\input ignoremce
\preface
\input epsf
\input calc
\def\problemskip{\vskip0.5in}
\def\ds{\displaystyle}

\title
\noindent
{\bf Math 226: Calculus IV}\hfill
Sign your Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam III}\quad{\sl April 19, 1996}\hfill
Print your Name:\hbox to 2.5truein{\hrulefill}\medskip\hfill
Section:\hbox to 2.5truein{\hrulefill}\medskip\hfill
TA:\hbox to 2.5truein{\hrulefill}
\bigskip\noindent
Do not turn this page until you are told to begin.
\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 16 multiple
choice questions worth 6 points each. You start with 10 points, and the highest 
possible score is 106.  Fill in the answers as you go along.  You will not be 
allowed to fill in the answers after the time is up.
\bigskip\noindent
You may use a calculator, but only the standard functions found on very inexpensive 
scientific calculators.  In particular you may not use graphing, integration, formula or 
program capabilities.  Doing so is a violation of the Honor Code.
\bigskip\noindent
You are required to hand in the answer sheet and the problems.  You may tear off the 
blank sheets at the back for scratch work. {\bf Please tear off these scratch sheets
gently, one page at a time.}

\def\answerskip{\vskip0.25in}

\note %1
\problem
 Find the exponents of the singularity . 

\correct
$2\ and -1$
\wrong
$\root\of{2}$\quad  and - $\root\of{2}$ 
\wrong
$2 and 1$ 
\wrong
$0 and 0$
\wrong
$3 and -1$




\note %2
\problem
The equation $x^2 y{\prime\prime}\ +\ (x^3 + x)y\prime - 4 y = 0$\ has a regular singular point at  $x = 0.$  For which values of $\gamma$ is there
 sure to be a series solution of the form

$$y (x) = |x|^\gamma\ \sum^{\infty}_{n=0} a_nx^n\ , \hbox{ with}\ a \ 0 \neq 0?$$   
\correct
 $2$ 
\wrong
 $2  and -2$ 
\wrong
$-{1\over 2}\ \pm\ \  \root\of {17\over2}$ 
\wrong
$1 and -4$
\wrong
1 



\note %3
\problem
The equation $x^2 y {\prime\prime} - x (x+3) y\prime + (x +3)\ y = 0$ has a regular singular point at $x = 0$ and the exponents of the singularity are 3 and 1.  The recurrence relation is
\smallskip
$$a_n = {(n + \gamma -2) a_n - 1\over (n + \gamma - 1)(n + \gamma - 3)}\ ,\ \ for \ m\geq1.$$
\smallskip
Find the first three nonzero terms of a series solution near $x = 0$, with $a_0 = 1.$

\correct
$x^3 (1 + {2x\over3}\ +\ {x^2 \over4} \ +  \ldots)$
\wrong
${2x4\over3} \ +\ {x^5 \over 4} \ +\ {s^6 \over15}\ +\ldots$
\wrong
$1 + {2x\over 3} +\ {3x^2 \over4}\ +\ldots.$
\wrong
$1 + {x\over 3} \ +\ {x^3 \over15}\ +\ldots$
\wrong
$x + {2x^2 \over 3}\ +\ {x^3 \over 4}\ +\ldots$



\note %4
\problem
The equation $2 x^2 \ y{\prime\prime}\ +\ 3 xy\prime\ -\ (1 + x)\ y = 0$ has a regular singular point at $x = 0$, with indicial equation $(\gamma + 1) (2 \gamma - 1) = 0.$  Find the recurrence relation for $n \geq 1$  for the coefficients of the series solution corresponding to $\gamma = {1\over2}$

\correct
$a_n = \ {a_n - 1\over2n^2 + 3 n}$
\wrong
$a_n = \ {a_n - 2\over n^2 + 3^n}$
\wrong
$a_n = \ {a_n - 1\over 2n + 1}$
\wrong
$a_n = \ {a_n - 1\over n (n + 3)}$
\wrong
$a_n = \ {a_n - 2\over xn (n + 1)}$


\note %5
\problem
For what values of c and d is the following system consistent?
$$\matrix{x_1  & + &x_2  & + &x_3   &   &    &= 2\cr
               &   &x_2  &   &      & + &x_4 &= 1\cr
          2x_1 & + &3x_2 & + &2x_3  & + &x_4 &= c\cr
           x_1 &   &     & + &x_3   & - &x_4 &= d\cr}$$ 

\correct
$c = 5, d = 1.$
\wrong
Consistent for every c and d.
\wrong
$c = 4, d = 2$
\wrong
$c = 0, d = 3$
\wrong
$c = \alpha$, $d = 1 - \alpha$ for $\alpha$ in $\Re$.


\note %6
\problem
Let W be the subspace of $\Re^5 $ spanned by the vectors (1, 0, 1, 0, 1), (1, 1, 1, 1, 1), (1,2, 1, 2, 1)  and (2,3, 2, 3, 2)  Find the dimension of W.

\correct
$2$
\wrong
$1$
\wrong
$3$
\wrong
$4$
\wrong
$5$


\note %7
\problem
Let A be a 7x11 matrix.  Suppose upon reduction to row echelen form, one obtains exactly 5 nonzero rows.  What is the dimension of the column space of A?

