\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 3} \cfoot{} \chead{} \rhead{\large March/April 1997 - page \ \thepage} \textwidth 6.8in \textheight 9.3in \setlength{\headheight}{20pt} \begin{document} \begin{enumerate} \item Which of the following has as its graph a parabola which opens left?$% \newline \qquad $A) \ $2y^{2}=11-3x% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ B) \ $5x=13+2y^{2}$ \ \ \ \ \ \ \ \ C) \ $3y=7+4x^{2}$% \newline \qquad D) \ $3x^{2}=8-2y$ \ \ \ \ \ \ \ \ E) \ $x^{2}=10-y^{2}$ \item If three distinct counting numbers have a sum of $10$ and a product of $20$, then what is their median?\newline \qquad A) \ $2$ \ \ \ \ \ \ \ \ B) \ $3$\ \ \ \ \ \ \ \ \ \ C) \ $4% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ D) \ $5$ \ \ \ \ \ \ \ \ \ E) \ There is not enough information given. \item If $L$ is a straight line passing through $\left( 8,11\right) $ with slope $-3$, \ then what is the $y-$coordinate of the intersection of $L$ with the line $x=5$?\newline \qquad A) \ $2$\ \ \ \ \ \ \ \ B) \ $10$ \ \ \ \ \ \ \ \ \ C) \ $12$\ \ \ \ \ \ \ \ \ \ D) \ $20% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ \ \ \ E) \ none of these \item If one root of \ $3x^{2}+j=kx$ \ is $2+\sqrt{3},$ where $j$ and $k$ are rational, then $j+k=$\newline \qquad A) \ $-15$\ \ \ \ \ \ \ B) \ $-9$ \ \ \ \ \ \ \ \ C) \ $9$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $15% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these \item If $AB$ is the diameter of a circle, and $C$ is on the circle such that $\overline{AC}=10$ \ and \ $\overline{BC}=16,$ then the area of the circle is\newline \qquad A) \ $89\pi %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ \ \ B) \ $2\pi \sqrt{89}$ \ \ \ \ \ \ \ \ \ \ C) \ $178\pi $% \ \ \ \ \ \ \ \ \ \ \ D) \ $356\pi $ \ \ \ \ \ \ \ \ E) \ none of these \item If the solution set for $f\left( x\right) <3$ \ is \ $\left[ 0,\infty \right) $, and the solution set for $f\left( x\right) >-2$ \ is $\left( -\infty ,5\right) ,$ then the solution set for \ $\left[ f\left( x\right) % \right] ^{2}\geq f\left( x\right) +6$ \ is\newline \qquad A) \ $\left( -\infty ,\infty \right) $\ \ \ \ \ \ \ \ \ B) \ $\left[ 0,5\right] $ \ \ \ \ \ \ C) \ $\left( -\infty ,0\right] $\ \ \ \ \ \ D) \ $% \left[ 5,\infty \right) $ \ \ \ \ \ \ \ E) \ $\left( -\infty ,0\right] \cup % \left[ 5,\infty \right) %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item If $\left( a,b\right) $ is a solution for $2a+3b\leq 12$ \ and \ $% 5a+2b\leq 20,$ where \ $a$ and $b$ are non-negative integers, then the maximum possible value of $a+b$ \ is\newline \qquad A) \ $2$\ \ \ \ \ \ \ \ \ B) \ $3$ \ \ \ \ \ \ \ C) \ $4$ \ \ \ \ \ \ D) \ $5% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ E) $6$ 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}{4.7824in% }{2.5789in}{0pt}{}{}{insert1.bmp}{\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 4.7824in;height 2.5789in;depth 0pt;original-width 13.3337in;original-height 8.6248in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert1.bmp';file-properties "XNPEU";}} \item How many solutions does the equation $g\left( x\right) =2$ have?% \newline \qquad A) \ $0$ \ \ \ \ \ \ \ \ \ B) \ $1$ \ \ \ \ \ \ \ \ \ \ C) \ $2$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $3$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $4% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item How many solutions does the equation $g\left( x\right) =\left\vert f\left( x\right) \right\vert $ \ have?\newline \qquad A) \ $0$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $1$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $2% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $4$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $% 6$ \item Suppos that $g$ is the restriction to $\left[ -6,6\right] $ \ of a periodic function $G$ \ whose domain is $\left( -\infty ,\infty \right) $ \ and whose period is $16$. \ The approximate value of $G\left( 100\right) $ \ is\newline \qquad A) \ $0$ \ \ \ \ \ \ \ \ \ B) \ $2$ \ \ \ \ \ \ \ \ \ \ C) \ $2.3$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $3% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $5$ \item Each of six balanced cubes is painted identically so that one letter of AMATYC is..on each face (giving each cube two faces with indistinguishable A's). \ If the six cubes are rolled (like dice), what is the probability, rounded to two decimal places, that AMATYC can be spelled by arranging the letters that appear on the top faces of the cubes? (M, T, Y,and C must appear once, A must appear twice).