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%TCIDATA{<META NAME="Title" CONTENT="AMATYC - November 1995">}
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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}
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\lhead{\large Exam 1}
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\rhead{\large November 1995 -  page   \ \thepage}
\textwidth 6.8in
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\begin{document}


\begin{enumerate}
\item The domain of $f\left( x\right) =\sqrt{\dfrac{x}{25-x^{2}}}$ is

A) \ $\left[ 0,5\right) $ \ \ \ \ \ \ \ \ B) \ $\left( -5,5\right) $ \ \ \ \
\ \ \ \ C) \ $\left( -\infty ,5\right) $ \ \ \ \ \ \ \ \ \ D) \ $\left[
0,\infty \right) $ \ \ \ \ \ \ \ \ E) \ $\left( -\infty ,-5\right) \cup %
\left[ 0,5\right) 
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$

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

A) \ $-\dfrac{5}{3}$ \ \ \ \ \ \ \ \ B) \ $-\dfrac{5}{6}$ \ \ \ \ \ \ \ \ \
\ C) \ $-1%
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$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $11$ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these

\item If \ $3x+2y=r$ \ and $5x-3y=s,$ then \ $4x+3y=$

A) \ $\dfrac{27r-s}{19}%
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$ \ \ \ \ \ \ \ \ B) \ $18r-12s$ \ \ \ \ \ \ \ \ C) \ $\dfrac{4r-3s}{5}$ \ \
\ \ \ \ \ \ \ D) \ $\dfrac{11r+2s}{6}$\ \ \ \ \ \ \ \ E) \ none of these

\item $\dsum\limits_{k=1}^{\infty }\left( \dfrac{2}{7}\right) ^{2k-1}$

A) \ $\dfrac{4}{49}$\ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{14}{45}%
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$ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{2}{5}$\ \ \ \ \ \ \ \ \ \ \ D) \ $7$ \ \
\ \ \ \ \ \ E) \ $\infty $

\item There are $360$ ways to arrange the six letters A, A, C, M, T, Y into
six-letter "words". \ (The two A's are indistinguishable.) \ If these are
placed in alphabetical order, in what position would "AMATYC" fall?$%
\allowbreak $

A) \ $48$th \ \ \ \ \ \ \ \ \ B) \ $52$nd \ \ \ \ \ \ \ \ \ \ C) \ $54$th \
\ \ \ \ \ \ \ \ \ \ \ D) \ $56$th \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$%
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$

\item The mean of five positive integers is $10$ and the (unique) mode is $%
20 $. \ The number of possible values for the median is

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

\item If the points $\left( 1,1\right) ,$ \ $\left( 4,5\right) ,$ and $%
\left( 9,y\right) $ \ lie on a straight line, then \ $y=$

A) \ $11$ \ \ \ \ \ \ \ \ \ B) \ $12$ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{35}{3}%
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$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{37}{3}$ \ \ \ \ \ \ \ \ \ \ \ \ E) \
none of these

Questions 8-10 refer to a function $f,$ whose graph is shown below and whose
domain is $\left[ -6,6\right] $.\FRAME{dtbpF}{5.2096in}{2.3713in}{0pt}{}{}{%
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\item How many solutions does the equation $f\left( x\right) +3=0$ \ have?

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

\item Solve for $x$: \ $\ \ f\left( 2x+1\right) =5$

A) \ $2$ \ \ \ \ \ \ \ \ \ B) \ $2.5%
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$ \ \ \ \ \ \ \ \ \ \ C) \ $6$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ There is one
solution, but it is none of these.

E) \ There is more than one solution.

\item Solve for $x:$ \ \ \ \ $f\left( x\right) =x+2$

A) \ $-5%
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$ \ \ \ \ \ \ \ \ \ B) \ $-2$ \ \ \ \ \ \ \ \ \ \ C) \ $0$\ \ \ \ \ \ \ \ \
\ \ \ D) \ There is one solution, but it is none of these.

E) \ There is more than one solution.

