\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}} \geometry{left=0.6in,right=0.7in,top=0.7in,bottom=0.7in} \pagestyle{fancy} \lhead{\large Exam 1} \cfoot{} \chead{} \rhead{\large November 1997 - page \ \thepage} \textwidth 6.8in \textheight 9.3in \setlength{\headheight}{20pt} \begin{document} \begin{enumerate} \item Referring to the Declaration of Independence (1776), Lincoln introduced his Gettysburg Addess in 1863 with "Four score and seven years ago..." \ How many years is a score of years? A) \ $5$ \ \ \ \ \ \ \ \ B) \ $10$ \ \ \ \ \ \ \ \ C) \ $20% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ D) \ $25$ \ \ \ \ \ \ \ \ E) $50\smallskip $ \item Let $R=\left\{ \left( x,y\right) \mid 3\leq x^{2}+y^{2}\leq 5\right\} $% . The area of $R$ is A) \ $32$ \ \ \ \ \ \ \ \ B) \ $20$ \ \ \ \ \ \ \ \ \ \ C) \ $4\pi $ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $2\pi %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$\smallskip $ \item At which equal sign does the error occur in the following proposed proof for $1=-1$? \ (Note: $i$ denotes $\sqrt{-1}$.)% \begin{equation*} 1\underset{\text{A}}{=}\sqrt{1}\underset{\text{B}}{=}\sqrt{-1\left( -1\right) }\underset{\text{C}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion }{=}\sqrt{-1}\cdot \sqrt{-1}=i\cdot i\underset{\text{D}}{=}i^{2}\underset{% \text{E}}{=}-1 \end{equation*}% $\smallskip $ \item Twenty years ago Jake built a house that is now half as old as Jake was when he built it. \ How old is Jake now? A) \ $30$\ \ \ \ \ \ \ \ \ \ B) \ $40$ \ \ \ \ \ \ \ \ \ \ C) \ $50$\ \ \ \ \ \ \ \ \ \ \ D) \ $80$ \ \ \ \ \ \ \ \ E) \ none of these$% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \smallskip $ \item Given that $f$ is a linear function with $f^{-1}=f$ \ and $f\left( 4\right) =10$, find $f\left( 9\right) $. A) \ $5% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ \ \ B) \ $6$ \ \ \ \ \ \ \ \ \ \ C) \ $7$\ \ \ \ \ \ \ \ \ \ \ D) \ $8$ \ \ \ \ \ \ \ \ E) \ $9\smallskip $ \item If a number is randomly selected from $\left\{ -10,-5-3,-2,-1,0,1,3,5,10\right\} $, find the probability that one more than the square of the selected number is a solution of $x^{2}+50=15x$. A) \ $0.1$ \ \ \ \ \ \ \ \ \ B) \ $0.2$ \ \ \ \ \ \ \ \ \ \ C) \ $0.3% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $0.4$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$\smallskip $ \item An angle of $23^{\circ }$ is inscribed in a circle of radius $9$. \ What is the length, to the nearest tenth, of the intercepted arc? A) \ $7.2% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ B) \ $6.5$ \ \ \ \ \ \ \ \ \ \ C) \ $6.1$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $5.4$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$\smallskip $ \item A $3\times 3$ \ magic square uses the integers $1,~2,$~$...,9$ \ once each in such a way that each column, each row, and each diagonal sums to $15$% . \ Find the value of $n$ for the magic square, a portion of which is shown below. \begin{tabular}[t]{|c|c|c|} \hline $8$ & ~~ & \\ \hline $n$ & & $7$ \\ \hline & & \\ \hline \end{tabular} A) $2$ \ \ \ \ \ \ \ \ \ B) \ $3% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ C) \ $4$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $5$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $6\smallskip $ \item Given that $f\left( x\right) =x^{2},$ \ \ $-10\leq x\leq -5,$ \ find $% f^{-1}\left( x\right) $. A) \ $f^{-1}\left( x\right) =\sqrt{x}$, \ \ $25\leq x\leq 100$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B) \ \ $f^{-1}\left( x\right) =\sqrt{-x}$, \ \ $25\leq x\leq 100$ C) \ \ $f^{-1}\left( x\right) =\sqrt{-x}$, \ \ $-100\leq x\leq -25$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ \ $f^{-1}\left( x\right) =-\sqrt{x}$, \ \ $25\leq x\leq 100% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ E) \ The inverso of $f$ is not a function $\smallskip $ \item Consider the points $\left( 5,7\right) ,$ $\left( -4,-4\right) ,$ $% \left( 3,2\right) ,$ $\left( 2,2\right) ,$ $\left( 7,-6\right) ,$ $\left( 8,1\right) ,$ \ and \ $\left( 4,4\right) $. \ How many distinct triangles with positive area can be made by choosing any three of these points as vertices? A) \ $34% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ B) \ $35$ \ \ \ \ \ \ \ \ \ \ C) \ $209$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $210$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these \item The mean of three numbers is ten more than the least of