\input calc
\def\problemspace{ .75truein}
\parindent=0pt

%% TITLE
{\bf Math 226: Calculus IV}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam I}\quad{\sl February 11, 1993}\hfill Section:\hbox to 2.5truein{\hrulefill}

\bigskip\bigskip\noindent
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this answer sheet. There are 15 multiple
choice questions worth 6 points each. You start with 10~points.

\def\answerspace{ .4truein}

%1
%% PROBLEM
Let $\A = \bmatrix{1&1\cr 1&1\cr}$, $\B = \bmatrix{1&-1\cr
-1&1\cr}$, and $\C = \bmatrix{3&-5\cr-7&9}$. Compute $\A\C - \B\C$.

%% CORRECT
$\bmatrix{-14& 18\cr 6& -10\cr}$
%% WRONG
$\bmatrix{3&-5\cr-7&9}$
%% WRONG
$\bmatrix{6&-10\cr-14&18}$
%% WRONG
$\bmatrix{-7&9\cr3&-5}$
%% WRONG
$\bmatrix{-4&4\cr-4&4\cr}$

%2
%% PROBLEM
Calculate the product
$\bmatrix{2&3\cr4&5\cr6&7\cr}\bmatrix{-1&0&3\cr0&2&0\cr}$.
%% CORRECT
$\bmatrix{-2&6&6\cr-4&10&12\cr-6&14&18\cr}$
%% WRONG
$\bmatrix{-2&0\cr0&10\cr}$
%% WRONG
$\bmatrix{-2&6\cr-4&10\cr}$
%% WRONG
$\bmatrix{-2&-4&-6\cr64&10&14\cr6&12&18\cr}$
%% WRONG
$\bmatrix{16&10\cr}$


%3
%% PROBLEM
Determine which of the following matrices are in echelon form.
$$\displaylines{
\A = \bmatrix{ 0 & 0 & 1\cr 0 & 1 & 0 \cr 1 & 0 & 0 \cr }\qquad
\B = \bmatrix{ 0 & 0 & 0 & 0 \cr 1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0\cr }\qquad
\C = \bmatrix{ 0 & 1 & 0 \cr 0 & 0 & 1\cr 0 & 0 & 0 \cr }\cr
\D = \bmatrix{ 1 & 0  & 2 & 4 \cr 0 & 0 & 1 & 2 \cr 0 & 0 & 0 & 1 \cr }\qquad
\E = \bmatrix{ 1 & 0\cr 0 & 1\cr 0 & 1\cr }
}$$

%% CORRECT
$\C$ and $\D$ only
%% WRONG
$\A$, $\B$, $\C$ and $\D$ only
%% WRONG
$\B$, $\C$ and $\D$ only
%% WRONG
$\A$, $\B$, $\C$, $\D$, and $\E$
%% WRONG
$\D$ only


%4
%% PROBLEM
Let $\A = \bmatrix{1&0&1\cr2&-1&3\cr0&1&-2\cr}$, $\X = \bmatrix{x_1\cr
x_2\cr x_3\cr}$, and $\B = \bmatrix{0\cr-1\cr3\cr}$. Determine which of
the following systems of equations is represented by $\A\X = \B$.

%% CORRECT
\vtop{\hbox{$x_1 + x_3 = 0$}
\hbox{$2x_1 - x_2 + 3x_3 = -1$}
\hbox{$x_2 - 2 x_3 = 3$}}
%% WRONG
\vtop{\hbox{$x_1 + 2x_2 = 0$}
\hbox{$- x_2 + 3x_3= -1$}
\hbox{$x_1 + 3x_2 - 2 x_3 = 3$}}
%% WRONG
\vtop{\hbox{$x_1 + x_2 + x_3 = 0$}
\hbox{$2x_1   + 3x_3 = -1$}
\hbox{$x_1 - x_2 - 2 x_3 = 3$}}
%% WRONG
\vtop{\hbox{$x_1 - x_3 = 0$}
\hbox{$2x_1 + x_2 - 3x_3 = 0$}
\hbox{$x_2 + 2 x_3 = 0$}}
%% WRONG
\vtop{\hbox{$x_1 - 2x_2 + x_3 = 0$}
\hbox{$x_1 + 3x_2= -1$}
\hbox{$-x_1 + x_2 + 2 x_3 = 3$}}


