\input calc
\def\problemspace{.25in}

%% TITLE
{\bf Math 226: Calculus IV}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam I}\quad{\sl February 13, 1992}\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 15 multiple
choice questions worth 6 points each. You start with 10~points.

%% PROBLEM
Let $\A = \bmatrix{1&0&-1\cr 0&2&0\cr 3&-1&4\cr}$ and $\B = \bmatrix{-2&3&0\cr 4&2&-1\cr 0&-1&1\cr}$. Compute $2\A - \B$.
%% CORRECT
$\bmatrix{4& -3& -2\cr -4& 2& 1\cr 6& -1& 7\cr}$
%% WRONG
$\bmatrix{2& 4& -1\cr 8& 4& -2\cr -10& 3& 5\cr}$
%% WRONG
$\bmatrix{4& 8& -2\cr 16& 8& -4\cr -20& 6& 10\cr}$
%% WRONG
$\bmatrix{-2& 3& 1\cr 4& 2& -1\cr 2& -2& 1\cr}$
%% WRONG
$\bmatrix{0& 3& -2\cr 4& 6& -1\cr 6& -3& 9\cr}$

%% PROBLEM
Let $\A = \bmatrix{ 1 & -1 \cr 3 & 1 \cr -1 & 3 \cr 2 & 0 \cr  }$ and $\B = \bmatrix{ 1 & -1 & 2 \cr -1 & 2 & 1 \cr  }$. Compute the matrix product $\A \B$.
%% CORRECT
$\bmatrix{2& -3& 1\cr 2& -1& 7\cr -4& 7& 1\cr 2& -2& 4\cr}$
%% WRONG
$\bmatrix{ 1 & 1 & 2\cr -3 & 2 &1\cr -1 & 3 & 0\cr 2 & 0 &0\cr  }$
%% WRONG
$\bmatrix{ 2 & 2 & -4 & 2 \cr -3 & -1 & 7 & -2 \cr 1 & 7 & 
  1 & 4 \cr  }$
%% WRONG
$\bmatrix{ 1 & -3 & -1 & 2 \cr 1 & 2 & 3 & 0 \cr  2 & 1 & 0 & 0 \cr}$
%% WRONG
{\sl Not defined.}

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


%% PROBLEM
Translate the following system of linear equations into an augmented matrix.
$$\eqalign{
x_1 - x_3 +4x_4 - x_5 = 7\cr
x_2 -x_3 + 2x_4 +6x_5 = 1\cr
2x_1 + 5x_2 + x_3 - 3x_5 = 4\cr
}$$
%% CORRECT
$\bmatrix{ 1 & 0 & -1 & 4 & -1 & : & 7 \cr 0 & 1 & -1 & 2 & 6
   & : & 1 \cr 2 & 5 & 1 & 0 & -3 & : & 4 \cr  }$
%% WRONG
$\bmatrix{ 1 & -1 & 4 & -1 & : &7 \cr 1 & 2 & -1 & 6 & : &1 \cr 2 & 
  5 & 1 & -3 & : &4 \cr  }$
%% WRONG
$\bmatrix{ 1 & 0 & 2 & : &7 \cr 0 & 1 & 5 & : &1 \cr -1 & -1 &1 
   & : &4 \cr  4 & 2 & 0 & : & 0\cr -1 & 6 & -3 & : & 0\cr}$
%% WRONG
$\bmatrix{ 1 & -1 & 4 & -1  \cr 1 & 2 & -1 & 6  \cr 2 & 5 & 1 & -3 \cr  }$
%% WRONG
$\bmatrix{ 1 & 0 & 2  \cr 0 & 1 & 5  \cr -1 & -1 & 1 \cr 
4 & 2 & 0\cr -1 & 6 & 3\cr}$


