<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% !TEX TS-program = XeLaTeX


\documentclass[12pt]{exam}
\usepackage{sexam}
%-----------------------------------------
\newcommand{\lycee}{\sffamily Ø«Ø§Ù†ÙˆÙŠØ© Ø§Ù„Ø¯ÙƒØªÙˆØ± Ø£Ø­Ù…Ø¯ Ø¹Ø±ÙˆØ©}
\newcommand{\annee}{2018-2017}
\newcommand{\examnum}{ Ø§Ù…ØªØ­Ø§Ù† Ø§Ù„ÙØµÙ„ Ø§Ù„Ø£ÙˆÙ„ Ù…Ø§Ø¯Ø© Ø§Ù„Ø±ÙŠØ§Ø¶ÙŠØ§Øª}
\newcommand{\examdate}{\date}
\newcommand{ \duree }{Ø³Ø§Ø¹ØªØ§Ù† \yagding[ifsymclock]{151}}
\newcommand{ \niveau }{Ø³Ù†Ø© Ø«Ø§Ù„Ø«Ø© ØªØ³ÙŠÙŠØ± ÙˆØ§Ù‚ØªØµØ§Ø¯ }
%--------------------------------------------
\begin{document}
%-------------------
{ \lycee}
%-----------------
\hfill
%-----------------
{\sffamily 
 Ø§Ù„Ø³Ù†Ø© Ø§Ù„Ø¯Ø±Ø§Ø³ÙŠØ©: 
 \annee
}
%-----------------
%-----------------
$\rule{\textwidth}{1pt}$

\vspace{9pt}
\centerline{\sffamily\large  \examnum}
%-----------------
$\rule{\textwidth}{1pt}$\\
{\sffamily
Ø§Ù„Ø´Ø¹Ø¨Ø©:
 \niveau  
 %-----------------
\hfill
%-----------------
   Ø§Ù„Ù…Ø¯Ø©:
    \duree 
}

\vspace{1cm}

%Ø¨Ø¯Ø§ÙŠØ© Ø§Ù„Ø§Ø³Ø¦Ù„Ø©
%----------------------------------------------
\begin{questions}
\pointsinrightmargin     % Ø¬Ø¹Ù„ Ø§Ù„ØªÙ†Ù‚ÙŠØ· Ø¹Ù„Ù‰ Ø§Ù„ÙŠØ³Ø§Ø±
\question [6]    %Ø§Ù„ØªÙ…Ø±ÙŠÙ† Ø§Ù„Ø£ÙˆÙ„
Ø§Ù„Ù…ØªØªØ§Ù„ÙŠØ© Ø§Ù„Ø¹Ø¯Ø¯ÙŠØ© 
$(U_n)$
 Ù…Ø¹Ø±ÙØ© ÙƒÙ…Ø§ ÙŠÙ„ÙŠ: 
 $U_0=6$
   Ùˆ Ù…Ù† Ø£Ø¬Ù„ ÙƒÙ„ Ø¹Ø¯Ø¯ Ø·Ø¨ÙŠØ¹ÙŠ  
  $n$ 
   ÙØ¥Ù† :
   $U_{n+1}=\dfrac{1}{4}U_n +3$.
\begin{parts}   %  Ø¨Ø¯Ø§ÙŠØ© Ø§Ù„Ø£Ø³Ø¦Ù„Ø© Ø§Ù„Ø®Ø§ØµØ© Ø¨Ø§Ù„ØªÙ…Ø±ÙŠÙ† Ø§Ù„Ø£ÙˆÙ„
\part[2]         % Ø¹Ù„ÙŠÙ‡ Ù†Ù‚Ø·ØªÙŠÙ† Ø§Ù„Ø³Ø¤Ø§Ù„ 1
Ø£Ø­Ø³Ø¨
$U_1$ , 
$ U_2$
Ùˆ
$U_3$
 .
\part      %   ØºÙŠØ± Ù…Ù†Ù‚Ø· Ø§Ù„Ø³Ø¤Ø§Ù„ Ø§Ù„Ø«Ø§Ù†ÙŠ
\begin{subparts}
\subpart[1]   % `\textarabic{Ø§Ù„Ø³Ø¤Ø§Ù„ Ø§Ù„ÙØ±Ø¹ÙŠ Ø§Ù„Ø£ÙˆÙ„ Ø§Ù„Ø®Ø§Øµ Ø¨Ø§Ù„Ø³Ø¤Ø§Ù„ Ø±Ù‚Ù… 1}` 
Ø£Ø«Ø¨Øª Ø¨Ø§Ù„ØªØ±Ø§Ø¬Ø¹ Ø£Ù†Ù‡ Ù…Ù† Ø£Ø¬Ù„ ÙƒÙ„ Ø¹Ø¯Ø¯ Ø·Ø¨ÙŠØ¹ÙŠ $n$ : 
$U_n \geq 4$ .