%5
%% PROBLEM
Determine the general solution to the system of equations associated to the
augmented matrix
$$\bmatrix{
1&-1& 3& 0& 1&:&4\cr
0& 0& 1&-2&-1&:&5\cr
0& 0& 0& 1& 1&:&3\cr
}$$
%% CORRECT
\vtop{\hbox{$x_1 = -29 + x_2 + 2x_5$}
\hbox{$x_3 = 11 - x_5$}
\hbox{$x_4 = 3 - x_5$}}
%% WRONG
\vtop{\hbox{$x_1 = 1 + x_2 - x_5$}
\hbox{$x_3 = 3 + x_5$}
\hbox{$x_4 = 3 - x_5$}}
%% WRONG
\vtop{\hbox{$x_1 = -28$}
\hbox{$x_3 = 10$}
\hbox{$x_4 = 2$}}
%% WRONG
\vtop{\hbox{$x_1 = 4 - x_2 + 3x_3 + x_5$}
\hbox{$x_3 = 5 - 2x_4 - x_5$}
\hbox{$x_4 = 3 + x_5$}}
%% WRONG
\vtop{\hbox{$x_1 = -29$}
\hbox{$x_3 = 11$}
\hbox{$x_4 = 3$}}


%6
%% PROBLEM
Transform the following matrix into row-echelon form.
$$\bmatrix{
1&1&1&1\cr
1&1&1&2\cr
1&1&3&3\cr
1&4&4&4\cr}$$

%% CORRECT
$\bmatrix{
1&1&1&1\cr
0&1&1&1\cr
0&0&1&1\cr
0&0&0&1\cr}$
%% WRONG
$\bmatrix{
1&1&1&1\cr
0&0&0&1\cr
0&0&0&0\cr
0&0&0&0\cr}$
%% WRONG
$\bmatrix{
1&1&1&1\cr
0&0&1&1\cr
0&0&0&1\cr
0&0&0&0\cr}$
%% WRONG
$\bmatrix{
1&1&1&1\cr
0&1&1&2\cr
0&0&1&3\cr
0&0&0&1\cr}$
%% WRONG
$\bmatrix{
1&1&1&1\cr
0&1&1/2&1\cr
0&0&1&1/2\cr
0&0&0&1\cr}$


%7
%% PROBLEM
Let $\A = \bmatrix{0&1&-1\cr-1&1&0\cr1&0&1\cr}$. Determine which of the
following is the first {\sl row} of the inverse matrix $\A^{-1}$.
%% CORRECT
$[1/2, -1/2, 1/2]$
%% WRONG
$[1, 1, 2]$
%% WRONG
$[1/4, -1/4, 3/4]$
%% WRONG
$[1, -2, -1]$
%% WRONG
$[-1/3, -1/3, 2/3]$


%8
%% PROBLEM
Choose the expression that give the determinant of the matrix $$\A = \bmatrix{
7&3&1&5\cr-2&6&-4&1\cr3&-1&2&1\cr6&2&-5&4\cr}$$
%% CORRECT
$7\det\bmatrix{6&-4&1\cr-1&2&1\cr2&-5&4\cr}
-3\det\bmatrix{-2&-4&1\cr3&2&1\cr6&-5&4\cr}
+ \det\bmatrix{-2&6&1\cr3&-1&1\cr6&2&4\cr}
-5\det\bmatrix{-2&6&-4\cr3&-1&2\cr6&2&-5\cr}$
%% WRONG
$7\det\bmatrix{6&-4&1\cr-1&2&1\cr2&-5&4\cr}
+3\det\bmatrix{-2&-4&1\cr3&2&1\cr6&-5&4\cr}
+ \det\bmatrix{-2&6&1\cr3&-1&1\cr6&2&4\cr}
+5\det\bmatrix{-2&6&-4\cr3&-1&2\cr6&2&-5\cr}$
%% WRONG
$7\det\bmatrix{6&-4&1\cr-1&2&1\cr2&-5&4\cr}
-3\det\bmatrix{-2&-4&1\cr3&2&1\cr6&-5&4\cr}
+ \det\bmatrix{6&-4&1\cr-1&2&1\cr2&-5&4\cr}
-5\det\bmatrix{-2&-4&1\cr3&2&1\cr6&-5&4}$
%% WRONG
$7\det\bmatrix{6&-4&1\cr-1&2&1\cr2&-5&4\cr}
-3\det\bmatrix{-2&6&1\cr3&-1&1\cr6&2&4\cr}
- \det\bmatrix{-2&6&1\cr3&-1&1\cr6&2&4\cr}
+5\det\bmatrix{6&-4&1\cr-1&2&1\cr2&-5&4\cr}$
%% WRONG
$-7\det\bmatrix{6&-4&1\cr-1&2&1\cr2&-5&4\cr}
+3\det\bmatrix{-2&-4&1\cr3&2&1\cr6&-5&4\cr}
- \det\bmatrix{-2&6&1\cr3&-1&1\cr6&2&4\cr}
+5\det\bmatrix{-2&6&-4\cr3&-1&2\cr6&2&-5\cr}$


