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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}
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\lhead{\large Exam 1}
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\rhead{\large November 1996 -  page   \ \thepage}
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\begin{document}


\begin{enumerate}
\item What is the ratio of the circumference ofa circle to the perimeter of
an inscribed square?

A) \ $\dfrac{\pi \sqrt{2}}{3}$ \ \ \ \ \ \ \ \ B) \ $\dfrac{\pi \sqrt{2}}{4}%
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$ \ \ \ \ \ \ \ \ C) \ $\dfrac{\pi }{2}$ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{\pi 
}{3}$ \ \ \ \ \ \ \ \ E) none of these

\item If $-200<t<-100$, \ which of the following has the greatest value?

A) \ $\sqrt{-t}%
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$ \ \ \ \ \ \ \ \ B) \ $-\sqrt[3]{t}$ \ \ \ \ \ \ \ \ \ \ C) \ $-\sin
^{-1}\left( \sin t\right) $ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $t^{3}$ \ \ \ \ \ \
\ \ \ \ \ E) \ $\dfrac{9t-1}{t}$

\item What is the domain of the identity $\log \left( x+3\right) +\log
\left( x-1\right) =\log \left( x^{2}+2x-3\right) $?

A) \ $\left( -3,1\right) $\ \ \ \ \ \ \ B) \ $\left( -\infty ,\infty \right) 
$ \ \ \ \ \ \ \ \ C) \ $\left( -3,\infty \right) $ \ \ \ \ \ \ \ \ \ D) \ $%
\left( -\infty ,-3\right) \cup \left( 1,\infty \right) $\ \ \ \ \ \ \ \ E) \ 
$\left( 1,\infty \right) 
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$

\item The number of positive integers less than $1000$ divisible by neither $%
5$ nor $7$ is

A) \ $630$\ \ \ \ \ \ \ B) \ $658$ \ \ \ \ \ \ \ \ C) \ $686%
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$ \ \ \ \ \ \ \ \ \ D) \ $688$\ \ \ \ \ \ \ \ E) \ $690$

\item Going into the last game of the season, a basketball team had averaged 
$83.8$ points per game. \ In their last game, they scored $97$ points,
raising their season average to $84.2$ points per game. \ How many games did
they play?

A) \ $27$\ \ \ \ \ \ \ \ \ \ B) \ $31$ \ \ \ \ \ \ \ \ \ \ C) \ $32$\ \ \ \
\ \ \ \ \ \ \ D) \ $33%
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$ \ \ \ \ \ \ \ \ E) \ none of these

\item For how many values of $k$ \ does the graph of $y=3x^{2}+kx+7$ have
its vertex on the $x-$axis?

A) \ $0$ \ \ \ \ \ \ \ \ \ B) \ $1$ \ \ \ \ \ \ \ \ \ \ C) \ $2%
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$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $4$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ infinitely
many

\item What is the negation of \ "Every good boy does fine"?

A) \ Some good boys do fine.\ \ \ \ \ \ \ \ \ \ B) \ Those who do not do
fine are not good boys.\ \ \ \ \ \ \ \ \ 

C) Some who do fine are not good boys.\ \ \ \ \ D) \ There is a good boy who
does \ not do fine.$%
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$ \ \ \ \ \ 

E) \ Everyone who does fine is a good boy.

Questions 8-10 refer to the functions $f$ and $g,$ whose graphs are shown
below and whose common domain is $\left[ -6,6\right] $.\FRAME{dtbpF}{5.2408in%
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\item The (approximate) value of $f\left( g\left( 2\right) \right) $ \ is \ 

A) \ $-0.9$ \ \ \ \ \ \ \ \ \ B) \ $-2.0$ \ \ \ \ \ \ \ \ \ \ C) \ $0.0$ \ \
\ \ \ \ \ \ \ \ \ \ D) \ $0.5%
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$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $1.7$

\item How many solutions does the equation $f\left( x\right) +g\left(
x\right) =0$ \ have?

A) \ $0$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $1$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $2%
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$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $3$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \
more than $3$

\item The range of $f\left( g\left( x\right) \right) $ \ is (approximately)

A) \ $\left[ 0,1\right] 
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$ \ \ \ \ \ \ \ \ \ B) \ $\left[ 0,2\right] $ \ \ \ \ \ \ \ \ \ \ C) \ $%
\left[ -4,2\right] $ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\left[ 0,5\right] $ \ \ \
\ \ \ \ \ \ \ \ \ E) \ $\left[ -4,5\right] $

\item Given $\sin A=M,$ \ $\cos T=Y,$ \ \ $M+Y=C,$ \ $A+T=\dfrac{\pi }{2},$
\ $A+4M=\dfrac{3\pi }{2},$ \ find $A+M+A+T+Y+C$.

