Math T104 Exam 1a

Individual Part

$%Math T104 Exam 1a %Given February 1999 % \hsize=6.2truein \input epsf \tolerance=3000 \parindent=0.8cm \rm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcount\qnno \def\qn {\noindent\strut\global\advance\qnno by1% \hbox to\parindent{\the\qnno.\hfil}\hangindent\parindent\ignorespaces} \def\qnn#1 {\noindent\strut \hbox to\parindent{\phantom{\the\qnno. }#1\hfil}\hangindent\parindent\ignorespaces} \def\qne#1 {\noindent\strut\global\advance\qnno by1% \hbox to\parindent{\the\qnno. #1\hfil }\hangindent\parindent\ignorespaces} \def\elem#1 #2 #3 #4 { $\vcenter{\halign{\hfil##$\,$&\hfil##\cr &${\tt #1}_{#4}$\cr $#2$&${\tt #3}_{#4}$\cr \noalign{\smallskip\hrule}}}$} \qn Write the following numbers in the specified bases. \medskip \qnn a. Write ${\tt 153}_{\rm six}$ in base ten. \bigskip \qnn b. Write ${\tt 2.03}_{\rm six}$ in base ten. \bigskip \qnn c. Write ${\tt 68}_{\rm ten}$ in base six. \bigskip \qnn d. Write ${\tt 4.5}_{\rm ten}$ in base six. \bigskip$

$\noindent Addition and multiplication tables for base six are given below. \medskip \hbox to\hsize{\hfill \vbox{\offinterlineskip \vbox{\halign{\strut\vbox to0.4cm{\vfil \hbox to0.6cm{\hfil\tt #\hfil}\vfil}\vrule &&\vbox to0.4cm{\vfil\hbox to0.6cm{\hfil\tt #\hfil}\vfil}\cr +&0&1&2&3&4&5\cr \noalign{\hrule} 0&0&1&2&3&4&5\cr 1&1&2&3&4&5&10\cr 2&2&3&4&5&10&11\cr 3&3&4&5&10&11&12\cr 4&4&5&10&11&12&13\cr 5&5&10&11&12&13&14\cr }}} \hfill \vbox{\offinterlineskip \vbox{\halign{\strut\vbox to0.4cm{\vfil \hbox to0.6cm{\hfil\tt #\hfil}\vfil}\vrule &&\vbox to0.4cm{\vfil\hbox to0.6cm{\hfil\tt #\hfil}\vfil}\cr $\times$&0&1&2&3&4&5\cr \noalign{\hrule} 0&0&0&0&0&0&0\cr 1&0&1&2&3&4&5\cr 2&0&2&4&10&12&14\cr 3&0&3&10&13&20&23\cr 4&0&4&12&20&24&32\cr 5&0&5&14&23&32&41\cr }} }\hfill}$

$\medskip \qn Perform the following additions, subtractions, multiplications, and divisions working in base six. Work neatly and show all work. \bigskip \halign{\indent\hfil#&&\hbox to 1in{\hfil}\hfil#\cr \hbox{a. \elem 20352 + 11431 {\rm six} }& \hbox{b. \elem 402 {\times} 35 {\rm six} }\cr \noalign{\bigskip} \hbox{c. \elem 2514 - 341 {\rm six} }& \hbox{d. ${\tt 2053}_{\rm six}\ \ {\div}\ \ {\tt 3}_{\rm six}$\ \ = }\cr} \bigskip$

$\qn Consider the operation $\otimes$ for the set of real numbers defined by\par $$ x\otimes y={x+y\over 2}. $$ \medskip \qnn a. Calculate ${\tt 3}\otimes({\tt 9}\otimes{\tt 5})$. \bigskip \qnn b. Calculate $({\tt 3}\otimes{\tt 9})\otimes{\tt 5}$. \bigskip \qnn c. Is this operation associative? \bigskip \qnn d. Is this operation commutative? \bigskip \qnn e. Does this operation have an identity? If so, what is the identity? \bigskip$

$\noindent Mr.~Jones runs a small store that does monogramming. One day he has four customers pick up items that they wanted monogrammed. He notices that each of the letters {\tt A}, {\tt B}, {\tt C} and {\tt D} were used as a first initial, a middle initial, and a last initial. From the following information, deduce each monogram. No one has two of the same letter in their monogram. Mr.~Adams picked up his order first, and said ``Hello" to his paperboy, Charlie, who was coming in as Mr.~Adams left. Charlie teased his friend about having the middlename Bartholomew when he saw him enter the store. The boys thought it was funny seeing their principal, Mr.~Douglas, come in to pick up a bathrobe with his initials on it. {\tt ABC} was not one of the monograms. \medskip \qn What were the four monograms? \bigskip$

Group Part

$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \qnno=0 \qne a. Find the next three terms in the sequence $$ 1, 2, 4, 8, 16, 32, \ldots $$ and explain the pattern. \bigskip \qnn b. Find the next three terms in the sequence $$ 1, 3, 7, 15, 31, \ldots$$ and explain the pattern. \bigskip \qnn c. Find a simple formula for computing the sum $$ 1+2+2^2+2^3+\cdots+ 2^n.$$ \bigskip$

$\noindent \qn Convert the following numbers written in base two to base four without first converting to base ten. \medskip \qnn a. $1011_{\rm two}$ \bigskip \qnn b. $1011011_{\rm two}$ \bigskip \qnn c. $101101000_{\rm two}$ \bigskip$

$\qn Develop a strategy for converting base two to base four without converting to base ten first. Use your strategy to convert $110110100010101_{\rm two}$ to base four. \bigskip \qn Describe your strategy. \bigskip \qn Explain why your strategy works. \bigskip$