\hfill \hfill } %} \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} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.6in,right=0.7in,top=0.7in,bottom=0.7in} \pagestyle{fancy} \lhead{\large Test 2} \cfoot{} \chead{} \rhead{\large February 1999 - page \ \thepage} \textwidth 6.8in \textheight 9.3in \setlength{\headheight}{20pt} \begin{document} \begin{enumerate} \item \textsf{Two bits, four bits, six bits, a dollar,}\newline \textsf{All for the home team, stand up an holler.}\newline If "dollar" completes an arithmetic sequence in this popular cheer, what is the value of a bit?\newline A) $\$0.02$ \ \ \ \ \ \ \ \ B) \ $\$0.05$ \ \ \ \ \ \ \ \ C) \ $\$0.10$ \ \ \ \ \ \ \ \ \ D) \ $\$0.125% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ E) \ $0.25\smallskip $ \item What is the minimum number of U.S. coins$^{\ast }$ required so that change could be provided for any purchase under $\$1.00$ when a $\$~1$ bill is presented for payment? \ $^{\ast }${\small pennies, nickels, dimes, quarters, half-dollars}\newline A) $7$ \ \ \ \ \ \ \ \ B) \ $9% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ C) \ $11$ \ \ \ \ \ \ \ \ \ D) \ $12$ \ \ \ \ \ \ \ \ E) \ none of these$\smallskip $ \item Find $m$ so that the area of the triangle bounded by $y=0$, \ $x=10$, \ and \ $y=mx$ \ is $3$.\newline A) \ $0.06% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ B) \ $0.08$ \ \ \ \ \ \ \ \ \ C) \ $0.09$ \ \ \ \ \ \ \ \ \ \ D) \ $0.10$\ \ \ \ \ \ \ \ \ \ E) \ $0.12\smallskip $ \item A flagpole and a nearby tree simultaneously cast shadows of length $52~% \unit{ft}$ and $88~\unit{ft}$ respectively. \ If the flagpole is known to be $39~\unit{ft}$ high, how tall is the tree?\newline A) \ $60~\unit{ft}$\ \ \ \ \ \ \ \ B) \ $66~\unit{ft}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ C) \ $70~\unit{ft}$ \ \ \ \ \ \ \ \ \ \ D) \ $75~\unit{ft}$\ \ \ \ \ \ \ \ \ \ E) \ $112~\unit{ft}\smallskip $ \item The product of four positive integers is equal to their sum. \ What is the median of the four numbers?\newline A) \ $1.5% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ B) \ $2$ \ \ \ \ \ \ C) \ $3$\ \ \ \ \ \ \ \ \ \ \ D) \ $4$ \ \ \ \ \ E) \ $8\smallskip $ \item What is the reminder when $1999^{1999}$ \ is divided by $10$?\newline A) \ $0$\ \ \ \ \ \ \ \ B) \ $1$ \ \ \ \ \ \ \ \ C) \ $5$\ \ \ \ \ \ \ \ \ D) \ $8$ \ \ \ \ \ \ \ \ E) \ $9% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \smallskip $ \item Lines $L_{1}$ and $L_{2}$ each pass through $\left( 3,4\right) $. \ $% L_{2}$ has positive slope $m$, and $L_{1}$ has slope $2m$. \ $L_{2}$ crosses the $y-$axis $k$ units above the point where $L_{1}$ crosses the $y-$axis. \ Find $k$.\newline A) \ $m$\ \ \ \ \ \ \ \ B) \ $3$ \ \ \ \ \ \ C) \ $m+3$\ \ \ \ \ \ \ \ \ \ \ D) \ $3m+4$ \ \ \ \ \ E) \ $3m% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \smallskip $ \item If the diameter of each tire is $26$ inches$,$ approximately how many revoutions per second are mad by each wheel when the car's speed is $60$ miles per hour?\newline A) $6$\ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $7$ \ \ \ \ \ \ \ \ \ \ \ C) \ $12$\ \ \ \ \ \ \ \ \ \ \ \ D) \ $13% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ E) \ $24\smallskip $ \item If $x+y+z=k,$ \ $x+2y+3z=2k,$ \ and \ $x+4y+6z=3k,$ \ then \ $3x+6y+2z= $\newline A) \ $-k% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ B) \ $-\dfrac{2}{3}k$ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{4% }{7}k$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{13}{11}k$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$\smallskip $ \item Suppose $f$ is a quadratic function with zeros $2$ and $\sqrt{2}$. \ If $f\left( 0\right) =1000$, in which of the following intervals lies $% f\left( \dfrac{2+\sqrt{2}}{2}\right) $?\newline A) \ $\left( -\infty ,-50\right) $ \ \ \ \ \ \ \ B) \ $\left[ -50,-20\right) %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ C) \ $\left[ -20,-10\right) $ \ \ \ \ \ \ \ \ \ D) \ $\left[ -10,0\right) $ \ \ \ \ \ \ \ \ E) \ $\left[ 0,\infty \right) \smallskip $ \item If \ $3\sin ^{2}\theta =5\sin \theta +2$ \ and \ $\cos \theta <0$, then \ $\sin 2\theta =$\newline A) \ $-\dfrac{2}{3}$\ \ \ \ \ \ \ \ \ \ B) \ $-\dfrac{\sqrt{3}}{2}$\ \ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{\sqrt{3}}{2}$\ \ \ \ \ \ D) \ $-\dfrac{4\sqrt{2}}{% 9}$\ \ \ \ \ \ E) \ $\dfrac{4\sqrt{2}}{9}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \smallskip $ \item Find the sum of the squares of the solutions for $\left( x^{2}-4\right) ^{\left( x^{2}-x-6\right) }=1$, \ \ $x\not=-2$.