% 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 III}} \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} % 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 Properties of $\Gamma\sse\mbox{Iso}(\R^2)$}) Solve the following parts: \noaddpoints % If you remove this line, the grading table will show 20 points for this problem. \begin{parts} \part[5] Let $L=\{(x,y)\in\R^2:x=y\}$ and $M=\{(x,y)\in\R^2:x=6\}$. Find an element $g\in\Gamma=\langle \overline{r}_{L},\overline{r}_{M}\rangle$ which has a {\it unique} fixed point. \vfill \part[5] Draw the $\Gamma$-orbit of the point $(0,0)$ where $\Gamma:=\langle t_{(2,3)},t_{(-1,0)}\rangle$. \vfill \part[5] Give an instance of a line $L\in\R^2$ such that its image under the quotient $\R^2\lr\R^2/\Gamma$ is finite, where $\Gamma:=\langle t_{(2,3)},t_{(-1,0)}\rangle$ as in $(b)$. \vfill \part[5] Show that any non-trivial isometry in $\Gamma:=\langle t_{(2,3)},\overline{r}\circ t_{(1,0)}\rangle$ cannot have fixed points. %\vfill %\part[5] Find two elements $g_1,g_2$ in the group %$$\Gamma:=\langle t_{(-4,6)},t_{(-3,9)},t_{(5,-15)},t_{(2,-3)},t_{(-1,3)},t_{(1,-1.5)}\rangle$$ which generate $\Gamma$, i.e. such that $\Gamma=\langle g_1,g_2\rangle$. \vfill \end{parts} \newpage \addpoints \question[20] ({\bf Reflections in $\R^2$}) Consider the two lines $L_0=\{y=0\},L_1=\{x=y\}\sse\R^2$ and the two lines $M_0=\{x=y+1\},M_1=\{x=-y+1\}\sse\R^2$. \noaddpoints % If you remove this line, the grading table will show 20 points for this problem. \begin{parts} \part[5] Show that the only fixed point of the isometry $\overline{r}_{L_1}\circ\overline{r}_{L_0}$ is $(0,0)$. \vfill \part[5] Prove that the isometry $\overline{r}_{M_1}\circ\overline{r}_{M_0}\circ\overline{r}_{L_1}\circ\overline{r}_{L_0}$ is a rotation. \vfill \part[5] Show that there exist two lines $N_0,N_1\sse\R^2$ such that $$\overline{r}_{M_1}\circ\overline{r}_{M_0}\circ\overline{r}_{L_1}\circ\overline{r}_{L_0}=\overline{r}_{N_1}\circ\overline{r}_{N_0}.$$ \vfill \part[5] Find the image of a point $(x,y)\in\R^2$ under the isometry $\overline{r}_{M_0}\circ\overline{r}_{L_1}$. \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. Solve the following parts: \noaddpoints % If you remove this line, the grading table will show 20 points for this problem. \begin{parts} \part[5] Consider the point $P=\frac{1}{\sqrt{3}}(1,1,1)$. Compute its spherical distance to each of the points $Q_1=(1,0,0),Q_2=(0,1,0)$ and $Q_3=(0,0,1)$. \vfill \part[5] Let $L\sse S^2$ be the unique line containing $P$ and $Q_1$, show that the point $\frac{1}{\sqrt{2}}(0,1,1)$ belongs to $L$.\\ \vfill \part[5] Let $E$ the unique line containing $Q_1$ and $Q_2$. Find the image of the unique line through $Q_1$ and $Q_3$ under the composition $\overline{r}_{L}\circ \overline{r}_{E}$.\\ \vfill \part[5] Find the fixed points of the isometry $f$ obtained by first applying $\overline{r}_{L}\circ \overline{r}_{E}$ and then applying the reflection $(x,y,z)\lr (-x,y,z)$.\\ \vfill \end{parts} \newpage \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] For any pair of points $P,Q\in K$ in the Klein bottle, there are infinitely many distinct lines $L\sse K$ containing $P,Q\in K$. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] There exist rotations $R_{P,\theta},R_{Q,\phi}\in\mbox{Iso}(\R^2)$ such that the composition $R_{P,\theta}\circ R_{Q,\phi}$ is {\it not} a rotation.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] Two lines $L,K\sse M$ in the twisted cylinder either intersect $0$,$1$ or infinitely many times.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] Let $\Gamma\sse\mbox{Iso}(\R^2)$ be generated by a finite number of translations. Then there exists a fundamental domain $D_\Gamma\sse\R^2$ of finite area.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] The quotient of the isometry $t_{(0,1)}:\R^2\lr\R^2$ gives a well-defined isometry in the twisted cylinder $\R^2/\langle t_{(1,0)}\circ\bar{r}\rangle$.\\ (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] There are no parallel lines in the cylinder. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] Every orientation-preserving isometry of the 2-sphere is a rotation. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] Every orientation-reserving isometry of the 2-sphere has a fixed point. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] There are parallel lines in the 2-sphere. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \part[2] The spherical distance between two points in the 2-sphere is the same as the Euclidean distance between these two points, as computed in $\R^3$. (1) True. \qquad \qquad \qquad \qquad (2) False. \vfill \end{parts} \end{questions} \end{document}