the three and fifteen less than the greatest of the three. If the median of the three numbers os $5$, find their sum. A) \ $5$ \ \ \ \ \ \ \ \ \ B) \ $20$ \ \ \ \ \ \ \ \ \ \ C) \ $25$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $30% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$\smallskip $ \item Thirteen unit circles are arranged with their centers equally spaced on a circle with radius $r$ in such a way that each \ of the unit circles is externally tangent to exactly two others. \ Find $r$ to the nearest hundredth. A) \ $4.06$ \ \ \ \ \ \ \ \ \ B) \ $4.14$ \ \ \ \ \ \ \ \ \ \ C) \ $4.18% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ D) \ $8.24$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $% 8.30\smallskip $ \item How many times does $y=15\sin \left( 180x\right) $ \ take on the value $5$ on the interval $\left[ 0,1\right] $? A) \ $56$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $58% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $59$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $60$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$\smallskip $ \item For any positive integer $n$, let $S\left( n\right) $ \ represent the least positive integer whose factorial is divisible by $n$. \ Find $S\left( 875\right) +S\left( 81\right) $. A) \ $16$ \ \ \ \ \ \ \ \ \ B) \ $9$ \ \ \ \ \ \ \ \ \ \ C) \ $24% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ D) \ $29$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$% \smallskip $ \item If $a$ is a solution for $\left\vert x-2\right\vert =8$ \ and \ $b$ is a solution for $x^{3}+x^{2}+36=24x$, then what is the maximum possible value for $ab$? A) \ $36$ \ \ \ \ \ \ \ \ \ B) \ $60$ \ \ \ \ \ \ \ \ \ \ C) \ $30$ \ \ \ \ \ \ \ \ \ \ D) \ $20% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$\smallskip $ \item If three people are chosen randomly from a large population, what is the probability that they were all born on different days of the week? A) \ $\dfrac{3}{5}$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{27}{49}$ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{1}{2}$ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{51}{152}$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $\dfrac{30}{49}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \smallskip $ \item A\ $4$ foot pole is $10$ feet from an $8$ foot pole. \ Each pole is vertical, and the ground is level. \ Suppose a stake is placed in the ground aligned between the two poles. \ Wires are then strung from the top of each pole and fastened taut to the stake at ground level. \ Considering all possible positions of the stake, find the maximum angle, to the nearest tenth of a degree, between the wires. A) \ $76.5^{\circ }$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $77.9^{\circ }$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $79.8^{\circ }$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $% 83.4^{\circ }$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $90.0^{\circ }% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \smallskip $ \item In standard decimal notation, how many positive integers less than $% 10\,000$ \ have at least one $3$ as a digit? A) \ $3438$ \ \ \ \ \ \ \ \ \ \ B) \ $3439% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ C) \ $3440$ \ \ \ \ \ \ \ \ \ \ \ D) \ $3441$\ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$\smallskip $ \item Suppose $f$ is a function whose domain includes the interval $\left[ -1,1\right] $. \ Then the graph of $y=f\left( \sin x\right) $ \ is necessarily symmetric with respect to A) \ the $y-$axis \ \ \ \ \ \ \ \ \ \ B) \ the vertical line $x=\pi $ \ \ \ \ \ \ \ \ C) \ the origin D) \ the vertical line $x=\dfrac{\pi }{2}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ \ \ E) \ none of these$\smallskip $ \item Given that $f\left( 1\right) =1$ \ and \ $f\left( t\right) =5+2f\left( t-1\right) $ \ for $t\geq 2,$ find $f\left( 100\right) $. A) \ $5+6^{99}$ \ \ \ \ \ \ \ \ \ B) \ $5+6\cdot 2^{100}$ \ \ \ \ \ \ \ \ C) \ $6\cdot 2^{99}-5% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ D) \ $5+6^{100}$\ \ \ \ \ \ \ \ \ \ E) \ none of these\pagebreak \end{enumerate} Answers \bigskip \begin{tabular}{lllll} 1. & C & & 11. & D \\ 2. & D & & 12. & C \\ 3. & C & & 13. & B \\ 4. & E & & 14. & C \\ 5. & A & & 15. & A \\ 6. & C & & 16. & E \\ 7. & A & & 17. & D \\ 8. & B & & 18. & B \\ 9. & D & & 19. & D \\ 10. & A & & 20. & C% \end{tabular} \end{document}