\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 1997 - page \ \thepage} \textwidth 6.8in \textheight 9.3in \setlength{\headheight}{20pt} \begin{document} \begin{enumerate} \item One billion seconds after birth, what was the age (in years) of a person on his/her previous birthday? A) \ $27$ \ \ \ \ \ \ \ \ B) \ $31% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ C) \ $48$ \ \ \ \ \ \ \ \ \ D) \ $69$ \ \ \ \ \ \ \ \ E) none of these \item If $\ \ a=2^{2}\cdot 3\cdot 5^{3}$ \ and \ $21\,000$ \ is the least common multiple of $a$ and $b$, then what is the smallest positive integer that $b$ can be? A) \ $14$ \ \ \ \ \ \ \ \ B) \ $56% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ C) \ $75$ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $168$ \ \ \ \ \ \ \ \ \ \ \ E) \ $21\,000$ \item Suppose that $f$ and $g$ are linear functions with $f\left( 3\right) =10,$ \ $g\left( 3\right) =15,$ \ and \ $\left( f+g\right) \left( 5\right) =20$. \ Then $\left( f+g\right) \left( 1\right) =$ A) \ $5$\ \ \ \ \ \ \ \ B) \ $25$ \ \ \ \ \ \ \ \ \ C) \ $30% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ D) \ $45$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ Not enough information is given \item Find the area of the quadrilateral whose vertices are at $\left( 0,0\right) ,$ $\left( 1,8\right) ,$ $\left( 3,2\right) ,$ and $\left( 6,5\right) $. A) \ $18.5$\ \ \ \ \ \ \ B) \ $20$ \ \ \ \ \ \ \ \ C) \ $21.5$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $23% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $26$ \item If \ $7^{2t+1}$ \ is written in the form $ka^{t},$ where \ $k$ and $a$ are positive constants, then $k+a=$ A) \ $14$\ \ \ \ \ \ \ \ \ \ B) \ $28$ \ \ \ \ \ \ \ \ \ \ C) \ $42$\ \ \ \ \ \ \ \ \ \ \ D) \ $56% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ E) \ $70$ \item The graphs of $y=f\left( x\right) $ \ and \ $y=g\left( x\right) $ \ are each straight lines, and these two lines are perpendicular, intersecting at the point $\left( 3,4\right) $. \ If \ $f\left( 5\right) =11,$ then \ $% g\left( 2\right) =$ A) \ $\dfrac{47}{11}$\ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{9}{2}$ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{23}{5}$\ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{13}{3}$ \ \ \ \ \ \ \ \ \ \ E) \ $\dfrac{30}{7}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item The set of points in the plane for which the distance from a fixed point is one-half the distance from a fixed line (not containing the fixed point) is A) \ a line\ \ \ \ \ \ \ \ \ \ B) \ a parabola\ \ \ \ \ \ \ C) an ellipse$% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ D) \ \ \ a hyperbola \ \ \ \ \ \ \ \ \ E) .none of these \ \ \ \ \ 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 $g\left( x\right) =f\left( x\right) +3$ have? A) \ $0$ \ \ \ \ \ \ \ \ \ B) \ $2$ \ \ \ \ \ \ \ \ \ \ C) \ $3$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $5$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $6% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item How many solutions does the equation $g\left( f\left( x\right) \right) =1$ \ have? A) \ $0$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $1$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $2$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $4% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $6$ \item The longest interval contained in the domain of $\sqrt{3-g\left( x\right) }$ has length A) \ $3$ \ \ \ \ \ \ \ \ \ B) \ $4$ \ \ \ \ \ \ \ \ \ \ C) \ $6$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $8% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $9$ \item $3x+5y=1$ is the equation of a line tangent to a circle with center at $\left( 5,4\right) $. \ What is the radius of the circle? A) \ $4\sqrt{2}$ \ \ \ \ \ \ \ \ \ B) \ $3\sqrt{5}$ \ \ \ \ \ \ \ \ \ \ C) \ $5$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $6$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item A cube is all white. \ Two sides are selected at random and painted red. \ What is the probability that the two red sides have a common edge? A) \ $\dfrac{5}{6}$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{4}{5}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{3}{4}$ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{2% }{3}$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $\dfrac{1}{2}$ \item Let $0\leq x\leq 1.