% 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 Midterm Examination}} \newcommand{\examdate}{May 1 2026} \newcommand{\timelimit}{50 Minutes} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \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[20] ({\bf Rotations in $\R^2$}) Consider the two points $P=(0,0),Q=(1,0)\in\R^2$ in the Euclidean plane. Solve the following parts: \noaddpoints \begin{parts} \part[5] Let $R_{P,\pi/2}$ be a rotation of angle $\pi/2$ centered at $P$. Compute the image $R_{P,\pi/2}(3,3)$ of the point $(3,3)\in\R^2$ under the isometry $R_{P,\pi/2}$. \vfill \part[5] Let $R_{Q,-\pi/2}$ be a rotation of angle $-\pi/2$ centered at $Q$. Compute the image $R_{Q,-\pi/2}(4,5)$ of the point $(4,5)\in\R^2$ under the isometry $R_{Q,-\pi/2}$. \vfill \part[5] Let $(x,y)\in\R^2$ be any point. Where does $(x,y)\in\R^2$ get send under the composition $R_{Q,-\pi/2}\circ R_{P,\pi/2}$ ? \vfill \part[5] Show that $R_{Q,-\pi/2}\circ R_{P,\pi/2}=t_{(1,1)}$. \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[20] ({\bf $\Gamma$-Geometry for the 2-Torus}) Let $T^2=\R^2/\Gamma$ be the Euclidean Torus, where $\Gamma=\langle t_{(0,1)},t_{(1,0)}\rangle\sse\mbox{Iso}(\R^2)$ is the group generated by the two translations $$t_{(0,1)},t_{(1,0)}:\R^2\lr\R^2.$$ \noaddpoints % If you remove this line, the grading table will show 20 points for this problem. \begin{parts} \part[5] Draw the $\Gamma$-orbits of the two points $P=(2,3),Q=(0.5,-7.5)\in\R^2$. \vfill \part[5] Find a fundamental domain $D_\Gamma\sse\R^2$ which contains $P\in\R^2$.\\ \vfill \part[5] Consider $P=(2,3),Q=(0.5,-7.5)\in\R^2/\Gamma$ as points in the 2-torus. Show that the line $\{(x,y)\in T^2:x=y\}\sse T^2$ contains both $P$ and $Q$.\\ \vfill \part[5] Find {\it all} lines $L\sse T^2$ such that $P,Q\in L$.\\ \vfill \end{parts} \newpage \addpoints \question[20] ({\bf Geometry in the Twisted Cylinder}) In this problem, {\it all} points and lines are considered in the twisted cylinder $M=\R^2/\Gamma$, where $\Gamma=\langle t_{(1,0)}\circ \overline{r}\rangle\sse\mbox{Iso}(\R^2)$. Solve the following parts:\\ \noaddpoints \begin{parts} \part[5] Consider the points $P=(0,0),Q=(0.9,0.2),R=(5.9,-0.2)\in M$. Find the three distances $d(P,Q),d(P,R),d(Q,R)\in M$. \vfill \part[5] Find the intersection points between the line $\{(x,y)\in M:x=0.5\}\sse M$ and the line $\{(x,y)\in M:x=-y\}\sse M$. \vfill \part[5] Find two lines $K,L\sse M$ such that $|L\cap K|=2$. \vfill \part[5] Show that given two points $S,T\in M$ in the complement of the line $H=\{(x,y)\in M:y=0\}\sse M$, there exists a continuous path $\gamma\sse M$ from $S$ to $T$ such that $|H\cap\gamma|=0$. \vfill \end{parts} \newpage \addpoints \question[20] ({\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] Show that there exists a unique line $L\sse S^2$ containing the points $(1,0,0)$ and $(0,1,0)$. \vfill \part[5] Prove that there are infinitely many lines in $S^2$ containing the points $(1,0,0)$ and $(-1,0,0)$. \vfill \part[10] Let $R_1:S^2\lr S^2$ be the rotation with center $(1,0,0)$ and angle $\pi/2$, and $R_2:S^2\lr S^2$ be the rotation with center $(0,0,1)$ and angle $\pi/2$. Determine whether the isometry $R_1\circ R_2$ is equal to $R_2\circ R_1$. \vfill \end{parts} \newpage \addpoints \addpoints \question[20] 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] Two lines $K,L\sse T^2$ cannot intersect at more than one point.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] Let $\Gamma\sse\mbox{Iso}(\R^2)$ be generated by translations. Then an isometry $g\in\Gamma$ cannot have fixed points.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] The composition of an even number of reflections cannot be a reflection.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] A glide reflection admits infinitely many fixed points.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] For any glide reflection $\overline{r}_1\in\mbox{Iso}(\R^2)$, there exists a glide reflection $\overline{r}_2\in\mbox{Iso}(\R^2)$ such that $\overline{r}_2\circ\overline{r}_1=\mbox{Id}$.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] An isometry $f:S^2\lr S^2$ is uniquely determined by the image of three points. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] There are no isometries in the M\"obius band except for the identity. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] The product of two rotations in $S^2$ is necessarily a rotation. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] Any isometry $f:S^2\lr S^2$ can be written as the product of at most three rotations. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \end{parts} \end{questions} \end{document}