\newline \qquad A) \ $0.03% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ B) \ $0.05$ \ \ \ \ \ \ \ \ \ \ C) \ $0.09$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $0.13$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $0.17$ \item A geometric sequence has a first term of $\tan t$ and a common ratio of $2\cos t$. \ What is the third term?\newline \qquad A) \ $4\sin t$ \ \ \ \ \ \ \ \ \ B) \ $2\sin 2t% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ C) \ $4\sec t$\ \ \ \ \ \ \ \ \ \ D) \ $4\cos 2t$ \ \ \ \ \ \ \ \ \ \ \ E) \ $4\cos ^{2}t$ \item Two straight lines, whose slopes are $m$ and $n$, intersect at the point $\left( m,n\right) ,$ where $m\not=n$. \ The line of slope $m$ has $y-$% intercept $m$, and the line of slope $n$ passes through the origin. \ The sum of all possible values for $m$ is\newline \qquad A) \ $-2$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $-1$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $1$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $2$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $0% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item Albert goes into a store and says "Lend me as much money as I already have and I will spend $\$20$ in your store." \ The owner agrees, and ALbert spends $\$20$. \ Albert does the same thing at a second, third, and fourth store, with the owner agreeing each time, and Albert spending $\$20$ each time. \ After this Albert has no money. \ What is the total of his debt to the four store owners?\newline \qquad A) \ $\$48.50$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $\$57.75$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $\$61.25% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\$69.75$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $\$75.00$ \item If $x^{4}+5x^{2}-19=0$, then what is the value of $% x^{7}-2x^{6}+5x^{5}-5x^{4}-19x^{3}+63x^{2}$?\newline \qquad A) \ $95% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ B) \ $8+4\sqrt{101}$ \ \ \ \ \ \ \ \ \ \ C) \ $12-4\sqrt{% 101}$ \ \ \ \ \ \ \ \ \ \ D) $0$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these \item Suppose all of the vehicles traveling on a certain interstate highway have either $18$ wheels on $5$ axles or $4$ wheels on two axles. \ In a five minute period, $224$ wheels on $88$ axles pass by. \ How many vehicles passed by during that period?\newline \qquad A) \ $18$ \ \ \ \ \ \ \ \ \ B) \ $23$ \ \ \ \ \ \ \ \ \ \ C) \ $29$ \ \ \ \ \ \ \ \ \ \ D) \ $31$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $35% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item Half the reciprocal of a real number is $2$ more than the square of the number. \ What is the least integer which is greater than ten times the number?\newline \qquad A) \ $1$ \ \ \ \ \ \ \ \ \ B) \ $3% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ C) \ $4$ \ \ \ \ \ \ \ \ \ \ D) \ $7$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these \item There are $100$ U.S. Senators, two from each of the $50$ states. \ If $% 20$ senators are chosen at random, what is the probability (rounded to two decimal places) that any given state is represented?\newline \qquad A) \ $0.29$ \ \ \ \ \ \ \ \ \ B) \ $0.31$ \ \ \ \ \ \ C) \ $0.34$ \ \ \ \ \ \ \ \ \ D) \ $0.36% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ \ \ \ \ E) \ $0.40$ \item Suppose $a>0.$ \ Solve for $s$: \ \ $a^{3x+1}=2a^{x}$. A) \ $-1+\log _{a}\left( \dfrac{3}{2}\right) $ \ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $-1+\log _{a}\left( \dfrac{2}{3}\right) $ \ \ \ \ \ \ \ \ \ \ \ C) \ $% \dfrac{-1+\log _{a}2}{2}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ D) \ $\log _{a}\sqrt{3}$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $\log _{a}\sqrt[3]{2}$ \item A picket fence has $100$ vertical boards, numbered $1$ through $100$. \ Any board whose number is in the form $2k-1$ is painted red ($% 1,3,5,7,...,99$). \ Next any \textbf{unpainted} board whose number is of the form $3k-1$ is painted blue ($2,8,14,...,98$). \ If this process is continued, by increasing the coefficient of $k$ by one and choosing a different color at each step, how many colors will be on the fence when every board has been painted?\newline \qquad A) \ $26% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ B) \ $29$ \ \ \ \ \ C) \ $32$ \ \ \ \ \ D) \ $35$\ \ \ \ \ \ E) none of these\pagebreak \end{enumerate} \bigskip Answers \bigskip \begin{tabular}{lllll} 1. & A & ~~~~~~~~~ & 11. & A \\ 2. & C & & 12. & B \\ 3. & D & & 13. & E \\ 4. & D & & 14. & C \\ 5. & A & & 15. & A \\ 6. & E & & 16. & E \\ 7. & D & & 17. & B \\ 8. & E & & 18. & D \\ 9. & C & & 19. & C \\ 10. & D & & 20. & A% \end{tabular} \end{document}