% Exam Template for UMTYMP and Math Department courses % % Using Philip Hirschhorn's exam.cls: http://www-math.mit.edu/~psh/#ExamCls % % run pdflatex on a finished exam at least three times to do the grading table on front page. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % These lines can probably stay unchanged, although you can remove the last % two packages if you're not making pictures with tikz. \documentclass[12pt]{exam} \RequirePackage{amssymb, amsfonts, amsmath, latexsym, verbatim, xspace, setspace} \RequirePackage{tikz, pgflibraryplotmarks} % By default LaTeX uses large margins. This doesn't work well on exams; problems % end up in the "middle" of the page, reducing the amount of space for students % to work on them. \usepackage[margin=1in]{geometry} % Here's where you edit the Class, Exam, Date, etc. \newcommand{\class}{University of California Davis} \newcommand{\term}{Euclidean Geometry MAT 141} \newcommand{\examnum}{\color{red}{Sample Final Examination}} \newcommand{\examdate}{June 11 2026} \newcommand{\timelimit}{2 Hours} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \renewcommand{\H}{\mathbb{H}} \newcommand{\sse}{\subseteq} \newcommand{\lr}{\longrightarrow} % For an exam, single spacing is most appropriate \singlespacing % \onehalfspacing % \doublespacing % For an exam, we generally want to turn off paragraph indentation \parindent 0ex \begin{document} % These commands set up the running header on the top of the exam pages \pagestyle{head} \firstpageheader{}{}{} \runningheader{}{\examnum\ - Page \thepage\ of \numpages}{\examdate} \runningheadrule \begin{flushright} \begin{tabular}{p{2.8in} r l} \textbf{\class} & \textbf{Name (Print):} & \makebox[2in]{\hrulefill}\\ \textbf{\term} & \textbf{Student ID (Print):} & \makebox[2in]{\hrulefill}\\ \\ \textbf{\examnum} && \textbf{\examdate}\\ \textbf{Time Limit: \timelimit} & \end{tabular}\\ \vspace{1cm} \end{flushright} \rule[1ex]{\textwidth}{.1pt} \vspace{1cm} This examination document contains \numpages\ pages, including this cover page, and \numquestions\ problems. You must verify whether there any pages missing, in which case you should let the instructor know. {\bf Fill in} all the requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated.\\ You may \textit{not} use your books, notes, or any calculator on this exam.\\ You are required to show your work on each problem on this exam. The following rules apply:\\ \begin{minipage}[t]{3.7in} \vspace{0pt} \begin{itemize} \item[(A)] \textbf{If you use a lemma, proposition or theorem which we have seen in the class or in the book, you must indicate this} and explain why the theorem may be applied. \item[(B)] \textbf{Organize your work}, in a reasonably neat and coherent way, in the space provided. Work scattered all over the page without a clear ordering will receive little credit. \item[(C)] \textbf{Mysterious or unsupported answers will not receive full credit}. A correct answer, unsupported by calculations, explanation, or algebraic work will receive little credit; an incorrect answer supported by substantially correct calculations and explanations will receive partial credit. \item[(D)] If you need more space, use the back of the pages; clearly indicate when you have done this. \end{itemize} Do not write in the table to the right. \end{minipage} \hfill \begin{minipage}[t]{2.3in} \vspace{0pt} %\cellwidth{3em} \gradetablestretch{2} \vqword{Problem} \addpoints % required here by exam.cls, even though questions haven't started yet. \gradetable[v]%[pages] % Use [pages] to have grading table by page instead of question \end{minipage} \newpage % End of cover page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % See http://www-math.mit.edu/~psh/#ExamCls for full documentation, but the questions % below give an idea of how to write questions [with parts] and have the points % tracked automatically on the cover page. % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %http://ksuweb.kennesaw.edu/~plaval/math4381/seqlimthm.pdf \begin{questions} % Question with parts \addpoints \question[18] ({\bf Euclidean $\R^2$}) Consider the two points $P=(1,2),Q=(-1,-5)\in\R^2$ in the Euclidean plane. Solve the following parts: \noaddpoints \begin{parts} \part[5] Find the Euclidean distance between $P$ and $Q$. \vfill \part[5] Let $M=\{(x,y)\in\R^2: x=y\}$ and consider the unique line $L\sse\R^2$ equidistant to $P$ and $Q$. Determine the image of $Q$ under the isometry $r_M\circ r_L$. \vfill \part[5] Find all the fixed points of the isometry $r_M\circ r_L$. \vfill \part[3] Show that there is no line $N\sse\R^2$ such that $r_N\circ r_M\circ r_L=\mbox{id}$. \vfill \end{parts} % If you want the total number of points for a question displayed at the top, % as well as the number of points for each part, then you must turn off the point-counter % or they will be double counted. \newpage \addpoints \question[18] ({\bf $\Gamma$-Geometry for the cylinder}) Let $C=\R^2/\Gamma$ be the Euclidean cylinder, where $\Gamma=\langle t_{(1,0)}\rangle\sse\mbox{Iso}(\R^2)$ is the group generated by the translation $$t_{(1,0)}:\R^2\lr\R^2.