From 9f6d0019e32ebfaf995f752eaedb7102d946c4ce Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Sun, 9 May 2021 17:24:55 +0200 Subject: [PATCH] =?UTF-8?q?Feat:=20DS2=20pour=20les=20maths=20compl=C3=A9m?= =?UTF-8?q?entaires?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Complementaire/DS/DS_21_05_10/DS_21_05_10.pdf | Bin 0 -> 107831 bytes Complementaire/DS/DS_21_05_10/DS_21_05_10.tex | 183 ++++++++++++++++++ 2 files changed, 183 insertions(+) create mode 100644 Complementaire/DS/DS_21_05_10/DS_21_05_10.pdf create mode 100644 Complementaire/DS/DS_21_05_10/DS_21_05_10.tex diff --git a/Complementaire/DS/DS_21_05_10/DS_21_05_10.pdf b/Complementaire/DS/DS_21_05_10/DS_21_05_10.pdf new file mode 100644 index 0000000000000000000000000000000000000000..00e4e0ce781817cd6e28ff37c9d1fdb5507ddbf9 GIT binary patch literal 107831 zcmce-V~}RevMBttHEr9rZEM=LjcMDqZQHgrZQJf?+uywBoE>o@?iX?IpS>%hYE@R| zs;FF2E%{7a!| 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+\usepackage{ifthen} + + +% Title Page +\title{DS 2} +\tribe{Math complémentaires} +\date{10 mai 2021} +\duree{1h} + +\newcounter{mycount} +\newcommand\tablelog{% +\setcounter{mycount}{0} + +\begin{multicols}{5} + \whiledo{\value{mycount}<250} + { + \stepcounter{mycount}% + \stepcounter{mycount}% + \makebox[4em]{\themycount}% steps the counter and typesets the value of t + \FPln{\natlogoft}{\themycount} + \FPeval{\res}{round(\natlogoft, 3)}\res\\ + }% calculates Ln(t) and typsets it +\end{multicols} +} + +\begin{document} +\maketitle + +Le barème est donné à titre indicatif, il pourra être modifié. + +\begin{exercise}[subtitle={Questions diverses}, points=5, tribe={complémentaire}, type={Exercise}, origin={ES - avril 2016 Pondichery}, tribe={1}] + Dans cet exercices les questions sont indépendantes et peuvent être répondues séparément. + \begin{enumerate} + \item Réaliser la multiplication $8\times 28$ en utilisant la table de logarithme ci-dessous (vous devrez détailler les étapes) + + \tablelog + \item Ci-dessous vous trouverez le nombre de personnes hospitalisées entre le moi de mai et de juillet 2020 à cause du covid. + + \begin{tabular}{|*{7}{c|}} + \hline + Date & 2020-05-01 & 2020-05-15 & 2020-06-01 & 2020-06-15 & 2020-07-01 & 2020-07-15 \\ + \hline + Hospitalisés & 25809 & 19801 & 14237 & 10707 & 8291 & 6873 \\ + \hline + \end{tabular} + \begin{enumerate} + \begin{multicols}{2} + \item Placer les données sur le graphique suivant + + \begin{tikzpicture}[baseline=(a.north), xscale=1.2, yscale=0.8] + \tkzInit[xmin=0,xmax=5,xstep=1, + ymin=0,ymax=30000,ystep=5000] + \tkzGrid + \tkzDrawY[label=Hospitalisés, right=1pt] + \tkzLabelY + \tkzDrawX[label=Date] + \end{tikzpicture} + \item Placer les données sur le graphique suivant en prenant le logarithme du nombre d'hospitalisations. + + \begin{tikzpicture}[baseline=(a.north), xscale=1.2, yscale=0.7] + \tkzInit[xmin=0,xmax=5,xstep=1, + ymin=0,ymax=6,ystep=1] + \tkzGrid + \tkzDrawY[label=$\log$(Hospitalisés), right=1pt] + \tkzLabelY + \tkzDrawX[label=Date] + \end{tikzpicture} + \end{multicols} + \item Nommer les deux courbes obtenues et expliquer en une phrase l'intérêt du graphique de droite. + \end{enumerate} + + \item Résoudre l'inéquation suivante + \[ + 3e^{x+1} - 4 = 10 + \] + + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Baccalauréat}, points=5, tribe={complémentaire}, type={Exercise}, origin={ES - avril 2016 Pondichery}, tribe={1}] + On dispose des renseignements suivants à propos du baccalauréat session 2015 : + + \setlength\parindent{1cm} + \begin{itemize} + \item[$\bullet~~$]49\,\% des inscrits ont passé un baccalauréat général, 20\,\% un baccalauréat technologique et les autres un baccalauréat professionnel; + \item[$\bullet~~$]91,5\,\% des candidats au baccalauréat général ont été reçus ainsi que 90,6\,\% des candidats au baccalauréat