\begin
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% This template of Exam sheet was created and customized by Prof. Kamel GUESMI from the University of Djelfa (Algeria) on 03/2024. It's Free Use License for everyone. Just make an invocation for him. May Allah bless you.
\documentclass[11pt]{exam}
\usepackage{amsmath, amssymb}
\usepackage{graphicx}
\usepackage{multirow}
\begin{document}
%-----Head of the Exam sheet
\begin{center}
\begin{tabular}{cc}
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\multirow{5}{*}{\includegraphics[scale=0.8]{Logo_uzad.png}} & \multicolumn{1}{c}{\Large{\textbf{University of Djelfa}} } \\
& \multicolumn{1}{l}{\textbf{Faculty: Sciences \& Technology}} \\
& \multicolumn{1}{l}{Departemnt: Electrical Engineering } \\
& \multicolumn{1}{l}{Exam.: Nonlinear Systems (2023/24)}\\
& \multicolumn{1}{l}{Duration: 01H30, Documents allowed \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad}\\ \hline
\end{tabular}
\end{center}
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\section*{Exercise 01 \small 1st style: Answers on separate sheet } %
Let's consider the nonlinear system:
$
\left\{
\begin{array}{l}
\dot{x}_1=x_1^2-x_1x_2 \\
\dot{x}_2=x_1^3-x_2
\end{array}
\right.
$
\begin{questions}
\question[2 \half] Compute the equilibrium points of this system.
\question[2] Determine the nature of each equilibrium point.
\question[2] Linearize the model of the system around each equilibrium point.
\end{questions}
\section*{Exercise 02 \small 2nd style: Answers on the same sheet } %
\begin{questions}
\question For \(x,y,z\) and \(n\) positive integers, show that \(x^n + y^n = z^n\).
\vspace{\stretch{40}}
\question Prove that the real part of all non-trivial zeros of the function \(\zeta(z)\) is \(\frac{1}{2}\)
\vspace{\stretch{40}}
\question Compute the following quantity \[\int_{0}^{\infty} \frac{\sin(x)}{x}\]
\vspace{\stretch{30}}
\end{questions}
\clearpage
\section*{Exercise 03 \small 3rd style: questions with subparts}%
\begin{questions}
\question Show that the equation \(x^n + y^n = z^n\) has a solution for \(x,y,z\) and \(n\) positive integers.
\begin{parts}
\part Determine the values of \(n\) according to the previous question?
\vspace{\stretch{1}}
\part If \(n=2\) show that the previous equation has a double solution.
\vspace{\stretch{1}}
\part Who is the first scientist established this equation
\vspace{\stretch{0.2}}
\begin{subparts}
\subpart Who actually proved the theorem?
\vspace{\stretch{0.2}}
\subpart How long did actually take to solve this problem?
\vspace{\stretch{0.2}}
\end{subparts}
\end{parts}
\question Prove that the real part of all non-trivial zeros of the function
\(\zeta(z)\) is \(\frac{1}{2}\)
\vspace{\stretch{1}}
\end{questions}
\section*{Exercise 04 \small 4th style: multiple choice and answers on the same sheet}
\begin{questions}
\question Which of these persons discoverd the water composition?
\begin{oneparchoices}
\choice Ibn khaldoun
\choice Ibn elhaithem
\choice El farabi
\choice None of the above
\end{oneparchoices}
\question Which of these persons was president of Algeria?
\begin{checkboxes}
\choice Houari Boumedienne
\choice Elyamine zeroual
\choice Kawther Zerogui
\choice I don't know
\end{checkboxes}
\end{questions}
\end{document}