\correct
$5$
\wrong
$6$
\wrong
$2$
\wrong
$1$
\wrong
$4$


\note %8
\problem
Let W be the set of solutions of A x = 0
$$A = \left[\matrix{1&0&1&-2&0\cr0&1&-3&0&2\cr0&0&0&0&1\cr}\right]$$
Which of the folowing is a basis for W?

\correct
$(-1, 3, 1, 0, 0) and (2, 0, 0, 1, 0)$
\wrong
$(1, 0, 1, -2, 0) and (0, 1, -3, 0, 2) and (0, 0, 0, 0, 1)$
\wrong
$(s, 3, 1, 1, 0)$
\wrong
$(1, -3, 0) and  (0, 2, 1)$
\wrong
$(1, -3, 1, 2, 0) and (2, 0, 0, 0, 1)$


\note %9
\problem
Find the inverse of the matrix
\smallskip
$$A = \left[\matrix{1 &1&0&\cr -1&0&0\cr-1&-1&1\cr}\right]$$
\smallskip 
It's first row is 

\correct
$(0, -1, 0)$.
\wrong
$(2, 0, 1 )$.
\wrong
It has no inverse.
\wrong
$(0, 1, 0)$.
\wrong
$(-1, -1, 1)$.


\note %10
\problem
For $A = \left[\matrix{2&1\cr3&0\cr1&-3 \cr}\right]$ and $B = \left[\matrix{2& 3\cr-1&4\cr}\right]$, find the last row of A B.

\correct
$(5, -9)$
\wrong
The product is not defined
\wrong
$(4, 6)$ 
\wrong
$(0, 8)$
\wrong
$(3, -12)$
 


\note %11
\problem
Let A x = 0 be an m x n system of honogeneous equations of rank $\gamma$.  Consider the following statements
\smallskip
A.  There is always at least one solution.
\smallskip
B.  If $\gamma < n$, there are infinitely many solutions
\smallskip 
C.  If $\gamma < m$, there are infinitely many soutions.
\smallskip
D.  If $\gamma = n$, there is exactly one solution.
\smallskip
E.  If $\gamma = m$, there is exactly one solution.


\correct
Only A, B, C and D are true 
\wrong
Only A is true
\wrong
Only A and B are true
\wrong
Only A, B and C are true
\wrong
All are true
 

\note %12
\problem
Let A = $\left[\matrix{2&0&3\cr1&-4&5}\right]$\ \ Find the middle row of $ A^t A.$

\correct
$(-4, 16, -20)$
\wrong
$A^t A$ is not defined.
\wrong
$(5, -4, 7)$
\wrong
$(10, -20, 34)$
\wrong
$(11, 16, 8)$


\note %13
\problem
If $A ^{-1} = \left[\matrix{1&0&1\cr0&1&0\cr1&0&2}\right]$ and  $B^{-1} = \left[\matrix{0&1&0\cr1&0&1\cr0&1&1}\right]$, find the bottom row of $(A B)^{-1}$ 

\correct
$(1, 1, 2)$
\wrong
$(0, 3, 2)$
\wrong
A B is not invertible
\wrong
$(2, 0, 3)$
\wrong
$(1, 2, 1)$


\note %14
\problem
Transform the following matrix into reduced echelon form.
$$\left[\matrix{1& 0&1&1\cr0&1&2&0\cr0&1&2&1\cr1&1&3&1}\right]$$

\correct
$\left[\matrix{1&0&1&0\cr0&1&2&1\cr0&0&0&1\cr0&0&0&0}\right]$
\wrong
$\left[\matrix{1&0&1&1\cr0&1&2&0\cr0&1&2&1\cr1&1&3&1}\right]$
\wrong
$\left[\matrix{1&0&1&1\cr0&1&2&0\cr0&1&2&1\cr0&1&2&0}\right]$
\wrong
$\left[\matrix{1&0&0&0\cr0&1&0&0\cr0&0&1&0\cr0&0&0&1}\right]$
\wrong
$\left[\matrix{1&0&0&1\cr0&0&1&2\cr0&0&0&0\cr0&0&0&0}\right]$



\note %15
\problem
Which of the subsets
\smallskip
$W_1 = \{(x_1, x_2, 0)  :  x_1^2 + x_2 ^2 = 1\} $
\smallskip
$W_2 = \{(x_1, x_2, x_3)  :  3x_1 + 4x_2 - 5x_3 = 0\} $
\smallskip
$W_3 = \{(x_1, x_2,x_3)  :  x_1 = 0 \gamma\  x_2 = 0\} $
\smallskip

are subspaces of $\Re^3$ ? 



\correct
Only $W_2$
\wrong
Only $W_1$
\wrong
Only $W_3$
\wrong
Only $W_1 \ and\  W_2$
\wrong
Only $W_2\ and \ W_3$


\note %16
\problem
Considered as vectors in $\cal M$ 2x2 , 
\smallskip
$$\left(\matrix{0&1\cr1&0}\right)\ , \left(\matrix{1&0\cr0&1}\right)\ , \left(\matrix{1&0\cr0&0}\right),\ \left(\matrix{1&1\cr1&k}\right)$$
\smallskip
are 
\correct
Linearly dependent for any choice of k.
\wrong
Linearly independent for any choice of k.
\wrong
Linearly dependent only if k = 1.
\wrong
Linearly independent only if k = 1.
\wrong
Linearly dependeant only for k = 0.


\end
