\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 Exam 2} \cfoot{} \chead{} \rhead{\large February 1985 - page \ \thepage} \textwidth 7.0in \textheight 9.2in \setlength{\headheight}{20pt} \begin{document} \begin{enumerate} \item If \ $y=2x$ \ and \ $z=2y$, \ then \ $x+y+z$ \ equals: A) \ $x$ \ \ \ \ \ \ \ \ B) \ $3x$ \ \ \ \ \ \ \ \ C) \ $5x$ \ \ \ \ \ \ \ \ \ D) \ $7x% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ E) \ $9x$ \item The diagonal of square I is $a+b$. \ The perimeter of square II with twice the area of square I is: A) \ $\left( a+b\right) ^{2}$ \ \ \ \ \ \ \ \ B) \ $\sqrt{2}\left( a+b\right) ^{2}$ \ \ \ \ \ \ \ \ C) \ $2\left( a+b\right) $ \ \ \ \ \ \ \ \ \ D) \ $\sqrt{8}\left( a+b\right) $ \ \ \ \ \ \ \ \ E) \ $4\left( a+b\right) %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item What is the $y-$intercept of the line passing through $\left( \sqrt{2}% ,1\right) $ \ and \ $\left( -2,2\right) $? A) \ $1$ \ \ \ \ \ \ \ \ B) \ $1.5$ \ \ \ \ \ \ \ \ C) \ $\sqrt{2}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ D) \ $\sqrt{3}$ \ \ \ \ \ \ \ \ E) \ $5-1$ \item For $x>0,$ \ $y>0,$ \ and \ $b>0$ \ where \ $b\not=1$ \ and \ $% y\not=1, $ \ let \ $\log _{b}x=m$ \ and \ $\log _{b}y=n$. \ Three of the following statements are true. \ Which one is, in general, not true? A) \ $\log _{b}xy=nm% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\ \ \ \ \ \ \ \ \ \ B) \ $\log _{b}x^{p}=pm$ \ \ \ \ \ \ \ \ \ \ C) \ $\log _{y}x=\dfrac{m}{n}$\ \ \ \ \ \ \ \ \ \ \ D) \ $\log _{b}\dfrac{x}{y}=m-n$ \ \item How many real solutions has the equation $\left\vert x^{2}-6x\right\vert =9$? A) \ $0$ \ \ \ \ \ \ \ \ \ B) \ $1$ \ \ \ \ \ \ \ \ \ \ C) \ $2$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $3% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $4$ \item Find the area of the triangle with vertices $\left( -3,1\right) ,$ \ $% \left( 1,2\right) ,$ \ and \ $\left( 2,-1\right) $. A) \ $6% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{13}{2}$ \ \ \ \ \ \ \ \ \ \ C) \ $7$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{15}{2}$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $8$ \item Find the positive number $x$ \ for which \ $\sqrt{x}=\sqrt[3]{y}$ \ and \ $\sqrt{y}=8$. A) \ $2$ \ \ \ \ \ \ \ \ \ B) \ $2\sqrt{2}$ \ \ \ \ \ \ \ \ \ \ C) \ $4$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $16% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $64$ \item When the three-digit numbers \ $6a3$ \ and \ $2b5$ \ are added together, the answer is a number divisible by $9$. \ The largest value of $% a+b$ \ is: A) \ $2$ \ \ \ \ \ \ \ \ \ B) \ $9$ \ \ \ \ \ \ \ \ \ \ C) \ $11% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $17$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $20$ \item Find the real part of $\dfrac{i}{1+\dfrac{i}{1+\dfrac{i}{1+i}}}$ A) \ $-\dfrac{1}{4}$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{1}{3}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{4}{5}$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $% \dfrac{2}{3}$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these \item The solution of the equation $\dfrac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}-% \sqrt{x-1}}=3$ \ is: A) \ $3$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{3}{5}$ \ \ \ \ \ \ \ \ \ \ C) \ $% \dfrac{4}{5}$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{5}{4}$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $\dfrac{5}{3}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item Which of the following describes the asymptotes for the hyperbola \ $% \dfrac{\left( x-8\right) ^{2}}{16}-\dfrac{\left( y-3\right) ^{2}}{4}=1$ A) \ $y-3=\pm \dfrac{1}{2}\left( x-8\right) %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ B) \ $y-1=\pm \dfrac{3}{8}\left( x-2\right) $ \ \ \ \ \ \ \ \ \ \ C) \ $y+2=\pm \dfrac{3}{8}\left( x-4\right) $ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $y-1=\pm \dfrac{1}{2}\left( x-2\right) $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $% y+2=\pm \dfrac{1}{4}\left( x-4\right) $ \item $\sin y+\sin \left( x-y\right) =\sin x$ \ for all \ $y$ \ provided that $x$ is: A) \ $60^{\circ }$ \ \ \ \ \ \ \ \ \ B) \ $90^{\circ }$ \ \ \ \ \ \ \ \ \ \ C) \ $180^{\circ }$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $270^{\circ }$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $360^{\circ }% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $\pagebreak \item In the figure $\overline{AB}=\overline{AC},$ \ angle $BAD=30^{\circ }$% , \ and \ $\overline{AE}=\overline{AD}.