technologique. + \end{itemize} + \setlength\parindent{0cm} + \begin{flushright} + \emph{Source: DEPP (juillet 2015)}\end{flushright} + + On choisit au hasard un candidat au baccalauréat de la session 2015 et on considère les évènements suivants :\index{probabilités} + + \setlength\parindent{1cm} + \begin{itemize} + \item[$\bullet~~$]$G$ : \og Le candidat s'est présenté au baccalauréat général \fg ; + \item[$\bullet~~$]$T$: \og Le candidat s'est présenté au baccalauréat technologique\fg ; + \item[$\bullet~~$]$S$: \og Le candidat s'est présenté au baccalauréat professionnel\fg ; + \item[$\bullet~~$]$R$: \og Le candidat a été reçu \fg. + \end{itemize} + \setlength\parindent{0cm} + + Pour tout évènement $A$, on note $P(A)$ sa probabilité et $\overline{A}$ son évènement contraire. + + De plus, si $B$ est un autre évènement, on note $P_B(A)$ la probabilité de $A$ sachant $B$. + + \medskip + + \begin{enumerate} + \item Préciser les probabilités $P(G), P(T), P_T(R)$ et $P_G(R)$. + \item Traduire la situation par un arbre pondéré. On indiquera les probabilités trouvées à la question précédente. Cet arbre pourra être complété par la suite.\index{arbre de probabilités} + \item Vérifier que la probabilité que le candidat choisi se soit présenté au baccalauréat technologique et l'ait obtenu est égale à \np{0,1812}. + \item Le ministère de l'Éducation Nationale a annoncé un taux global de réussite pour cette session de 87,8\,\% pour l'ensemble des candidats présentant l'un des baccalauréats. + \begin{enumerate} + \item Vérifier que la probabilité que le candidat choisi se soit présenté au baccalauréat professionnel et l'ait obtenu est égale à \np{0,24845}. + \item Sachant que le candidat s'est présenté au baccalauréat professionnel, déterminer la probabilité qu'il ait été reçu. On donnera une valeur approchée du résultat au millième. + \end{enumerate} + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Toboggan}, points=7, tribe={complémentaire}, type={Exercise}, origin={ES - mai 2016 Liban}, tribe={1}] + Soit $f$ la fonction définie sur l'intervalle [3~;~13] par : + + \[ f(x) = - 2x + 20 - \text{e}^{-2x + 10}.\] + + \textbf{Partie A : Étude de la fonction \boldmath $f$ \unboldmath} + + \medskip + + \begin{enumerate} + \item + \begin{enumerate} + \item Montrer que la fonction dérivée $f'$, de la fonction $f$, définie pour tout $x$ de l'intervalle [3~;~13], a pour expression :\index{dérivée} + + \[f'(x) = - 2 + 2\text{e}^{-2x+10}\] + + \item Résoudre dans l'intervalle [3~;~13] l'inéquation: $-2 + 2e^{-2x + 10}$ + \item En déduire les solutions de $f'(x) \geqslant 0$. + \item En déduire le signe de $f'(x)$ sur l'intervalle [3~;~13] et dresser le tableau de variations de $f$ sur cet intervalle. + \end{enumerate} + \item + \begin{enumerate} + \item Démontrer que $F(x) = x^2 + 20x + 0.5e^{-2x + 10}$ est une primitive de $f(x)$. + \item (Bonus) Calculer l'intégrale $\displaystyle\int_3^{13} f(x)\:\text{d}x$.\index{intégrale} + + \end{enumerate} + \end{enumerate} + + \bigskip + + \textbf{Partie B : Application} + + \medskip + + Une usine fabrique et commercialise des toboggans. Sa capacité mensuelle de production est + comprise entre 300 et \np{1300}. On suppose que toute la production est commercialisée. + + Le bénéfice mensuel, exprimé en milliers d'euros, réalisé pour la production et la vente de $x$ centaines + de toboggans est modélisé sur l'intervalle [3~;~13] par la fonction $f$. + + En utilisant la partie A, répondre aux questions suivantes : + + \medskip + + \begin{enumerate} + \item Déterminer le bénéfice réalisé pour la productions et la vente de 10 toboggans. + \item Déterminer le nombre de toboggans que l'usine doit produire pour obtenir un bénéfice maximal et + donner ce bénéfice, arrondi à l'euro. + \end{enumerate} + +\end{exercise} + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: