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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 1}
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\rhead{\large November 1998 -  page   \ \thepage}
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\begin{document}


\begin{enumerate}
\item $\left[ 3,7\right) \cup \left\{ \left( -8,4\right] \cap \left(
-2,5\right) \right\} $

A) \ $\left( -2,5\right) $ \ \ \ \ \ \ \ \ B) \ $\left[ 3,5\right) $ \ \ \ \
\ \ \ \ C) \ $\left[ 3,4\right] $ \ \ \ \ \ \ \ \ \ D) \ $\left( -2,7\right) 
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$ \ \ \ \ \ \ \ \ E) none of these

\item How many of the following could not be the sine of an acute angle? \ $%
\dfrac{\sqrt{11}}{5},~e-2,~\dfrac{\pi }{4},~\dfrac{1-\sqrt{2}}{7}$

A) \ $0$ \ \ \ \ \ \ \ \ B) \ $1$ \ \ \ \ \ \ \ \ \ \ C) \ $2$ \ \ \ \ \ \ \
\ \ \ \ \ D) \ $3%
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$ \ \ \ \ \ \ \ \ \ \ \ E) \ $4$

\item The sum of the squares of three consecutive positive is  $21\,170$. \
What is the sum of the cubes of those three consecutive positive integers?

A) \ $332\,064$ \ \ \ \ \ \ B) \ $1\,778\,616%
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$ \ \ \ \ \ \ \ \ C) \ $2\,985\,984$ \ \ \ \ \ \ \ \ D) \ $3\,080\,217$\ \ \
\ \ \ \ \ \ E) \ $16\,003\,008$

\item An airplane flies a roundtrip distance of $L$ miles each way. \ The
average speed on the outward leg is $160$ miles per hour, the average speed
on the return leg is $240$ miles per hour. \ Find the average speed for the
entire trip.

A) \ $192$ mph$%
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$\ \ \ \ \ \ \ \ B) \ $196$ mph\ \ \ \ \ \ \ \ \ C) \ $200$ mph\ \ \ \ \ \ \
\ \ D) \ $204$ mph \ \ \ \ \ \ \ \ E) \ $208$ mph

\item Find the reflection (or mirror image) of the point $\left( -3,0\right) 
$ about the line $y=3x-1$.

A) \ $\left( 3,0\right) $\ \ \ \ \ \ \ \ B) \ $\left( 3,-3\right) $ \ \ \ \
\ \ \ \ C) \ $\left( 3,-\sqrt{2}\right) $\ \ \ \ \ \ \ \ \ D) \ $\left(
3,-1\right) $ \ \ \ \ \ \ \ \ E) \ $\left( 3,-2\right) 
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$

\item An equilateral triangle is inscribed in a circle. \ What percentage of
the area of the circle is covered by the triangle? \ Round to the nearest
whole percent.

A) \ $59\%$ \ \ \ \ \ \ \ \ \ B) \ $53\%$ \ \ \ \ \ \ \ \ \ \ C) \ $44\%$ \
\ \ \ \ \ \ \ \ \ \ \ D) \ $41\%%
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$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $38\%$

\item Lines $L_{1}$ \ and \ $L_{2}$ \ each pass through $\left( 3,4\right) ,$
but the slope of $L_{1}$ \ is one more than the slope of $L_{2}$. \ $L_{2}$
\ crosses the $y-$axis $k$ units above the point where $L_{1}$ crosses the $%
y-$axis. \ Find $k$.

A) \ $1$ \ \ \ \ \ B) \ $2$ \ \ \ \ \ C) \ $3%
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$\ \ \ \ \ D) \ $4$ \ \ \ \ \ \ \ E) \ There is not enough information given
to determine $k$.

\item How many positive numbers are there such that \ $x^{\left(
x^{x}\right) }=\left( x^{x}\right) ^{x}$

A) $1$ \ \ \ \ \ \ \ B) \ $2%
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$ \ \ \ \ \ \ C) \ $3$\ \ \ \ \ \ \ \ \ D) \ more than $3,$ but finitely
many \ \ \ \ \ \ \ \ \ E) \ infinitely many

\item What is the reminder when $3^{3333}$ \ is divided by $5$?

A) \ $0$ \ \ \ \ \ \ \ \ \ B) \ $1$ \ \ \ \ \ \ \ \ \ \ C) \ $2$ \ \ \ \ \ \
\ \ \ \ \ \ D) \ $3%
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$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $4$

\item How many of the following intervals are subsets of the domain of $%
f\left( x\right) =\dfrac{\sqrt{3x^{2}-7x-6}}{x-4}$ ?

$\left( -\infty ,-1\right) ,~~~\left( -7,0\right) ,~~~\left[ \pi ,4\right)
,~~~\left[ 3,10\right) ,~~~\left( 8,\infty \right) $

A) \ $1$ \ \ \ \ \ \ \ \ \ B) \ $2$\ \ \ \ \ \ \ \ \ \ C) \ $3%
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$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $4$ \ \ \ \ \ \ \ \ \ \ E) \ $5$

\item If $\theta $ is an acute angle with $\sin 3\theta =\cos 2\theta $, \
find $\sin 5\theta $.

A) \ $\dfrac{\sqrt{5}-1}{4}$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{\sqrt{2}}{2}$ \
\ \ \ \ \ \ \ \ \ C) \ $\dfrac{\sqrt{3}}{2}$\ \ \ \ \ \ \ \ \ \ \ \ D) \ $1%
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$ \ \ \ \ \ \ \ \ \ \ E) \ none of these

\item A square has sides of $10$ inches each. \ A second square is inscribed
in the original square by connecting the midpoints of the sides. \ A third
square is then formed by connecting the midpoints of the sides of the second
squar. \ This process is continued endlessly. \ What is the sum of the areas
of the infinite sequence of squares?

A) \ $150$ sq in\ \ \ \ \ B) \ $200$ sq in$%
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$\ \ \ \ C) \ $150+50\sqrt{2}$ sq in\ \ \ \ \ D) \ $300$ sq in \ \ \ \ E) \
The sum is infinite.

\item How many of the following equations have infinitely many solutions?

I. \ $\cot ^{2}x=\left( \cot x\right) ^{2}$ \ \ \ \ \ \ \ \ \ \ II. \ $%
\left( \sin x\right) ^{-1}=\csc x$ \ \ \ \ \ \ \ \ \ \ \ III. \ $\cos
^{-1}x=\sec x$

IV. \ $\sin ^{-1}x=\left( \sin x\right) ^{-1}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ V. \ $\tan ^{-1}x=\cot x$

A) \ $1$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $2$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $3%
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$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $4$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $%
5$

\item Which of the following is the best graph of $2^{x+y}=3^{x+2y}$?

\ \ \ \ \ \ A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
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\item Let $y_{1}=f\left( x\right) =\dfrac{x+1}{x-1}$; \ $y_{2}=f\left(
y_{1}\right) $; \ $y_{3}=f\left( y_{2}\right) $; \ $y_{n+1}=f\left(
y_{n}\right) $ \ \ for $n=1,2,3,4,...$ \ \ Find \ $y_{100}$.

A) \ $x%
%TCIMACRO{\TeXButton{correct answer}{\color{black}}}%
%BeginExpansion
\color{black}%
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$ \ \ \ \ \ \ \ \ B) \ $\dfrac{x+1}{x-1}$\ \ \ \ \ \ \ \ \ C) \ $\dfrac{x+100%
}{x-100}$ \ \ \ \ \ \ \ D) \ $\dfrac{100x+1}{100x-1}$ \ \ \ \ \ \ \ \ \ E) \
none of these

\item Ten writers covering a basketball league vote for the Most Valuable
Player of the league by listing their top three choices. \ A first place
vote earns five points, a second place vote earns three points, and a third
place vote earns one point. Julie Jubilee received $37$ points. \ How many
writers listed her on their ballots?

A) \ $7$ \ \ \ \ \ B) \ $8$ \ \ \ \ \ C) \ $9%
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\color{black}%
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$\ \ \ \ \ D) \ $10$ \ \ \ \ \ \ \ E) \ There is not enough information to
determine this.

\item A circle is inscribed inside the parabola $y=x^{2}$ \ with points of
tangency at $x=\pm 1$. \ The slope of the line tangent to both the parabola
and the circle at the point $\left( 1,1\right) $ is $2$. \ Find the $y-$%
coordinate of the point of the circle nearest the $x-$axis.

A) \ $\dfrac{3-\sqrt{5}}{2}%
%TCIMACRO{\TeXButton{correct answer}{\color{black}}}%
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$ \ \ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{2}{5}$\ \ \ \ \ \ \ \ \ \ \ \ C) \ $%
\dfrac{\sqrt{3}}{4}$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{\sqrt{5}}{5}$ \ \
\ \ \ \ \ \ \ \ \ \ E) \ none of these

\item A chord in a unit circle cuts off a region of area one. \ What is the
measure to the nearest degree of the central angle which subtends the chord?

A) \ $135^{\circ }$ \ \ \ \ \ \ \ \ B) \ $139^{\circ }$ \ \ \ \ \ \ C) \ $%
142^{\circ }$ \ \ \ \ \ \ \ \ \ D) \ $146^{\circ }%
%TCIMACRO{\TeXButton{correct answer}{\color{black}}}%
%BeginExpansion
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%EndExpansion
$\ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $150^{\circ }$

\item Let  $f\left( x\right) =a\sqrt{x}+\dfrac{b}{\sqrt{x}}$, \ where $%
a,b,x>0$. \ Find the range of $f$.

A) \ $\left( a+b,\infty \right) $ \ \ \ \ \ \ \ \ \ \ \ B) \ $\left[
2ab,\infty \right) $ \ \ \ \ \ \ \ \ \ \ C) \ $\left[ \sqrt{a}+\sqrt{b}%
,\infty \right) $ \ \ \ \ \ \ \ \ \ D) \ $\left[ a^{2}+b^{2},\infty \right) $

E) $\ \left[ 2\sqrt{ab},\infty \right) 
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$

\item After wearing a pair of socks a dozen times, what is the probability
to the nearest hundrtedth that each sock has been on each foot exactly a
half dozen times?

A) \ $0.23%
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$\ \ \ \ \ \ \ \ B) \ $0.25$\ \ \ \ \ \ \ \ C) \ $0.27$ \ \ \ \ \ \ \ \ D) \ 
$0.37$\ \ \ \ \ \ \ \ E) \ $0.50$\pagebreak 
\end{enumerate}

Answers

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

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