%9
%% PROBLEM
Let $$\bmatrix{1&2&3&4\cr0&0&1&2\cr0&2&3&4\cr0&4&5&7\cr}\quad D = 7$$
be an intermediate stage in the method of pivotal condensation (to compute the
determinant). After further row operations the matrix becomes
$$\bmatrix{1&2&3&4\cr0&1&3/2&2\cr0&0&1&2\cr0&0&-1&-1\cr}$$
What is the value of $D$ at this stage.
%% CORRECT
$-14$
%% WRONG
$-7$
%% WRONG
$7/2$
%% WRONG
$14$
%% WRONG
$-7/2$


%10
%% PROBLEM
Compute the inverse of the matrix $\bmatrix{x&1-x\cr1&{1\over x}}$ for any
$x\ne 0$.

%% CORRECT
$\bmatrix{{1\over x^2}&1 - {1\over x}\cr -{1\over x}& 1\cr}$
%% WRONG
$\bmatrix{{1\over x^2}&{1\over1- x}\cr\strut 1& x\cr}$
%% WRONG
$\bmatrix{{1\over x}&0\cr\strut 0& x\cr}$
%% WRONG
$\bmatrix{{1\over x^2}&x - 1\cr\strut -1& x\cr}$
%% WRONG
$\bmatrix{{1\over x}&x - 1\cr -{1\over x}& 1\cr}$


%11
%% PROBLEM
Let $\X_0 = [-11,9,11,2]$, $\X_1 = [2,-1,1,3]$, $\X_2 = [-1,1,3,2]$, $\X_3 = [0,1,-1,1]$.
If $c_1$, $c_2$, and $c_3$ are constants such that $\X_0 = c_1\X_1 + c_2\X_2 + c_3\X_3$, then
%% CORRECT
$c_1 = -3$
%% WRONG
$c_1 = 5$
%% WRONG
$c_1 = 1$
%% WRONG
$c_1 = -2$
%% WRONG
{\sl no such constant exists}


%12
%% PROBLEM
Use the Gram-Schmidt process to construct an orthogonal set of vectors from the vectors
$\X_1 = [1,0,0,1,0]$, $\X_2 = [1,0,2,0,-1]$, $\X_3 = [-1,-1,0,1,1]$.
%% CORRECT
$\Y_1 = [1,0,0,1,0]$, $\Y_2 = [1,0,4,-1,-2]$, $\Y_3 = [-9, -11, 8, 9, 7]$
%% WRONG
$\Y_1 = [1,0,0,1,0]$, $\Y_2 = [1,0,4,-1,0]$, $\Y_3 = [-2,7,1,2,0]$
%% WRONG
$\Y_1 = [1,0,0,1,0]$, $\Y_2 = [1,0,0,-1,0]$, $\Y_3 = [0, -1, 1, 0, -1]$
%% WRONG
$\Y_1 = [1,0,0,1,0]$, $\Y_2 = [1,0,2,-1,4]$, $\Y_3 = [-2, -1, 0, 2, 1]$
%% WRONG
$\Y_1 = [1,0,0,1,0]$, $\Y_2 = [1,0,2,-1,-1]$, $\Y_3 = [-8, -11, 4, 8, -8]$


%13
%% PROBLEM
Determine which of the following differential equations are linear:
$${\rm (A)}\quad y'' - {y'\over x} = \cos(x),\qquad {\rm (B)}\quad y'' - {y'\over x} = \cos(y),\qquad
{\rm (C)}\quad x y' - y = 0,\qquad {\rm (D)}\quad y y' - x = 0$$
%% CORRECT
A and C only
%% WRONG
C only
%% WRONG
C and D only
%% WRONG
A, C, and D only
%% WRONG
A, B, C, and D


%14
%% PROBLEM
Find the largest interval containing $x = 1$ on which a solution to the differential equation
$$x(x^2-4)y' + 2y = x^3$$
is guaranteed to exist.
%% CORRECT
$(0,2)$
%% WRONG
$(-2,2)$
%% WRONG
$(-1,2)$
%% WRONG
$(0,\infty)$
%% WRONG
$(-\infty, \infty)$


%15
%% PROBLEM
Suppose $y(x)$ satisfies the differential equation $y' + \sin(x) y = \sin(x)$ and $y(0) = 0$. Find $y(\pi)$.
%% CORRECT
$1 - e^{-2}$
%% WRONG
$1 - e^{-1}$
%% WRONG
$1$
%% WRONG
$1 + e$
%% WRONG
$1 + e^2$


%% END
\end