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


%% PROBLEM
Solve the system of equations defined by $\A \X = \B$ where
$$\A = \bmatrix{ 1 & 0 & -1 \cr 1 & 1 & -1 \cr 1 & -1 & 0 \cr  }\qquad \X = \bmatrix{x_1\cr x_2\cr x_3\cr}
   \qquad \B = \bmatrix{1\cr 2\cr 3\cr}$$
%% CORRECT
\vtop{
\hbox{$x_1 = 4$}
\hbox{$x_2 = 1$}
\hbox{$x_3 = 3$}
}
%% WRONG
\vtop{
\hbox{$x_1 = 3$}
\hbox{$x_2 = 1$}
\hbox{$x_3 = 2$}
}
%% WRONG
\vtop{
\hbox{$x_1 =  -2$}
\hbox{$x_2 =  1$}
\hbox{$x_3 =  -3$}
}
%% WRONG
\vtop{
\hbox{$x_1 = -1$}
\hbox{$x_2 = 1$}
\hbox{$x_3 = -2$}
}
%% WRONG
{\sl No solution exists}

%% PROBLEM
Find the inverse of the matrix
$$\bmatrix{ 1 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 1 \cr  }$$
%% CORRECT
$\bmatrix{ 0 & 1 & 0 \cr -1 & 1 & 1 \cr 1 & -1 & 0 \cr  }$
%% WRONG
$\bmatrix{ 1 & 0 & -1 \cr -1 & 0 & 0 \cr 0 & -1 & 1 \cr  }$
%% WRONG
$\bmatrix{ 0 & 0 & 1 \cr -1 & 1 & 2 \cr 1 & 0 & -1 \cr  }$
%% WRONG
$\bmatrix{ 1 & 1 & -1 \cr 0 & 1 & 0 \cr 0 & -1 & 1 \cr  }$
%% WRONG
$\bmatrix{ 0 & -1 & 1 \cr 0 & 1 & 0 \cr 1 & 2 & -1 \cr  }$

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


%% PROBLEM
Which of the following gives the determinant of the matrix
$$\bmatrix{ 6 & 1 & -2 & 1 \cr 4 & -3 & 1 & 9 \cr 0 & 1 & 0
   & -2 \cr 1 & -7 & 8 & 1 \cr  }$$
%% CORRECT
$2 \det\bmatrix{ 6 & 1 & -2 \cr 4 & -3 & 1 \cr 1 & -7 & 8 \cr  } -
\det\bmatrix{ 6 & -2 & 1 \cr 4 & 1 & 9 \cr 1 & 8 & 1 \cr  }$
%% WRONG
$6 \det\bmatrix{ -3 & 1 & 9 \cr 1 & 0 & -2 \cr -7 & 8 & 1 \cr  }$
%% WRONG
$6 \det\bmatrix{ -3 & 1 & 9 \cr 1 & 0 & -2 \cr -7 & 8 & 1 \cr  } +
\det\bmatrix{ 4 & 1 & 9 \cr 0 & 0 & -2 \cr 1 & 8 & 1 \cr  } - 
2\det\bmatrix{ 4 & -3 & 9 \cr 0 & 1 & 0 \cr 1 & -7 & 8 \cr  }$
%% WRONG
$ \det\bmatrix{ 1 & -2 & 1 \cr -3 & 1 & 9 \cr 1 & 0 & -2 \cr  } +
4 \det\bmatrix{ 1 & -2 & 1 \cr 1 & 0 & -2 \cr -7 & 8 & 1 \cr  }$
%% WRONG
{\sl None of the above}

%% PROBLEM
Calculate the determinant of the matrix
$$\bmatrix{ 1 & 2 & 3 & 4 \cr 2 & 1 & 2 & 3 \cr 3 & 2 & 1 & 
  2 \cr 4 & 3 & 2 & 1 \cr  }$$
%% CORRECT
$-20$
%% WRONG
$-80$
%% WRONG
$40$
%% WRONG
$120$
%% WRONG
$0$


