_3 &= \sqrt{2\over3}[1,{1\over2},0,{1\over2}]\cr
}$$}}
%% WRONG
{\hsize=1.5truein\vtop{$$\eqalign{
\Q_1 &= [0,-1,0,1]\cr
\Q_2 &= [0,0,1,0]\cr
\Q_3 &= [2,1,0,1]\cr
}$$}}
%% WRONG
{\hsize=1.5truein\vtop{$$\eqalign{
\Q_1 &= [1,0,0,0]\cr
\Q_2 &= [0,1,0,0]\cr
\Q_3 &= [0,0,1,0]\cr
}$$}}
%% WRONG
{\hsize=1.5truein\vtop{$$\eqalign{
\Q_1 &= {1\over\sqrt{2}}[0,-1,0,1]\cr
\Q_2 &= {1\over\sqrt{3}}[0,1,1,-1]\cr
\Q_3 &= {1\over\sqrt{3}}[1,1,-1,0]\cr
}$$}}
%% WRONG
{\hsize=1.5truein\vtop{$$\eqalign{
\Q_1 &= {1\over\sqrt{2}}[0,-1,0,1]\cr
\Q_2 &= {1\over3}[0,2,1,-2]\cr
\Q_3 &= {1\over\sqrt{2}}[1,1,0,0]\cr
}$$}}

%% PROBLEM %15
Find the general solution to the differential equation
$$ x^2y' + y = e^{\ds 1\over\ds x},\quad x > 0$$
%% CORRECT
{$y = ce^{\ds 1\over\ds x}-{1\over x}e^{{\ds 1\over\ds x}}$}
%% WRONG
{$y = -{c\over x^2}e^{\ds 1\over\ds x}-{1\over x}$}
%% WRONG
{$y = ce^{\ds 1\over\ds x}-{1\over x}$}
%% WRONG
{$y = {c\over x}e^{-{\ds 1\over\ds x}}+ e^{{\ds 1\over\ds x}}$}
%% WRONG
{$y = {c\over x}-e^{-{\ds 1\over\ds x}}$}

%% PROBLEM %16
On which of the following intervals does a unique solution exists to the
initial value problem
$$(x^2 - 2x)y' + xy = \cos(x),\quad y(1) = 0$$
%% CORRECT
{$(0,2)$}
%% WRONG
{$(-\infty,2)$}
%% WRONG
{$(0,\infty)$}
%% WRONG
{$(-\infty,\infty)$}
%% WRONG
{{\sl A unique solution does not exist}}

%% END
\bye