A) \ $0$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{\pi }{2}$ \ \ \ \ \ \ \ \ \ \ C) \ $%
\pi $ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $2\pi 
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$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $4\pi $

\item Which of these numbers is the greatest? \ (The subscript indicates the
base.)

A) \ $0.10_{\text{two}}$ \ \ \ \ \ \ \ \ \ B) \ $0.12_{\text{three}}%
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$ \ \ \ \ \ \ \ \ \ \ C) \ $0.21_{\text{five}}$ \ \ \ \ \ \ \ \ \ \ D) \ $%
0.42_{\text{nine}}$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $0.53_{\text{twelve}}$

\item Approximately what percentage of all families with four children have
an equal gender split (two boys and two girls)?

A) \ $33.3\%$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $37.5\%%
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$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $50.0\%$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ 
$62.5\%$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $66.7\%$

\item How many real numbers are there such that the $5$th power of the
number is the sum of the $4$th and $3$rd powers of the number?

A) \ $1$ \ \ \ \ \ \ \ \ \ B) \ $2$ \ \ \ \ \ \ \ \ \ \ C) \ $3%
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$ \ \ \ \ \ \ \ \ \ \ D) \ $5$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these

\item Given $2^{x}=8^{y+1}$ \ and \ $9^{y}=3^{x-9},$ then the value of $x+y$
\ is

A) \ $9$ \ \ \ \ \ \ \ \ \ B) \ $18$ \ \ \ \ \ \ \ \ \ \ C) \ $24$ \ \ \ \ \
\ \ \ \ \ D) \ $27%
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$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $30$

\item How many of the following could be the intersection of a plane and the
surface of a cube: empty set, line segment, triangle, quadrilateral,
pentagon, hexagon?

A) \ $2$ \ \ \ \ \ \ \ \ \ B) \ $3$ \ \ \ \ \ \ \ \ \ \ C) \ $4$ \ \ \ \ \ \
\ \ \ \ D) \ $5$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $6%
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$

\item If $x$ and $y$ are two real numbers such that $xy$, \ $\dfrac{x}{y}$,
\ and \ $x-y$ \ are all equal, \ then $x+y=$

A) \ $-\dfrac{3}{2}%
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$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $-\dfrac{1}{2}$ \ \ \ \ \ \ \ \ \ \ \ \ \ C)
\ $0$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{1}{2}$ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ E) \ $\dfrac{3}{2}$

\item A circle has center $\left( 6,7\right) $. \ Find the area of the
triangle formed by the coordinate axes and the tangent line to the circle at
the point $\left( 2,5\right) $ \ on the circle.

A) \ $\dfrac{121}{6}$ \ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{81}{4}%
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$ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{169}{8}$ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{%
49}{2}$\ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these

\item Two hikers are $\dfrac{3}{5}$ of the way through a mile-long tunnel
when they hear a train approaching from ahead. \ In a panic, they take off
running, but in opposite directions. One heads for the near end of the
tunnel, averaging $13$ mph, while the other heads back to the far end,
averaging $11$ mph. \ The train, traveling at a constant speed, roars by
each hiker just as he or she escapes from the tunnel. What was the speed of
the train (nearest mph)?

A) \ $18$ mph \ \ \ \ \ B) \ $42$ mph \ \ \ \ \ C) \ $55$ mph \ \ \ \ \ D) \ 
$67$ mph$%
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$\ \ \ \ \ \ E) $80$ mph

\item A statue is $50~\unit{ft}$ high and it sits on a pedestal which is $25~%
\unit{ft}$ high. \ \ How far back (along level ground) should an observer,
whose eyes are $5~\unit{ft}$ above ground level, stand in order to have
equal angles made at her eyes by the statue and by the pedestal on which it
sits?

A) \ $25~\unit{ft}$ \ \ \ \ \ \ \ \ \ \ \ B) \ $35~\unit{ft}$ \ \ \ \ \ \ \
\ \ \ \ \ \ C) \ $40~\unit{ft}%
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$ \ \ \ \ \ \ \ \ \ \ \ D) \ $50~\unit{ft}$\ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $%
75~\unit{ft}$\pagebreak
\end{enumerate}

Answers - November 1996

\bigskip

\begin{tabular}{lllll}
1. & B &  & 11. & D \\ 
2. & A &  & 12. & B \\ 
3. & E &  & 13. & B \\ 
4. & C &  & 14. & C \\ 
5. & D &  & 15. & D \\ 
6. & C &  & 16. & E \\ 
7. & D &  & 17. & A \\ 
8. & D &  & 18. & B \\ 
9. & C &  & 19. & B \\ 
10. & A &  & 20. & C%
\end{tabular}

\end{document}