$$ \noaddpoints % If you remove this line, the grading table will show 20 points for this problem. \begin{parts} \part[5] Draw the $\Gamma$-orbits $\Gamma P$ and $\Gamma Q$ in $\R^2$ of the two points $$P=(-1,4),Q=(3.25,-7.75)\in\R^2.$$ \vfill \part[5] Compute the distance in $C$ from $\Gamma P$ to $\Gamma Q$.\\ \vfill \part[5] Show that there are infinitely many lines in $C$ through $\Gamma P$ and $\Gamma Q$.\\ \vfill \part[3] Consider the projections $\pi(L_1),\pi(L_2)$ to $C$ of the two lines $$L_1=\{x=0\}\sse \R^2,\quad L_2=\{47x+y=4\}\sse\R^2.$$ Explicitly find all the intersections points of $\pi(L_1)$ and $\pi(L_2)$ in $C$.\\ \vfill \end{parts} \newpage \addpoints \question[18] ({\bf Spherical geometry}) Consider the 2-sphere $$S^2:=\{(x,y,z)\in\R^3:~x^2+y^2+z^2=1\},$$ endowed with the spherical distance. \noaddpoints % If you remove this line, the grading table will show 20 points for this problem. \begin{parts} \part[5] Consider the points $P=(1,0,0)\in S^2$ and $Q=\frac{1}{\sqrt{3}}(1,-1,1)$. Compute the spherical distance $d_{S^2}(P,Q)$ from $P$ to $Q$. \vfill \part[5] Let $R_{P,\pi/2},R_{Q,\pi/2}\in\mbox{Isom}(S^2)$ be the rotations of angle $\pi/2$ centered at $P$ and $Q$. Show that $R_{P,\pi/2}\circ R_{Q,\pi/2}$ is a rotation and find its center and angle. \vfill \part[5] Determine whether $R_{P,\pi/2}\circ R_{Q,\pi/2}$ is equal to $R_{Q,\pi/2}\circ R_{P,\pi/2}$. \vfill \part[3] Is there a spherical isometry $f\in\mbox{Isom}(S^2)$ such that $f\circ(R_{P,\pi/2},R_{Q,\pi/2})$ has no fixed points? \vfill \end{parts} \newpage \addpoints \addpoints \question[18] ({\bf Hyperbolic distances and lines in $\mathbb{H}^2$}) Let $P=i,Q=3+4i\in\H^2$ be points in the hyperbolic upper-half plane $\H^2$. Solve the following parts: \noaddpoints \begin{parts} \part[5] Show that $L=\{z\in\H^2:|z-4|^2=17\}\sse\H^2$ is the unique hyperbolic line through the points $P$ and $Q$. \vfill \part[5] Compute the hyperbolic distance $d_{\H^2}(P,Q)$. \vfill \part[5] Find the unique hyperbolic line $M$ equidistant to $P$ and $Q$. \vfill \part[3] Let $r_L,r_M\in\mbox{Isom}(\H^2)$ be the hyperbolic inversions along $L$ and $M$. Compute the image of $P$ and $Q$ under the composition $r_M\circ r_L$. \vfill \end{parts} % If you want the total number of points for a question displayed at the top, % as well as the number of points for each part, then you must turn off the point-counter % or they will be double counted. \newpage \addpoints \question[18] ({\bf Hyperbolic isometries in $\mathbb{H}^2$}) Consider the map $f,g:\H^2\lr\H^2$ $$f(z)= \frac{\bar{z} + 9}{3\bar{z} - 5}.$$ \noaddpoints \begin{parts} \part[5] Show that $f$ is a hyperbolic isometry and it has no fixed points. \vfill \part[5] Let $g:\H^2\lr\H^2$ be a hyperbolic isometry such that $$g\left(-\frac{21}{17} + \frac{16}{17}i\right)=i,\quad g\left(-\frac{33}{61} + \frac{64}{61}i\right)=2i,\quad g\left(-\frac{17}{13} + \frac{32}{13}i\right)=1+i.$$ Determine the image of the point $2+3i$ under the composition $f\circ g$. \vfill \part[5] Find a hyperbolic line $L\sse\H^2$ such that $f(L)=L$.\\ \vfill \part[3] Show that there exists no $n\in\N$ such that $f^n=\mbox{id}$. \vfill \end{parts} % If you want the total number of points for a question displayed at the top, % as well as the number of points for each part, then you must turn off the point-counter % or they will be double counted. \newpage \addpoints \question[10] For each of the ten sentences below, circle whether they are {\bf true} or {\bf false}. You do {\it not} need to justify your answer.\\ \noaddpoints % If you remove this line, the grading table will show 20 points for this problem. \begin{parts} \part[2] \small Two different lines in the twisted cylinder cannot intersect only at two points.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] Any orientation-reversing hyperbolic isometry must have a fixed point.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] A spherical isometry is determined by the image of three distinct points.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] For any two points $P,Q\in S^2$, there exists a unique line between them.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] Any hyperbolic triangle with vertices in $\H^2$ must have area less than $\pi$.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] There exists a regular hyperbolic hexagon whose angles are all $\pi/2$. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] For any two distinct hyperbolic lines $L,M\sse\H^2$, there exists a hyperbolic isometry $f:\H^2\lr\H^2$ with $f(L)=M$. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] If $L\sse\R^2$ is the line equidistant to $P,Q\in\R^2$, then any isometry fixing the points of $L$ must send $P$ to $Q$. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] If two hyperbolic lines $L,M$ do not intersect, then there exists a unique hyperbolic line that is perpendicular to both $L$ and $M$. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \end{parts} \end{questions} \end{document}