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Bertrand Benjamin 2020-11-10 09:34:58 +01:00
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\documentclass[a4paper,12pt]{article}
\usepackage{myXsim}
% Title Page
\title{DS 3}
\tribe{Terminale STI2D}
\date{12 novembre 2020}
\duree{1h}
\setlength{\columnseprule}{0}
\begin{document}
\maketitle
Le barème est donné à titre indicatif, il pourra être modifié.
\begin{exercise}[subtitle={Automatismes}, points=6]
Dans cet exerice les questions sont indépendantes.
\end{exercise}
\begin{exercise}[subtitle={Complexes}, points=6]
On note i le nombre complexe de module 1 et d'argument $\dfrac{\pi}{2}$ qui vérifie $i^2 = -1$.
\medskip
On note $z_A$, $z_B$ et $z_C$ les nombres complexes suivants
\[
z_A = -2 - 2i \qquad \qquad z_B = 2i + 4 \qquad \qquad z_C = -1 + \sqrt{3}i
\]
\begin{enumerate}
\item Calculer le conjugué de $z_A$
\item Calculer les quantités suivantes
\[
z_D = z_A + z_B \qquad z_E = z_B \times z_A \qquad z_F = \frac{z_A}{z_B}
\]
\item Calculer le module et l'argument de $z_C$.
\item Soit $Z$ le nombre complexe de module $r = 3$ et d'argument $\theta = \dfrac{2\pi}{3}$.
\item Placer les points $z_A$, $z_B$, $z_C$ et $Z$ sur le plan complexe mis en annexe.
\end{enumerate}
\end{exercise}
\begin{exercise}[subtitle={Citerne}, points=1]
\begin{minipage}{0.6\textwidth}
On se propose de fabriquer avec le moins de tôle possible une citerne fermée en forme de parallélépipède rectangle dont le volume intérieur doit être de $12m^3$. La longueur est aussi fixée à $3m$ par le cahier des charges.
On peut donc faire varier uniquement la largeur (notée $x$) et la hauteur (notée $h$) de la cuve.
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
\includegraphics[scale=0.8]{./fig/citerne}
\end{minipage}
\begin{enumerate}
\item Expliquer pourquoi quand la largeur $x$ change, la hauteur $h$ doit elle aussi changer pour respecter les contraintes.
\item Démontrer que l'on doit avoir $h = \dfrac{4}{x}$.
\item On note $S(x)$ l'aire totale de la citerne (c'est à dire la somme des aires des six faces). Montrer que l'on peut écrire
\[
S(x) = 6x + 8 + \frac{24}{x}
\]
\item Démontrer que
\[
S'(x) = \frac{6x^2-24}{x^2}
\]
\item Démontrer que
\[
S'(x) = \frac{6(x-2)(x+2)}{x^2}
\]
\item En déduire le tableau de variation de $S(x)$ sur $\intOF{0}{10}$ est .
\begin{tikzpicture}[baseline=(a.north)]
\tkzTabInit[lgt=3,espcl=3]{$x$/1, $S(x)$/2}{$0$, $2$, $10$}
\tkzTabVar{D+/ , -/ , +/ }
\end{tikzpicture}
\item Déterminer les valeurs de $x$ et $h$ correspondant à une utilisation minimal de tôle.
\end{enumerate}
\end{exercise}
\pagebreak
\center{\Large Annexe}
\vfill
\begin{tikzpicture}[baseline=(a.north), xscale=1.5, yscale=1.5]
\tkzInit[xmin=-5,xmax=5,xstep=1,
ymin=-5,ymax=5,ystep=1]
\tkzGrid
\draw (1, 0) node [below right] {1};
\draw (0, 1) node [above left] {$i$};
\draw [->, very thick] (-5, 0) -- (5, 0);
\draw [->, very thick] (0, -5) -- (0, 5);
%\tkzAxeXY
\foreach \x in {0,1,...,5} {
% dots at each point
\draw[black] (0, 0) circle(\x);
}
\end{tikzpicture}
\vfill
\end{document}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "master"
%%% End:

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