%% PROBLEM
If $c_1$, $c_2$, and $c_3$ are constants such that $B = [5,7,2]$ is a
linear combination, $B = c_1 V_1 + c_2 V_2 + c_3 V_3$, of the vectors
$V_1 = [1,0,0]$, $V_2 = [1,1,0]$, and $V_3 = [1,1,1]$, then
%% CORRECT
$c_1 = -2$
%% WRONG
$c_1 = 5$
%% WRONG
$c_1 = 0$
%% WRONG
$c_1 = 3$
%% WRONG
{\sl No such constants exist}


%% PROBLEM
Determine a maximal subset of linearly independent vectors from the set of
vectors
$$\displaylines{
\X_1 = [1,2,3,5],\qquad \X_2 = [0,0,0,0],\qquad \X_3 = [1,3,2,5],\cr
\X_4 = [2,3,7,10],\qquad \X_5 = [0,1,-1,0]\cr
}$$
%% CORRECT
$\X_1$, $\X_3$
%% WRONG
$\X_1$, $\X_3$, $\X_5$
%% WRONG
$\X_1$, $\X_3$, $\X_4$, $\X_5$
%% WRONG
$\X_1$, $\X_3$, $\X_4$
%% WRONG
$\X_1$, $\X_2$, $\X_3$


%% PROBLEM
Find an orthonormal set of vectors from the following vectors using the Gram-Schmidt method. 
$$\X_1 = [1,0,0,1],\qquad \X_2 = [1,0,1,0],\qquad \X_3 = [0,1,1,0]$$

%% CORRECT
\vtop{
\hbox{$\Q_1 = {1\over\sqrt{2}}[1,0,0,1]$}
\hbox{$\Q_2 = {1\over\sqrt{6}}[1,0,2,-1]$}
\hbox{$\Q_3 = {1\over2\sqrt{3}}[-1,3,1,1]$}
}
%% WRONG
\vtop{
\hbox{$\Q_1 = {1\over\sqrt{2}}[1,0,0,1]$}
\hbox{$\Q_2 = {1\over\sqrt{2}}[1,0,1,0]$}
\hbox{$\Q_3 = {1\over2}[1,1,-1,-1]$}
}
%% WRONG
\vtop{
\hbox{$\Q_1 = {1\over\sqrt{2}}[1,0,0,1]$}
\hbox{$\Q_2 = {1\over\sqrt{6}}[1,0,2,-1]$}
\hbox{$\Q_3 = {1\over\sqrt{3}}[-1,0,1,1]$}
}
%% WRONG
\vtop{
\hbox{$\Q_1 = {1\over\sqrt{2}}[1,0,0,1]$}
\hbox{$\Q_2 = {1\over\sqrt{7}}[1,2,-1,-1]$}
\hbox{$\Q_3 = {1\over\sqrt{7}}[1,1,2,-1]$}
}
%% WRONG
\vtop{
\hbox{$\Q_1 = {1\over\sqrt{2}}[1,0,0,1]$}
\hbox{$\Q_2 = {1\over\sqrt{6}}[1,1,2,-1]$}
\hbox{$\Q_3 = {1\over\sqrt{14}}[1,3,-2,0]$}
}



%% PROBLEM
Determine the general solution of the differential equation
$$y' - {1\over x} y = x^2$$
%% CORRECT
$y = {1\over2}x^3 + cx$
%% WRONG
$y = \ln(x) + {1\over 3}x^3 + c$
%% WRONG
$y = {1\over3}x^4 + {c\over x}$
%% WRONG
$y = e^{1/x} x^2 + c e^{1/x}$
%% WRONG
$y = {1\over3}x^3e^{-1/x} + c e^{-1/x}$




%% PROBLEM
Find the interval on which a unique solution to the following initial value
problem exists.
$$\cos(x) y' + \sin(x) y = e^x,\qquad y(0) = 1$$
%% CORRECT
$(-\pi/2, \pi/2)$
%% WRONG
$(-\pi, \pi)$
%% WRONG
$(-2\pi, 2\pi)$
%% WRONG
$(-\infty,\infty)$
%% WRONG
$(0,\pi)$



%% END
\bye
