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%TCIDATA{Created=Wednesday, July 12, 2006 00:27:03}
%TCIDATA{LastRevised=Saturday, February 23, 2013 10:15:25}
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%TCIDATA{<META NAME="Title" CONTENT="AMATYC - March 1986">}
%TCIDATA{<META NAME="DocumentShell" CONTENT="Scientific Notebook\Booklet #1 - with Instructions">}
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\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
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\lhead{\large Exam 3 - Answers}
\cfoot{}
\chead{}
\rhead{\large March 1986 }
\textwidth 7.4in
\textheight 9.6in 
\setlength{\headheight}{20pt}

\begin{document}


\begin{enumerate}
\item If $f\left( x\right) =\dfrac{2x-3}{5},$ \ then \ $f^{-1}\left(
x\right) =$ ?

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A) \ $\dfrac{3x+2}{5}$

B) \ $5\left( 3x+2\right) $

C) \ $5\left( \dfrac{x}{2}+3\right) $

D) \ $\dfrac{5x+3}{2}%
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\color{black}%
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$

E) \ none of these%
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\item The volume of a sphere of radius $6\unit{cm}$ is:

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A) \ $144\pi \unit{cm}^{3}$

B) \ $288\pi \unit{cm}^{3}%
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$

C) \ $216\pi \unit{cm}^{3}$

D) \ $864\pi \unit{cm}^{3}$

E) \ none of these%
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\item Which of the following numbers is the greatest?

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A) \ $17_{\text{ten}}$

B) \ $1110_{\text{two}}$

C) \ $102_{\text{four}}%
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$

D) \ $24_{\text{five}}$

E) \ $21_{\text{eight}}$ 
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\item Suppose that $m$ varies directly as the square of $p$ and inversely as 
$q$. \ Suppose also that $m=8$ when $p=2$ and $q=6.$ \ Find $m$ if $p=6$ and 
$q=10$.

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A) \ $43.2%
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$

B) \ $120$

C) \ $14.4$

D) \ $20.4$

E) \ none of these%
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\item Given that $\left( x,y\right) $ satisfies $x^{2}+y^{2}\leq 49$, what
is the probability that $\left( x,y\right) $ also satisfies%
\begin{equation*}
\left\{ 
\begin{array}{c}
x\geq 1 \\ 
0<y\leq 5 \\ 
5x+3y\leq 30%
\end{array}%
\right. 
\end{equation*}%
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A) \ $\dfrac{1}{6}$

B) \ $\dfrac{1}{2\pi }$

C) \ $\dfrac{40}{95\pi }$

D) \ $\dfrac{20}{49\pi }$

E) \ none of these%
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\item Rationalize the denominator and simplify: \ $\dfrac{4}{1+\sqrt[3]{5}}$

A) \ $-1-\sqrt[3]{5}-\sqrt[3]{25}$ \ \qquad B) \ $\dfrac{2\left( 1-\sqrt[3]{5%
}+\sqrt[3]{25}\right) }{3}%
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\qquad $C) \ $\dfrac{2\sqrt[3]{5}}{3}$ \qquad D) \ $-1+\sqrt[3]{5}$ \qquad
E) \ none of these

\item Solve $\left\vert 2x-1\right\vert +\left\vert x\right\vert =3$. \ The
sum of the solutions is:

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A) \ $\dfrac{2}{3}%
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$

B) \ $\dfrac{4}{3}$

C) \ $2$

D) \ $\dfrac{8}{3}$

E) \ none of these%
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\item A farmer bought 100 head of animals: horses, sheep, and pigs. \ He
bought at least one of each animal and spent exactly $\$100$. \ Horses,
sheep, and pigs cost $\$10,$ $\$3,$ and $\$0.50$ each, respectively. \ How
many horses did he buy?

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A) \ $2$

B) \ $5%
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$

C) \ $7$

D) \ $8$

E) \ none of these%
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\item If $x+y=1,$ then the largest value of $xy$ is:

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A) \ $1$

B) \ $0.5$

C) \ $\dfrac{\sqrt{5}}{5}$ 

D) \ $\dfrac{\sqrt{3}}{3}$ \ 

E) \ none of these$%
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$ 
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\item Solve \ $\tan ^{2}x\sec ^{2}x-5\tan ^{2}x+3=0$ \ for $0\leq x\leq 2\pi 
$. \ The number of solutions is:\medskip 

A) \ $2\qquad \qquad \qquad \ \ \ $B) \ $4\qquad \ \ \ ~\qquad \qquad $C) \ $%
6\qquad \ \ \ \ \qquad \qquad $D) \ $8%
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$ \qquad\ \ \qquad \qquad E) \ none of these\pagebreak 

\item A function $y=f\left( x\right) $ \ is described by the graph below:%
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1;num-x-ticks 5;constrained TRUE;numpoints 100;plotstyle "patch";axesstyle
"normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name
\TEXUX{$y$};function \TEXUX{$\sqrt{1-x^{2}}$};linecolor "black";linestyle
1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range
"-1,1";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle
"Line";rangeset"X";valid_file "T";tempfilename
'MIBSZV06.wmf';tempfile-properties "XPR";}}which of the following represents
the graph of the translation $y-2=f\left( x+1\right) $?