\subpart         % Ø§Ù„Ø³Ø¤Ø§Ù„ Ø§Ù„ÙØ±Ø¹ÙŠ ØºÙŠØ± Ù…Ù†Ù‚Ø·
Ø¨ÙŠÙ‘Ù† Ø£Ù†Ù‘ Ø§Ù„Ù…ØªØªØ§Ù„ÙŠØ© 
$(U_n)$
Ù…ØªÙ†Ø§Ù‚ØµØ© . Ù‡Ù„ 
$(U_n)$
Ù…ØªÙ‚Ø§Ø±Ø¨Ø©ØŸ
Ø¹ÙŠÙ‘Ù† Ù†Ù‡Ø§ÙŠØªÙ‡Ø§.
\end{subparts}   % Ù†Ù‡Ø§ÙŠØ© Ø§Ù„Ø£Ø³Ø¦Ù„Ø© Ø§Ù„ÙØ±Ø¹ÙŠØ© Ø§Ù„Ø®Ø§ØµØ© Ø¨Ø§Ù„Ø³Ø¤Ø§Ù„ 2

\part      % Ø§Ù„Ø³Ø¤Ø§Ù„ Ø§Ù„Ø«Ø§Ù„Ø«
$(V_n)$
Ø§Ù„Ø¹Ø¯Ø¯ÙŠØ© Ø§Ù„Ù…Ø¹Ø±ÙØ© Ù…Ù† Ø£Ø¬Ù„ ÙƒÙ„ Ø¹Ø¯Ø¯ Ø·Ø¨ÙŠØ¹ÙŠ $n$ ÙƒÙ…Ø§ ÙŠÙ„ÙŠ :
$V_n=U_n-4$. 
\begin{subparts}
\subpart[1]   % `\textarabic{Ø§Ù„Ø³Ø¤Ø§Ù„ Ø§Ù„ÙØ±Ø¹ÙŠ Ø§Ù„Ø£ÙˆÙ„ Ø§Ù„Ø®Ø§Øµ Ø¨Ø§Ù„Ø³Ø¤Ø§Ù„ Ø±Ù‚Ù… 3}` 
Ø¨ÙŠÙ‘Ù† Ø£Ù† Ø§Ù„Ù…ØªØªØ§Ù„ÙŠØ© 
$(V_n)$
Ù…ØªØªØ§Ù„ÙŠØ© Ù‡Ù†Ø¯Ø³ÙŠØ© Ø£Ø³Ø§Ø³Ù‡Ø§ 
$q=\dfrac{1}{4}$
ÙˆØ­Ø¯Ù‡Ø§ Ø§Ù„Ø£ÙˆÙ„  
$V_0$
ØŒ Ø«Ù… Ø£ÙƒØªØ¨ Ø¹Ø¨Ø§Ø±Ø© Ø­Ø¯Ù‡Ø§ Ø§Ù„Ø¹Ø§Ù…. 
\subpart
Ø¨ÙŠÙ‘Ù† Ø£Ù†Ù‘Ù‡ Ù…Ù† Ø£Ø¬Ù„ ÙƒÙ„ Ø¹Ø¯Ø¯ Ø·Ø¨ÙŠØ¹ÙŠ $n$ Ù„Ø¯ÙŠÙ†Ø§ :
$U_n=2\left(\dfrac{1}{4}\right)^n +4$.
Ø«Ù… Ø£Ø­Ø³Ø¨
$\lim\limits_{n \rightarrow + \infty} U_n$
.
\subpart
Ø£Ø­Ø³Ø¨ Ø¨Ø¯Ù„Ø§Ù„Ø© $n$ Ø§Ù„Ù…Ø¬Ù…ÙˆØ¹ 
$S_n$
Ø­ÙŠØ« :
$S_n=U_0 + U_1 + U_2 + \ldots + U_n$.
\end{subparts}   % 

