% 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.
%
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% These lines can probably stay unchanged, although you can remove the last
% two packages if you're not making pictures with tikz.
\documentclass[11pt]{exam}
\RequirePackage{amssymb, amsfonts, amsmath, latexsym, verbatim, xspace, setspace}
%\RequirePackage{tikz, pgflibraryplotmarks}
\RequirePackage{tikz}
\usetikzlibrary{plotmarks}
% 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{tabularx}
\usepackage[margin=2.5cm,a4paper]{geometry}
% Here's where you edit the Class, Exam, Date, etc.
\newcommand{\class}{YAP501E - Engineering Mathematics}
\newcommand{\term}{Fall 2014}
\newcommand{\examnum}{MIDTERM EXAM}
\newcommand{\examdate}{November 4th, 2014}
\newcommand{\timelimit}{60 Minutes}
\newcommand{\questionspace}{0.7cm}
% 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{\class}{\examnum\ - Page \thepage\ of \numpages}{\examdate}
\runningheadrule
\begin{tabularx}{\textwidth}{X r}
\textbf{\class} & \textbf{\examdate} \\
\textbf{\term}&\\
\textbf{Time allowed: \timelimit} &\\
\end{tabularx}\\
\rule[1ex]{\textwidth}{.1pt}
This exam contains \numquestions\ problems. Check to see if any of the questions are missing. Enter all requested information on the top of your answer sheet(s). You are required to show your work on each problem on the answer sheet. You may not use your books and notes during the exam.
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%
% 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.
%
%
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\vspace{0.3cm}
\begin{center}
\bfseries \examnum
\end{center}
\vspace{0.1cm}
\begin{questions}
% -----------------------------------------------------------------
\addpoints
\question[20] Find the solution $y(x)$ for the following initial-value problem and state whether it is unique or not:
\begin{equation*}
x^2 y^\prime = x y^\prime +2x-1
\end{equation*}
with the condition: $y(-1)=2\ln (2)$.
\vspace{\questionspace}
% -----------------------------------------------------------------
\addpoints
\question[20] Reduce the given system to the Echelon form:
\begin{equation*}
\mathbf{A} = \left[\begin{array}{rrrrr}
1 & 2 & 3 & 4 & -1 \\
1 & -2 & 3 & -2 & 1 \\
2 & 0 & 6 & 2 & 0 \\
1 & 1 & 1 & 1 & 1 \\
\end{array}\right]
\end{equation*}
\vspace{\questionspace}
% -----------------------------------------------------------------
\addpoints
\question[30] Solve the given coupled differential equations (Think of it as an eigenvalue problem):
\begin{equation*}
\dot{x} = x + 4y
\end{equation*}
\begin{equation*}
\dot{y} = x + y
\end{equation*}
where $\dot{x}=\frac{dx}{dt}$ and $\dot{y}=\frac{dy}{dt}$.
\vspace{\questionspace}
% -----------------------------------------------------------------
\addpoints
\question[30] A body of mass, $m$, is dropped from rest, at time $t=$0, with its displacement $x(t)$ measured downwards from the point of release, the equation of motion is $m\cdot\frac{d^2x}{dt^2}=m \cdot g$, where $g$ is the acceleration of gravity and $t$ is the time (Free-fall problem). Find the solution $x(t)$ for this system.
\vspace{\questionspace}
\end{questions}
\end{document}