Math T104 Final a

Individual Part

$%Math T104 Final a %Given March 1999 % \input epsf \hsize=6.2truein \tolerance=3000 \parindent=1cm \rm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcount\qnno %\headline={\rm T104 Final a\hfil {\it Individual Part}} \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 Find a $10$ digit number which is divisible by $4$ but not divisible by $6$. \medskip \qn Suppose $a$ and $b$ satisfy the inequalities $$0<a<1\qquad\hbox{and}\qquad b>2.$$ Decide whether the following statements are true or false. If a statement is true, explain why. If it's false, give a counter example. \medskip \qnn a. $\displaystyle{b\over a}<b$. \medskip \qnn b. $\displaystyle{a\over b}<a$. \medskip \qnn c. $\displaystyle{a\over b}+{b\over a}=1$. \medskip$

$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent \def\abx{\vbox{\hrule\hbox{\vrule height.7cm\hskip.9cm\vrule}\hrule}} \qne a. Finish filling in the multiplication table for base~$5$ arithmetic. \medskip $$ \vbox{\offinterlineskip \vbox{\halign{\strut\vbox to0.73cm{\vfil \hbox to1.1cm{\hfil\tt #\hfil}\vfil}\vrule &&\vbox to0.73cm{\vfil\hbox to1.1cm{\hfil\tt #\hfil}\vfil}\cr $\times$&1&2&3&4&10&11\cr \noalign{\hrule} %1&1&2&3&4&10&11\cr %2&2&4&11&13&20&22\cr %3&3&11&14&22&30&33\cr %4&4&13&22&31&40&44\cr %10&10&20&30&40&100&110\cr %11&11&22&33&44&110&121\cr 1&1&2&3&4&10&11\cr 2&2&4&\abx&13&\abx&22\cr 3&\abx&11&\abx&22&30&33\cr 4&4&13&\abx&31&\abx&44\cr 10&\abx&\abx&30&\abx&100&110\cr 11&11&22&33&44&110&121\cr }}} $$$

$\qn Perform the following arithmetic problems in the specified base. \smallskip \qnn b. \elem 304 {\times} 21 {\rm five} \medskip \qnn c. ${\tt 3013}_{\rm five}\ \ {\div}\ \ {\tt 11}_{\rm five}$\ \ =$

$\qne a. Find the greatest common divisor of $15$ and $50$. \medskip \qnn b. Find the least common multiple of $14$ and $35$.$

$\qne a. Calculate $3\times 3+5$ mod $8$. \medskip \qnn b. Calculate $3\times (3+5)$ mod $8$.$

$\qn Consider the triangle $$ \vbox{\epsfxsize=3.7truein\epsfbox{trian.eps}} $$ \smallskip \qnn a. Which angle is opposite the longest side? \medskip \qnn b. Is is possible to construct a triangle with side lengths of $4\,$cm, $5\,$cm and $10\,$cm? \medskip \qnn c. If a triangle has sides with length $a$, $b$ and $c$ where $c$ is the longest. What must be true about the lengths $a+b$? Why?$

$\qn Determine the perimeter and area of the following object. $$ \vbox{\epsfxsize=3truein\epsfbox{pick1.eps}} $$$

$\qne a. Put the numbers $$ {1\over 2},\quad {100\over 99},\quad {2\over 3},\quad {1\over 3},\quad {99\over 100} $$ in order from least to greatest. \medskip \qnn b. Put the numbers $$ -3.23, \quad -3.2\overline 3, \quad -3.\overline{23}, \quad -3.\overline{2}, \quad -3.2$$ in order from least to greatest. \medskip \qn Given $\displaystyle{x\over y}>0$, find a fraction between 0 and $\displaystyle{x\over y}$. \medskip$

Group Part

$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\headline={\rm T104 Final a\hfil {\it Group Part}} \qnno=0 \qn Find the sum of the measures of the angles $\angle 1$, $\angle 2$, $\angle 3$, $\angle 4$ and $\angle 5$ in any five pointed star. $$ \vbox{\epsfxsize=3truein\epsfbox{star.eps}} $$$

$\qne a. Convert $1/7$ to a decimal. \medskip \qnn b. You should have found that $1/7$ is a repeating decimal with $6$ digits in the repeating block. In general, if $1/n$ can be represented as a repeating decimal, then there will be no more that $n-1$ digits in the repeating block. Explain why this is true. (You might want to convert some more fractions to repeating decimals. 1/17 is a good but tedious example.) \medskip\medskip \qnn c. Convert $2/7$ and $3/7$ to decimals and compare to your decimal representation of $1/7$. What do you notice? \medskip \qnn d. Based upon your observations above and without dividing find the decimal representations for $4/7$, $5/7$ and $6/7$.$

$\qn In an attempt to sell more specialized tools, General Motors plans to manufacture strange headed screws not the usual ones like \hbox to 3em{\hfil\epsfysize=1.5em\epsfbox{stan.eps}\hfil} and \hbox to 3em{\hfil\epsfysize=1.5em\epsfbox{phil.eps}\hfil} to use in the transmission of some of their cars. When Matt made the following suggestion, his boss asked him to find the area of the shaded region. He, in turn, passed the problem on to you. $$ \vbox{\epsfxsize=2.5truein\epsfbox{screw.eps}} $$$

$\qn Marvin the Thief went to a bakery to steal some pies. After he loaded the pies in his car he thought that he should taste the pies to see if they were really as good as he had heard. He ate part of one pie which was delicious and decided to leave. On his way driving out, he was stopped by bakery guards three times. Each time, he bribed his way out by giving away one-third of all the pies he currently had plus three more pies. In the end Marvin escaped with only one pie. How many pies did Marvin steal and how much of the pie that he tasted did he eat?$