\newline A) \ $5$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $9$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $10$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $14$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $% 19% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\pagebreak \item Suppose the earth were a perfect sphere with a perfectly fitting belt of $24\,000$ miles surrounding it along a great circular path. \ Suppose the belt was cut, and one hundred feet of additional material was added to the belt, with the "loose fit" evenly distributed around the earth so that the new belt was still circular with its center at the center of the earth. \ Which of the following \textbf{best} describes the resulting situation?% \newline A) \ You colud slip a piece of paper between the belt and the earth.\newline B) \ You could get your fingers under the belt.\newline C) \ You could crawl under the belt.\newline D) \ You could walk upright under the belt.\newline E) \ You could drive a truck under the belt.$% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \smallskip $ \item If one car is one mile behind another car traveling at a constant rate of $65$ miles per hour on an interstate highway \ with plenty of open road ahead, how fast must the first car travel in order to overtake the second car in ten minutes?\newline A) \ $68$ mph \ \ \ \ \ \ \ \ \ B) \ $69$ mph \ \ \ \ \ \ \ \ \ \ C) \ $70$ mph\ \ \ \ \ \ \ \ \ \ D) \ $71$ mph$% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $75$ mph$\smallskip $ \item Consider the following proposed casino game. \ A player pays $\$2$ and roll two dice.$^{\ast }$ \ If the sum of dots on the upper faces of the two dice totals at least ten, the player receives $\$10$, for a gain of $\$8.$ \ Otherwise, the player gets nothing, for a loss of $\$2$. \ If this game is played $3000$ times, what is the expected outcome (or mathematical expectation) for the OPERATOR of this game? $^{\ast }${\small A die is a cube whose faces are marked with from one to six dots, with each number of dots occuring exactly once. \ The plural of die is dice.}\newline A) \ gain $\$3000$\ \ \ \ \ \ B) \ gain $\$1000% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ C) \ lose $\$2000$\ \ \ \ \ \ D) \ lose $\$3000$\ \ \ \ \ E) \ none of these \item What is the area (to the nearest hundredth of square unit) \ of a regular pentagon with a sode length of one unit?\newline A) \ $1.48$\ \ \ \ \ \ \ B) \ $1.54$\ \ \ \ \ \ \ \ C) \ $1.60$\ \ \ \ \ \ \ \ D) \ $1.64$\ \ \ \ \ \ \ \ \ \ E) \ $1.72% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item Let \ $f\left( x\right) =x^{2}-10$. \ How many real solutions does $% f\left( f\left( f\left( x\right) \right) \right) =100$ \ have?\newline A) \ $0$ \ \ \ \ \ \ \ \ \ B) \ $1$ \ \ \ \ \ \ \ \ \ \ C) \ $2$ \ \ \ \ \ \ \ \ \ D) \ $4% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ \ \ \ \ E) \ $8$ \item An isosceles triangle is inscribed in a circle of radius one. \ Two equal sides of the triangle are twice as long as the third side. \ Find the area of the triangle. \newline A) \ $\dfrac{15\sqrt{15}}{64}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{8\sqrt{5}}{15}$ \ \ \ \ \ \ \ \ \ C) \ $% \dfrac{5\sqrt{3}}{8}$\ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{16-\sqrt{5}}{64}$\ \ \ \ \ \ \ \ \ \ E) \ none of these \item How many real solutions does \ $\left( x-50\right) ^{2}=1000+7\sin \left( 50x\right) $ \ have?\newline A) \ $2$\ \ \ \ \ \ \ \ B) \ $6$ \ \ \ \ \ \ \ C) \ $8% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ D) \ $10$\ \ \ \ \ \ \ \ E) \ none of these \item Patty Point takes a random walk through the rectangular coordinate plane as follows. \ She tosses a dime and a nickel. \ She then moves one unit in the direction as indicated in the chart below. \ \ \ \ \ \ \begin{tabular}{c|c|c} dime & nickel & direction \\ \hline heads & heads & of the positive $x-$axis ("east") \\ heads & tails & of the negative $x-$axis ("west") \\ tails & heads & of the positive $y-$axis ("north") \\ tails & tails & of the negative $y-$axis ("south")% \end{tabular} If she starts at the origin, what is the probability that she is in the first quadrant after three tosses? {\small \ (Note: \ POints on a coordinate axis are not in any quadrant.)}\newline A) \ $\dfrac{3}{32}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ B) \ $\dfrac{5}{64}$ \ \ \ \ \ C) \ $\dfrac{3}{16}$ \ \ \ \ \ D) \ $\dfrac{1}{8}$\ \ \ \ \ \ E) \ none of these\pagebreak \end{enumerate} Answers \bigskip \begin{tabular}{lllll} 1. & D & ~~~~~~~~~ & 11. & E \\ 2. & B & & 12. & E \\ 3. & A & & 13. & E \\ 4. & B & & 14. & D \\ 5. & A & & 15. & B \\ 6. & E & & 16. & E \\ 7. & E & & 17. & D \\ 8. & D & & 18. & A \\ 9. & A & & 19. & C \\ 10. & B & & 20. & A% \end{tabular} \bigskip \end{document}