\end{parts}
 %  Ù†Ù‡Ø§ÙŠØ© Ø§Ù„ØªÙ…Ø±ÙŠÙ† Ø§Ù„Ø£ÙˆÙ„

\vspace{2cm}


\question [7]      % Ø¨Ø¯Ø§ÙŠØ© Ø§Ù„ØªÙ…Ø±ÙŠÙ† Ø§Ù„Ø«Ø§Ù†ÙŠ Ø¹Ù„ÙŠÙ‡ 7 Ù†Ù‚Ø§Ø· Ù…Ø«Ù„Ø§
Ø§Ù„Ø¬Ø¯ÙˆÙ„  Ø§Ù„ØªØ§Ù„ÙŠ ÙŠØ¹Ø·ÙŠ Ù…Ø³Ø§ÙØ© Ø§Ù„ØªÙˆÙ‚Ù Ø¨Ø§Ù„Ø£Ù…ØªØ§Ø± Ø¹Ù†Ø¯ Ø§Ù„Ø¶ØºØ· Ø¹Ù„Ù‰ Ø§Ù„Ù…ÙƒØ¨Ø­ Ù„Ø³ÙŠØ§Ø±Ø© Ù…Ø§ Ø­Ø³Ø¨ Ø§Ù„Ø³Ø±Ø¹Ø© Ø§Ù„Ù…Ø³ØªØ¹Ù…Ù„Ø©\\
 Ùˆ Ø§Ù„Ù…Ù‚Ø¯Ø±Ø© Ø¨Ù€ :
$\mathtt{Km/h}$.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 
$100$ &amp; $90$ &amp; $80$ &amp; $70$ &amp; $60$ &amp; $50$ &amp; $40$ &amp; Ø§Ù„Ø³Ø±Ø¹Ø© $x_i$ \\ 
\hline 
$85.4$ &amp; $70.7$ &amp; $57.5$ &amp; $46$ &amp; $35.7$ &amp; $26.5$ &amp; $18.6$ &amp; Ø§Ù„Ù…Ø³Ø§ÙØ© $y_i$ \\ 
\hline 
\end{tabular} 
\end{table}
\begin{parts}     % Ø¨Ø¯Ø§ÙŠØ© Ø§Ù„Ø£Ø³Ø¦Ù„Ø©
\part[2]     %Ø§Ù„Ø³Ø¤Ø§Ù„ Ø§Ù„Ø£ÙˆÙ„  Ù„Ù‡ Ø¹Ù„Ø§Ù…ØªØ§Ù†
Ù…Ø«Ù„ Ø³Ø­Ø§Ø¨Ø© Ø§Ù„Ù†Ù‚Ø· ÙÙŠ Ù…Ø¹Ù„Ù… Ù…ØªØ¹Ø§Ù…Ø¯ Ùˆ Ù…ØªØ¬Ø§Ù†Ø³ 
$(O;\vec{i};\vec{j})$
ØŒ
Ø§Ù„ÙˆØ­Ø¯Ø© :
\quad
$
\begin{cases}
1 \mathtt{Cm} &amp;\rightarrow 10 \, \mathtt{ Km/h}\\
1 \mathtt{Cm} &amp;\rightarrow 10 \,\mathtt{ m}