$ \ Then $x$ equals:\FRAME{dtbpF}{% 2.1188in}{1.1in}{0pt}{}{}{insert1.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.1188in;height 1.1in;depth 0pt;original-width 2.0799in;original-height 1.0672in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert1.bmp';file-properties "XNPEU";}} A) \ $7\dfrac{1}{2}^{\circ }$ \ \ \ \ \ \ \ \ \ B) \ $10^{\circ }$ \ \ \ \ \ \ \ \ \ \ C) \ $12\dfrac{1}{2}^{\circ }$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $% 15^{\circ }$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $20^{\circ }$ \item Of the following, which fraction is an integer multiple of each of the fractions $\dfrac{6}{7}$, \ $\dfrac{5}{14}$, $\dfrac{10}{21}$ ? A) \ $\dfrac{7}{30}$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{7}{15}$ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{15}{7}$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\dfrac{30}{7}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $\dfrac{80}{21}$ \item Fifteen billiard balls are lying on a table in such a way that they are just squeezed inside an equilateral triangular frame whose inside perimeter is $876$. \ The radius of a billiard ball is: A) \ $\dfrac{73}{2}$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{146}{4+\sqrt{3}}$ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{146}{2+\sqrt{3}}$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $% \dfrac{146}{3+\sqrt{3}}% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these \item The least positive integer which has reminders $1$, $1$, and $5$ when divided by $3$, $5$, and $7$ respectively, is: A) \ $166$ \ \ \ \ \ \ \ \ \ B) \ $151$ \ \ \ \ \ \ \ \ \ \ C) \ $145$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $131$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these$% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \item Each valve $A,$ $B,$ and $C,$ when open, releases water into a tank at its own constant rate. \ With all three valves open, the tank fills in one hour, with only valves $A$ and $C$ open it takes $1$ hour and $20$ minutes, and with only valves $B$ and $C$ open it takes $2$ hours. \ The time it takes to fill the tank with only valves $A$ and $B$ open is: A) \ $\dfrac{2}{3}$ hr \ \ \ \ \ \ \ \ \ B) \ $\dfrac{4}{3}$ hr$% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{3}{2}$ hr \ \ \ \ \ \ \ \ \ \ \ \ D) \ $2$ hr \ \ \ \ \ \ \ \ \ \ \ \ E) \ $\dfrac{9}{4}$ \ hr \item In how many different arrangements can a careless office boy place $5$ letters into $5$ mailboxes so that no one gets the right letter? A) \ $32$ \ \ \ \ \ \ \ \ \ B) \ $44% %TCIMACRO{\TeXButton{correct answer}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ \ \ \ \ \ \ \ \ \ \ C) \ $60$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $120$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $225$ \item In the given diagram, points $B$, $C$, and $T$ \ are on the circle, and $AT\ \ $is \ tangent to the circle at $T$ \ \ If \ $AB=3$ \ and \ $BC=4$% , \ find $\dfrac{AB+AT}{AT+AC}$.\FRAME{dtbpF}{1.6561in}{0.9997in}{0pt}{}{}{% insert2.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.6561in;height 0.9997in;depth 0pt;original-width 1.6198in;original-height 0.9669in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert2.bmp';file-properties "XNPEU";}} \item Find the limiting value of $\dfrac{1}{9}+\dfrac{3}{27}+\dfrac{5}{81}+% \dfrac{7}{243}+...+\dfrac{2k-1}{3^{k+1}}$ \end{enumerate} \end{document}