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\ \ A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C)$%
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$

\FRAME{itbpFX}{1.2047in}{1.2047in}{0in}{}{}{Plot}{\special{language
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"normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name
\TEXUX{$y$};function \TEXUX{$-1-\sqrt{1-\left( x-1\right) ^{2}}$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
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"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file
"T";tempfilename 'MIBSZX07.wmf';tempfile-properties "XPR";}} \ \ \ 

\ \ \ \ \ \ \ B)

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1;num-x-ticks 5;constrained TRUE;numpoints 100;plotstyle "patch";axesstyle
"normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name
\TEXUX{$y$};function \TEXUX{$\dfrac{1}{2}+\sqrt{1-x^{2}}$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "-1,1";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file
"T";tempfilename 'MIBSZX08.wmf';tempfile-properties "XPR";}} \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \FRAME{itbpFX}{1.2047in}{1.2047in}{0in}{}{}{Plot}{\special%
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\TEXUX{$\dfrac{1}{2}+\sqrt{1-\left( x+1\right) ^{2}}$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes
"Solid";var1range "-2,2";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file
"T";tempfilename 'MIBSZX09.wmf';tempfile-properties "XPR";}}\ 

D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E)

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1;num-x-ticks 5;constrained TRUE;numpoints 100;plotstyle "patch";axesstyle
"normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name
\TEXUX{$y$};function \TEXUX{$\dfrac{3}{2}-\sqrt{1-\left( x-1\right)
^{2}}$};linecolor "black";linestyle 1;pointstyle "point";linethickness
3;lineAttributes "Solid";var1range "-1,2";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file
"T";tempfilename 'MIBSZY0A.wmf';tempfile-properties "XPR";}}\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \FRAME{itbpFX}{1.2047in}{1.2047in}{0in%
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"T";tempfilename 'MIBSZY0B.wmf';tempfile-properties "XPR";}}

\item Solve $\ \log \left( 3+x\right) =\log 3+\log x$

A) \ $\dfrac{2}{9}$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{1}{333}$ \ \ \ \ \ \ \ \
\ \ C) \ $\dfrac{1}{1000}$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $1000$ \ \ \ \ \ \ \
\ \ \ \ \ E) none of these$%
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$

\item Bob is as old as Joe was when Bob was as old as Joe had been when Bob
was half as old as Joe is. \ Their combined age is $44.$ \ How old is Joe?

A) \ $24%
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$ \ \ \ \ \ \ \ \ \ B) \ $28$ \ \ \ \ \ \ \ \ \ \ C) \ $34$ \ \ \ \ \ \ \ \
\ \ \ \ D) \ $36$ \ \ \ \ \ \ \ \ \ \ \ \ E) none of these

\item From a two-digit number $N$ we subtract the number with the digits
reversed and find that the result is a positive perfect cube. \ Then:

A) \ $N$ cannot end in $5$ \ \ \ \ \ B) \ $N$ can not end in any digit other
than $9$ \ \ \ \ \ \ \ C) \ $N$ does not exist\ \ \ \ \ \ D) \ There are
exactly seven values for $N%
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$\ \ \ \ \ \ E) \ None of these.

\item Find the limiting sum of \ $\dfrac{1}{3\cdot 6}+\dfrac{1}{6\cdot 9}+%
\dfrac{1}{9\cdot 12}+.....$

A) \ $\dfrac{5}{54}$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{1}{9}%
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$ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{7}{54}$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $%
\dfrac{1}{7}$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these

\item Three \ marksmen simultaneously shoot at and hit a rapidly spinning
spherical target. \ What is the probability that the three points of impact
are on the same hemisphere?

A) \ $\dfrac{3}{4\pi }$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{2}{3\pi }$ \ \ \ \ \
\ \ \ \ \ C) \ $\dfrac{6}{\pi ^{2}}$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{1}{%
4}$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$%
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$
\end{enumerate}

\end{document}