\end{cases}
$
\part
\begin{subparts}
\subpart[1]
Ø¹ÙŠÙ† Ø§Ø­Ø¯Ø§Ø«ÙŠØ§ Ø§Ù„Ù†Ù‚Ø·Ø© Ø§Ù„Ù…ØªÙˆØ³Ø·Ø© 
$G$ 
ØŒ
 Ø«Ù… Ù…Ø«Ù„Ù‡Ø§ ÙÙŠ Ù†ÙØ³ Ø§Ù„Ù…Ø¹Ù„Ù… . 
\subpart[2]
Ø¨ÙŠÙ‘Ù† Ø£Ù†Ù‘ Ù…Ø¹Ø§Ù…Ù„ ØªÙˆØ¬ÙŠÙ‡ Ù…Ø³ØªÙ‚ÙŠÙ… Ø§Ù„Ø§Ù†Ø­Ø¯Ø§Ø± Ø¨Ø§Ù„Ù…Ø±Ø¨Ø¹Ø§Øª Ø§Ù„Ø¯Ù†ÙŠØ§ Ù‡Ùˆ  
$a=1.11$
 ØŒ
 Ø§Ù†Ø´Ø¦ Ù‡Ø°Ø§ Ø§Ù„Ù…Ø³ØªÙ‚ÙŠÙ….
\end{subparts}
\part     \hfill     %  \hfill Ø§Ø°Ø§ Ø§Ø±Ø¯Ù†Ø§ Ø§Ù„Ø³Ø¤Ø§Ù„ Ø§Ù„ÙØ±Ø¹ÙŠ ÙŠÙƒÙˆÙ† ÙÙŠ Ø³Ø·Ø± Ù…Ø³ØªÙ‚Ù„ Ù†Ø³ØªØ¹Ù…Ù„ 
\begin{subparts}
\subpart[1]
ÙƒÙ… Ø³ØªÙƒÙˆÙ† Ù…Ø³Ø§ÙØ© Ø§Ù„ØªÙˆÙ‚Ù Ø¹Ù†Ø¯ Ø§Ø³ØªØ¹Ù…Ø§Ù„ Ø§Ù„Ø³Ø±Ø¹Ø© 
$160\, \mathtt{Km/h}$ØŸ
\subpart[1]
 Ø£ÙˆÙ‚ÙØª Ø§Ù„Ù…ØµØ§Ù„Ø­ Ø§Ù„Ù…Ø®ØªØµØ© Ø£Ø­Ø¯ Ø§Ù„Ø³Ø§Ø¦Ù‚ÙŠÙ† Ùˆ Ø¨Ø¹Ø¯ ØªØ³Ø¨Ø¨Ù‡ ÙÙŠ Ø­Ø§Ø¯Ø« Ù…Ø±ÙˆØ± Ùˆ Ø¨Ø¹Ø¯ Ø­Ø³Ø§Ø¨ Ø§Ù„Ù…Ø³Ø§ÙØ©\\
  ÙˆØ¬Ø¯ÙˆÙ‡Ø§ 
 $230\,\mathtt{m}$
\\
$-$ 
\,
Ø¨Ø§Ø³ØªØ¹Ù…Ø§Ù„ Ø§Ù„ØªØ¹Ø¯ÙŠÙ„ Ø§Ù„Ø³Ø§Ø¨Ù‚ Ø£ÙˆØ¬Ø¯ Ø§Ù„Ø³Ø±Ø¹Ø© Ø§Ù„ØªÙŠ ÙƒØ§Ù† ÙŠØ³ÙˆÙ‚ Ø¨Ù‡Ø§ Ø§Ù„Ø³Ø§Ø¦Ù‚         
\quad
                  ( ØªØ¯ÙˆØ± Ø§Ù„Ù‚ÙŠÙ… Ø¥Ù„Ù‰  $10^{-2}$ ).         
\end{subparts}
\end{parts}


