\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 1998 - page \ \thepage} \textwidth 7.2in \textheight 9.3in \setlength{\headheight}{20pt} \begin{document} \begin{enumerate} \item If $\left( -4,-3\right) $, $\left( -1,4\right) $, and $\left( 1,-2\right) $ are three of the vertices of a paarallelogram, then the fourth vertex is A) \ $\left( 2,4\right) $ \ \ \ \ \ \ \ \ B) \ $\left( 3,4\right) $ \ \ \ \ \ \ \ \ C) \ $\left( 5,4\right) $ \ \ \ \ \ \ D) \ $\left( 5,1\right) $ \ \ \ \ \ \ \ \ E) \ $\left( 4,5\right) %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item Let \ $f$ \ be a linear function of the form \ $f\left( x\right) =mx+b$% . \ If \ $f\left( 5\right) =-3$ \ \ and \ $f\left( 8\right) =4,$ \ then $m+b= $ A) \ $-14$ \ \ \ \ \ \ \ \ B) \ $-12$\ \ \ \ \ \ \ \ \ \ C) \ $-\dfrac{37}{3}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ D) \ $17$ \ \ \ \ \ \ \ \ \ E) \ none of these \item Which of the following equations best fits the given data? \begin{tabular}{c|cccc} $x$ & $y$ & ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ & A) & $x+2y=1$ \\ \cline{1-2} $0.27$ & $0.30$ & & B) & $2x+y=1$ \\ $0.51$ & $0.12$ & & C) & $3x-2y=1$ \\ $0.73$ & $-0.05$ & & D) & $3x+4y=2% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \\ $1.04$ & $-0.28$ & & E) & $4x+3y=2$ \\ $1.39$ & $-0.54$ & & & \end{tabular} \item Let $f$ be a polynomial function such that, for all real $x$, $\ f\left( x-1\right) =x^{2}-3x+5$. \ Then, for all real $x$, \ $f\left( x+1\right) =$ A) \ $x^{2}+x+3% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ B) \ $x^{2}-x+3$ \ \ \ \ \ \ \ \ C) \ $x^{2}+x$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $x^{2}-3x+7$ E) \ none of these \item If three-eights of a number is two-sevenths more than four-fifth of the number, then in which of the following intervals lies three-sevenths less than three times the cube of the number? A) \ $\left( -\infty ,-2\right] $\ \ \ \ \ \ \ \ \ \ B) \ $\left( -2,-1\right) $ \ \ \ \ \ \ \ \ \ \ C) \ $\left[ -1,1\right] %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ \ \ \ D) \ $\left( 1,2\right) $ \ \ \ \ \ \ \ \ E) \ $\left[ 2,\infty \right) $ \item A family of seven has a dining area with seven distinct seats. \ Which of the following is the best estimate of how long it would take the family to exhaust all of the possible seating assignments, assuming they eat one meal per day together and always use a new seating assignment each day? A) \ $7$ months\ \ \ \ \ \ \ \ B) \ $3$ years \ \ \ \ \ \ C) \ $8$ years\ \ \ \ \ \ D) \ $14$ years$% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ E) \ $37$ years \item Figures $1,~2,$ and $3$ consist of $5$, $13,$ and $25$ unit squares, respectively. \ If the pattern is continued, figure $100$ would consist of how many square units?\FRAME{dtbpF}{4.0326in}{1.5108in}{0pt}{}{}{insert.bmp}{% \special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 4.0326in;height 1.5108in;depth 0pt;original-width 4.3465in;original-height 1.9666in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert.bmp';file-properties "XNPEU";}} A) \ $10\,401$\ \ \ \ \ \ \ \ \ B) \ $19\,801$ \ \ \ \ \ \ \ C) \ $20\,201% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ D) \ $39\,801$ \ \ \ \ \ \ \ \ \ E) none of these \item If $f$ is a quadratic function of the form $f\left( x\right) =ax^{2}+bx+c,$ \ \ $a\not=0,$ such that $f\left( 3\right) =f\left( 7\right) =0$, \ find $\dfrac{f\left( 4\right) +f\left( 10\right) -f\left( 6\right) }{% f\left( -2\right) -f\left( 12\right) +f\left( 0\right) }$. A) \ $-1$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{4}{5}$ \ \ \ \ \ \ \ \ \ \ C) \ $1% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $2$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these \item Given that $a>0$ \ and \ $\left[ \begin{array}{cc} a & -2 \\ 1 & d% \end{array}% \right] ^{2}=\left[ \begin{array}{cc} 1 & 0 \\ 0 & 1% \end{array}% \right] ,$ \ find $d$. A) \ $-\sqrt{2}$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $-1$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $1$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\sqrt{3}$\ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item Consider the graphs of $y=\sin x,$ \ $y=\cos x,$ \ $y=\tan x,$ \ $% y=\cot x,$ \ $y=\sec x,$ \ and \ $y=\csc x$. \ Let $R=\left\{ \left( x,y\right) \mid 0\leq x\leq \dfrac{\pi }{2},0\leq y\leq 100\right\} $. \ How many points of $R$ are on at least two of the graphs? A) \ $10$ \ \ \ \ \ \ \ \ \ B) \ $11% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ C) \ $12$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $14$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $15$ \item Which of the following has a graph with exactly two vertical asymptotes? A) \ $\ln \left\vert x^{2}-x-6\right\vert %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ B) \ $y=\tan x$ \ \ \ \ \ \ \ \ \ \ C) \ $y=\dfrac{% x^{2}-1}{x^{2}+1}$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $y=\dfrac{x+5}{x^{2}-6x+9}$ E) \ $y=\dfrac{x+6}{x^{2}+5x-6}$ \item For $\left\vert x\right\vert \leq 1$, \ $\cos \left( 2\cos ^{-1}x\right) =$ A) \ $\dfrac{2}{\sqrt{1-x^{2}}}$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{1}{\sqrt{% 1-4x^{2}}}$\ \ \ \ \ \ \ \ \ C) \ $2x$\ \ \ \ \ \ \ \ \ \ D) \ $2x^{2}-1% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these \item $49^{3\log _{7}4}=$ A) \ $2^{12}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $2^{98}$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $% 7^{12}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $7^{21}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these \item For each person taking this exam, an ordered triple could be made of the form $\left( c,w,b\right) $, \ where $c$ \ is the number of questions answered correctly, $w$ \ is the number answered wrong, and $b$ is the number left blank. \ Clearly, $c+w+b=20$. \ How many such ordered triples are possible? A) \ $95$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $231% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ C) $242$\ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $8000$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $9261$ \item Let $ABC$ \ be a right triangle with $AC=BC=6$. \ Let $D$ and $E$ \ be on $AC$ and $BC,$ respectively, with $CD=CE=4$. \ Let $AE$ and $BD$ \ intersect at $F$. \ Find the area of triangle $ABF$. A) \ $3.5$\ \ \ \ \ \ \ \ B) \ $3.6% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ C) \ $3.75$\ \ \ \ \ \ \ \ \ \ D) $\ 4$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $4.2$ \item Let \ \ $P\left( x\right) =4x^{5}-12x^{4}+21x^{3}-8x^{2}-31x-10$. \ Find the product of the distinct complex zeros of $P$. A) \ $10$ \ \ \ \ \ \ \ \ B) \ $-1$\ \ \ \ \ \ \ \ \ \ C) \ $-\dfrac{5}{2}$ \ \ \ \ \ \ \ \ \ D) \ $-5% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ \ \ \ \ E) \ $-10$ \item Point $D$ is on side $AB$ of triangle $ABC,$ with $\angle ACD=\angle BCD=60^{\circ }$, \ $AC=5,$ and $BC=15$. \ $CD=$ A) \ $3$ \ \ \ \ \ \ \ \ \ B) \ $3.25$ \ \ \ \ \ \ \ \ \ C) \ $2\sqrt{3}$ \ \ \ \ \ \ \ \ \ \ D) \ $3.5$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $3.75% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item If $\left( -1,4\right) $ \ and \ $\left( 2,4\right) $ \ are the two foci of an ellipse, and $\left( -2,2\right) $ \ is a point on the ellipse, what is the maximum value for the $y-$coordinates of all of the points on the ellipse? A) \ $4+2\sqrt{2}$ \ \ \ \ \ \ \ \ \ B) \ $7% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ C) \ $4+\sqrt{10}$ \ \ \ \ \ \ \ \ \ D) \ $4+2\sqrt{5}$\ \ \ \ \ \ \ \ E) \ none of these \item A square of area one is surrounded by four distinct circles, each with area one and each externally tangent to the square at a midpoint of the side of the square. \ Find the area, to the nearest hundredth of a square unit, of the circle circumscribing the four circles. A) \ $8.33% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ \ B) \ $8.47$\ \ \ \ \ \ \ \ \ C) \ $8.51$\ \ \ \ \ \ \ \ \ D) \ $8.59$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $8.67$ \item What is the probability that a solution for \ $x^{2}+3x<10$ \ is also \ $x^{2}>5$? A) \ $\dfrac{2+\sqrt{5}}{14}$ \ \ \ \ B) \ $\dfrac{\sqrt{5}}{7}$ \ \ \ \ \ C) \ $\dfrac{5-\sqrt{5}}{7}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ D) \ $\dfrac{2\sqrt{5}}{7}$\ \ \ \ \ \ E) none of these\pagebreak \end{enumerate} Answers \bigskip \begin{tabular}{lllll} 1. & E & ~~~~~~~~~ & 11. & A \\ 2. & C & & 12. & D \\ 3. & D & & 13. & A \\ 4. & A & & 14. & B \\ 5. & C & & 15. & B \\ 6. & D & & 16. & D \\ 7. & C & & 17. & E \\ 8. & C & & 18. & B \\ 9. & E & & 19. & A \\ 10. & B & & 20. & C% \end{tabular} \end{document}