course_sem_exam1_library.tex

\documentclass{article}
% \usepackage{multicol}
\usepackage{amsmath,amssymb}
\usepackage[margin=3cm]{geometry}
\usepackage{graphicx,color}

\newcounter{zone}
\setcounter{zone}{0}
\newcommand{\zone}{\clearpage\refstepcounter{zone}\section*{Zone \arabic{zone}}}
\newcounter{question}
\setcounter{question}{0}
\newcounter{variant}
\newcounter{questionpoints}
\newcommand{\question}[1]{\newpage \refstepcounter{question} \setcounter{variant}{0} \setcounter{questionpoints}{#1}}
\newcommand{\variant}{\vspace{4em}\refstepcounter{variant}\noindent \arabic{question}/\arabic{variant}. (\arabic{questionpoints} point\ifnum \thequestionpoints > 1 s\fi) }
\newenvironment{answers}{\begin{enumerate}}{\end{enumerate}}
% \newenvironment{columnanswers}{\begin{multicols}{2}\begin{enumerate}}{\end{enumerate}\end{multicols}}
\newcommand{\answer}{\item }
\newcommand{\correctanswer}{\item $\bigstar$ }
\renewcommand{\theenumi}{\Alph{enumi}}
\newenvironment{solution}{{\bf Solution.} }{\vspace*{.3in}\hrule}

\begin{document}

\begin{center}
\textbf{\Large Exam}
\end{center}

\bigskip
\noindent
\begin{itemize}
\item There are 3 problems worth points as shown in each question.
\item You must not communicate with other students during this test.
\item No books, notes, calculators, or electronic devices allowed.
\item This is a 20 minute exam.
\item Do not turn this page until instructed to.
\item There are several different versions of this exam.
\end{itemize}

\bigskip\bigskip
\noindent
\textbf{\Large 1. Fill in your information:}

\bigskip
{\Large\bf
\begin{tabular}{ll}
Full Name: & \underbar{\hskip 8cm} \\[0.5em]
UIN (Student Number): & \underbar{\hskip 8cm} \\[0.5em]
NetID: & \underbar{\hskip 8cm}
\end{tabular}
}

\bigskip
\bigskip
\noindent
\textbf{\Large 2. Fill in the following answers on the Scantron form:}

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\zone

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\question{3} % worth 3 points

\variant
What is $\displaystyle
\sum_{n=0}^\infty
\frac{3^{n+1}}{4^n}$?
\begin{answers}
  \answer $1/4$
  \correctanswer $12$
  \answer $3/4$
  \answer $4$
  \answer $36$
\end{answers}
\begin{solution}
  \[ = 3 \sum_{n=0}^\infty
  \left(\frac{3}{4}\right)^n
  = 3 \frac{1}{1 - 3/4}
  = 12 \]
\end{solution}

\variant
What is $\displaystyle
\sum_{n=0}^\infty
\frac{2^{n+1}}{3^n}$?
\begin{answers}
  \answer $3$
  \answer $1/3$
  \answer $2/3$
  \correctanswer $6$
  \answer $12$
\end{answers}
\begin{solution}
  \[ = 2 \sum_{n=0}^\infty
  \left(\frac{2}{3}\right)^n
  = 2 \frac{1}{1 - 2/3}
  = 6 \]
\end{solution}

\variant
What is $\displaystyle \sum_{n=0}^\infty \frac{3^{n+1}}{5^n}$?
\begin{answers}
  \answer $2/5$
  \answer $5$
  \answer $1/5$
  \correctanswer $15/2$
  \answer $10$
\end{answers}
\begin{solution}
  \[ = 3 \sum_{n=0}^\infty \left(\frac{3}{5}\right)^n = 3 \frac{1}{1 - 3/5} = 15/2 \]
\end{solution}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\question{1}

\variant
Find the average value of the function $f(x) = \displaystyle x\,
\sqrt{2x^2+1}$ on the interval $[0,2]$.
\begin{answers}
\correctanswer $\frac{13}{6}$
\answer $\frac{\sqrt2}{3}$
\answer $\frac{\sqrt2}{6}$
\answer $\frac{13}{3}$
\answer $\frac{26}{3}$
\end{answers}
\begin{solution}
\end{solution}

\variant
Find the average value of the function $f(x) = \displaystyle x\,
\sqrt{5x^2-4}$ on the interval $[1,2]$.
\begin{answers}
\correctanswer $\frac{21}{10}$
\answer $\frac{21}{5}$
\answer $\frac{2\sqrt2}{15}$
\answer $\frac{\sqrt2}{15}$
\answer $21$
\end{answers}
\begin{solution}
\end{solution}

\variant
Find the average value of the function $f(x) = \displaystyle
\sqrt{8x+1}$ on the interval $[1,3]$.
\begin{answers}
\correctanswer $\frac{49}{12}$
\answer $\frac{3\sqrt3-1}{24}$
\answer $\frac{3\sqrt3-1}{12}$
\answer $\frac{49}{6}$
\answer $\frac{98}{3}$
\end{answers}
\begin{solution}
\end{solution}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\question{1}

\variant
Evaluate $\displaystyle\int_{\pi}^{2 \pi} x^2\cos x \, dx$.
\begin{answers}
\answer $-6\pi$
\answer $-2\pi$
\answer $0$
\answer $2\pi$
\correctanswer $6\pi$
\end{answers}
\begin{solution}
\end{solution}

\variant
Evaluate $\displaystyle\int_{\pi/2}^{3\pi/2} x^2\sin x \, dx$.
\begin{answers}
\correctanswer $-4\pi$
\answer $-8\pi$
\answer $8\pi$
\answer $0$
\answer $4\pi$
\end{answers}
\begin{solution}
\end{solution}

\end{document}