\newpage   % Ø§Ù„ØªÙ…Ø±ÙŠÙ† Ø§Ù„Ø«Ø§Ù„Ø« ÙŠÙƒÙˆÙ† ÙÙŠ ØµÙØ­Ø© Ø¬Ø¯ÙŠØ¯Ø© Ù…Ø«Ù„Ø§

\question[7]     % Ø§Ù„ØªÙ…Ø±ÙŠÙ† Ø§Ù„Ø«Ø§Ù„Ø« 7 Ù†Ù‚Ø§Ø·
Ù„ØªÙƒÙ† Ø§Ù„Ø¯Ø§Ù„Ø© 
$f$
Ø§Ù„Ù…Ø¹Ø±ÙØ© Ø¹Ù„Ù‰    
$\left] 1,+\infty \right[ $
  Ø¨Ø§Ù„Ø¹Ø¨Ø§Ø±Ø©: 
  $$f(x)=\dfrac{-x^2+4x-1}{x-1}$$
  Ùˆ Ù„ÙŠÙƒÙ† $(C_{f})$ ØªÙ…Ø«ÙŠÙ„Ù‡Ø§ Ø§Ù„Ø¨ÙŠØ§Ù†ÙŠ ÙÙŠ Ù…Ø³ØªÙˆÙŠ Ù…Ù†Ø³ÙˆØ¨ Ø§Ù„Ù‰ Ù…Ø¹Ù„Ù… Ù…ØªØ¹Ø§Ù…Ø¯ Ùˆ Ù…ØªØ¬Ø§Ù†Ø³
$(O;\vec{i};\vec{j})$ 
  .
 \begin{parts} 
\part
Ø¹ÙŠÙ† Ù†Ù‡Ø§ÙŠØªÙŠ Ø§Ù„Ø¯Ø§Ù„Ø©  $f$ Ø¹Ù†Ø¯ Ø£Ø·Ø±Ø§Ù Ù…Ø¬Ø§Ù„ ØªØ¹Ø±ÙŠÙÙ‡Ø§ .
\part
Ø¹ÙŠÙ† Ø§Ù„Ø§Ø¹Ø¯Ø§Ø¯ Ø§Ù„Ø­Ù‚ÙŠÙ‚ÙŠØ© 
$b$ , $a$
Ùˆ 
$c$
Ø¨Ø­ÙŠØ« ÙŠÙƒÙˆÙ† Ù…Ù† Ø£Ø¬Ù„ ÙƒÙ„ 
$x$
Ù…Ù†
$] 1,+\infty [ $
:
  $$f(x)=ax+b+\dfrac{c}{x-1}$$
   \part
   Ø¨ÙŠÙ† Ø£Ù†Ù‡ Ù…Ù† Ø£Ø¬Ù„ ÙƒÙ„ $x$ 
   Ù…Ù†
$\left] 1,+\infty \right[ $ :
$$f'(x)=\dfrac{-x^2+2x-3}{(x-1)^2}$$
\part   
  Ø£Ø¹Ø· Ø¬Ø¯ÙˆÙ„ ØªØºÙŠØ±Ø§Øª Ø§Ù„Ø¯Ø§Ù„Ø© 
  $f$.
   \part
    Ø£Ø«Ø¨Øª Ø£Ù†Ù‘ Ø§Ù„Ù…Ø³ØªÙ‚ÙŠÙ… $(D)$ Ø°Ùˆ Ø§Ù„Ù…Ø¹Ø§Ø¯Ù„Ø© 
   $y=-x+3$
   Ù…Ø³ØªÙ‚ÙŠÙ… Ù…Ù‚Ø§Ø±Ø¨ Ù„Ù…Ù†Ø­Ù†ÙŠ Ø§Ù„Ø¯Ø§Ù„Ø©   
    $f$
Ø¨Ø¬ÙˆØ§Ø± 
$+\infty$    
    .
    \part
Ø£Ø¯Ø±Ø³ Ø§Ù„ÙˆØ¶Ø¹ Ø§Ù„Ù†Ø³Ø¨ÙŠ Ù„Ù„Ù…Ù†Ø­Ù†ÙŠ $(C_{f})$  Ùˆ Ø§Ù„Ù…Ø³ØªÙ‚ÙŠÙ…  $(D)$ 
.
       \part
  Ø¨ÙŠÙ† Ø£Ù† Ø§Ù„Ù…Ø¹Ø§Ø¯Ù„Ø© :
  $f(x)=0$
   ØªÙ‚Ø¨Ù„ Ø­Ù„Ø§Ù‹  ÙˆØ­ÙŠØ¯Ù‹Ø§
   $\alpha$ 
  ÙÙŠ Ø§Ù„Ù…Ø¬Ø§Ù„ 
  $\left[ 3.5;4 \right] $.
        \part
     Ø£Ø±Ø³Ù… Ø§Ù„Ù…Ø³ØªÙ‚ÙŠÙ…
     $ (D)$
   Ùˆ  Ø§Ù„Ù…Ù†Ø­Ù†ÙŠ 
     $(C_{f})$ 
. 
  \end{parts}
\end{questions}     % Ù†Ù‡Ø§ÙŠØ© Ø§Ù„ØªÙ…Ø§Ø±ÙŠÙ†
\end{document}</pre></body></html>