$ \ \ $1-\cos \left( \sin ^{-1}\sqrt{x}\right) $ A) \ $x$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $x^{2}$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $\sqrt{x}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{x}{2-x}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $1-\sqrt{1-x}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item If $r$ represents the time for a population experiencing exponential growth to double, then what is the time needed for the population to triple? A) \ $\dfrac{3}{2}r$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $r^{3/2}$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{r\ln 3}{\ln 2}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{3\ln r}{\ln 2}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $r\ln \dfrac{3}{2}$ \item If the $x-$intercepts of a parabola are $\left( 2\pm \sqrt{3},0\right) $ \ and the point $\left( 0,10\right) $ \ is on the parabola, then which of the following points is on the parabola? A) \ $\left( 1,-30\right) $ \ \ \ \ \ \ \ \ \ B) \ $\left( 2,-50\right) $ \ \ \ \ \ \ \ \ \ \ C) \ $\left( 3,-20\right) %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ D) \ $\left( 4,20\right) $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $\left( 5,50\right) $ \item The smallest angle of a triangle with sides of length $3,$ $5,$ and $6$ is A) \ $\sin ^{-1}\left( \dfrac{2\sqrt{14}}{15}\right) %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ B) \ $\sin ^{-1}\left( \dfrac{13}{15}\right) $ \ \ \ \ \ \ \ \ \ \ C) \ $\tan ^{-1}\left( \dfrac{\sqrt{13}}{15}\right) $ \ \ \ \ \ \ \ \ \ \ D) \ $\cos ^{-1}\left( \dfrac{14}{15}\right) $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $% \cos ^{-1}\left( \dfrac{3\sqrt{7}}{15}\right) $ \item Find the sum of the real solutions for the equation \ $% 8^{3x+1}=4^{x^{2}-2}$. A) \ $-4$ \ \ \ \ \ \ \ \ \ B) \ $0$ \ \ \ \ \ \ \ \ \ \ C) \ $3$ \ \ \ \ \ \ \ \ \ \ D) \ $4.5% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $6$ \item Approximately how high would a stack of $10!$ pennies be? A) \ $3000~\unit{ft}$ \ \ \ \ \ \ \ \ \ B) \ $3$ miles$% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ C) \ $30$ miles \ \ \ \ \ \ \ \ \ D) \ $300$ miles\ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $3000$ miles \item A collection of $53$ coins has value $\$~7.05$ and consists entirely of nickels, dimes, and quarters. \ Which of the following could not be the number of nickels in the collection? A) \ $1$ \ \ \ \ \ \ \ \ \ \ B) \ $11% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ C) \ $19$ \ \ \ \ \ \ \ \ \ \ \ D) \ $25$\ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $31$ \item A car is traveling at a constant speed of $65$ mph along a road parallel to a railroad track. \ The car overtakes a mile long train (traveling in the same direction) traveling at a constant speed of 60 mph. \ How long does it take from the time the car passes the rear of the train until it passes the engine at the head of the train? A) \ $6$ min \ \ \ \ \ B) \ $10$ min \ \ \ \ \ C) \ $12$ min$% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ D) \ $13$ min\ \ \ \ \ \ E) $20$ min\pagebreak \end{enumerate} Answers \bigskip \begin{tabular}{lllll} 1. & B & ~~~~~~~~~ & 11. & E \\ 2. & B & & 12. & B \\ 3. & C & & 13. & E \\ 4. & D & & 14. & C \\ 5. & D & & 15. & C \\ 6. & E & & 16. & A \\ 7. & C & & 17. & D \\ 8. & E & & 18. & B \\ 9. & D & & 19. & B \\ 10. & D & & 20. & C% \end{tabular} \end{document}