\item Isosceles right \ triangles are cut from the four corners of a square
piece of paper $s$ inches by $s$ inches so that a regular octagon is
produced. \ The length of the legs of the isosceles right triangles, in
inches, is

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

\item The probability that a randomly chosen integer from the interval $%
\left( -2.5,7.5\right) $ satisfies the inequality $x+2<x^{2}$ is

A) \ $0.5$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $0.6%
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$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $0.7$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $%
0.8$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $0.9$

\item A right triangle has perimeter $k$ units, area $k$ square units, and
one leg of length $\sqrt{k}$ units. \ $k=$

A) \ $13+8\sqrt{3}$ \ \ \ \ \ \ \ \ \ B) \ $19\sqrt{2}$ \ \ \ \ \ \ \ \ \ \
C) \ $27$ \ \ \ \ \ \ \ \ \ \ D) \ $14+6\sqrt{5}%
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$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these

\item An asymptote of $2x^{3}-\left( y+3\right) x^{2}+2\left( x-1\right) -y=0
$ \ is

A) \ $x=-1$ \ \ \ \ \ \ \ \ \ B) \ $x=1$ \ \ \ \ \ \ \ \ \ \ C) \ $y=2$ \ \
\ \ \ \ \ \ \ \ D) \ $y=-3$

E) \ None of these are asymptotes.$%
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$

\item If the two zeros of a quadratic function $P$ are $z_{1}$ and $z_{2}$ \
and \ $P\left( 0\right) =k,$ then $P\left( r\right) =$

A) \ $k\left( r-z_{1}\right) \left( r-z_{2}\right) $ \ \ \ \ \ \ \ \ \ \ \ \
B) \ $z_{1}r^{2}+z_{2}r+k$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $%
r^{2}+\left( z_{1}+z_{2}\right) r+k$

D) \ $r^{2}-\left( z_{1}+z_{2}\right) r+k$\ \ \ \ \ \ \ \ \ \ \ E) \ $\dfrac{%
k}{z_{1}z_{2}}\left( r-z_{1}\right) \left( r-z_{2}\right) 
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$

\item Four points, $A,$ $B,$ $C$, and $D$ are situated in a plane such that $%
\overline{AB}=4,$ \ $\overline{BC}=3,$ \ $\overline{CD}=2,$ \ and \ $%
\overline{DA}=1$. \ The greatest possible value for the measure $%
\measuredangle ABC$ is

A) \ $\dfrac{\pi }{3}$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{\pi }{4}$ \ \ \
\ \ \ \ \ \ \ \ \ \ C) \ $\tan ^{-1}\left( \dfrac{3}{4}\right) $ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ D) \ $\tan ^{-1}\left( \dfrac{4}{3}\right) $ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ E) \ $\cos ^{-1}\left( \dfrac{2}{3}\right) $

\item $\sqrt{9+\sqrt{9+\sqrt{9+\sqrt{9+...}}}}=$

A) \ $\pi $ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{13}{4}$ \ \ \ \ \ \ \ \ \ \
\ \ \ C) \ $\sqrt{10}$ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{1+\sqrt{37}}{2}%
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$\ \ \ \ \ \ \ \ \ \ \ \ \ E) $\dfrac{3+\sqrt{10}}{2}$

\item The sum of $n$ consecutive odd whole numbers is $1477,$ where $n>1$.
What is the least of these $n$ whole numbers?

A) \ $53$ \ \ \ \ \ B) \ $91$ \ \ \ \ \ C) \ $137$ \ \ \ \ \ D) \ $205%
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$\ \ \ \ \ \ E) There is not enough information given.

\item $\vartriangle ABC$ has $\measuredangle C=90^{\circ }$ \ and \ $%
\overline{AC}=5$. \ Point $D$ is located on $BC$ so that $\overline{CD}=1$
and \ $\measuredangle DAC=\measuredangle DAB$. \ Find $\overline{BD}$.\ 

A) \ $1$ \ \ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{13}{12}$ \ \ \ \ \ \ \ \ \ \ \ \
\ C) \ $\dfrac{11}{10}$ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{7}{6}$\ \ \ \ \ \
\ \ \ \ \ \ \ E) \ $\dfrac{6}{5}$

\item A certain kind of coated candy bits comes in $n$ colors. \ The
probability that a randomly chosen bit has color $c_{i}$ \ is \ $p_{i}$,
where $\dsum\limits_{i=1}^{n}p_{i}=1$. \ If \ $n$ \ bits are chosen at
random from a very large vat of candy bits, what is the probability that all 
$n$ colors are represented in the sample?

A) \ $n!\dprod\limits_{i=1}^{n}p_{i}%
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$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $\dsum\limits_{i=1}^{n}c_{i}p_{i}$ \ \ \ \ \
\ \ \ \ \ \ \ \ C) \ $\dprod\limits_{i=1}^{n}c_{i}p_{i}$ \ \ \ \ \ \ \ \ \ \
\ D) \ $\dsum\limits_{i=1}^{n}p_{i}^{n}$\ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $%
\dsum\limits_{i=1}^{n}\dsum\limits_{j=1}^{n}p_{i}^{n-1}p_{j}$
\end{enumerate}

\end{document}