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a/TST/DS/DS_20_09_30/DS_20_10_02.tex b/TST/DS/DS_20_09_30/DS_20_10_02.tex new file mode 100644 index 0000000..86eb4ab --- /dev/null +++ b/TST/DS/DS_20_09_30/DS_20_10_02.tex @@ -0,0 +1,31 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DS 2 \hfill TST3} +\tribe{TST3} +\date{2 octobre 2020} +\duree{1h} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=2, + type=Exercise, +} + + +\begin{document} +\maketitle + +Le barème est donné à titre indicatif, il pourra être modifié. + +\input{exercises.tex} +\printcollection{banque} +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: + diff --git a/TST/DS/DS_20_09_30/DS_20_10_02_QF.pdf b/TST/DS/DS_20_09_30/DS_20_10_02_QF.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c41c418871e5f7a8e666dc03d2cf4e1d0674b6a2 GIT binary patch literal 56080 zcmce+W2|t^)-AYg8_%|F+qP}n*0XKfwr$(CZCm}G@7|NXNq5ry=bTE`N@1?b8W}TV 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b/TST/DS/DS_20_09_30/DS_20_10_02_QF.tex @@ -0,0 +1,48 @@ +\documentclass[a4paper,10pt, landscape, twocolumn]{article} +\usepackage{myXsim} + +% Title Page +\title{DS 2 \hfill TST3} +\date{2 octobre 2020} +\duree{15 minutes} + +\pagestyle{empty} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=2, + type=automatismes, +} + +\begin{document} +\input{exercises.tex} + +\maketitle + +\medskip +{\Large Nom - Prénom: \dotfill} + +\medskip +Calculatrice non autorisée + +\printcollection{banque} + +\pagebreak + +\maketitle + +\medskip +{\Large Nom - Prénom: \dotfill} + +\medskip +Calculatrice non autorisée + +\printcollection{banque} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: + diff --git a/TST/DS/DS_20_09_30/exercises.tex b/TST/DS/DS_20_09_30/exercises.tex index 80b2f2d..422e841 100644 --- a/TST/DS/DS_20_09_30/exercises.tex +++ b/TST/DS/DS_20_09_30/exercises.tex @@ -1,5 +1,5 @@ \collectexercises{banque} -\begin{exercise}[subtitle={Automatismes}, points=5, step=1, type={Auto}] +\begin{exercise}[subtitle={Automatismes}, points=5, step=1, type={automatismes}] \begin{enumerate} \item Un smartphone coûte 200€. Calculer sont prix après une réduction de 30\%. \vfill @@ -14,6 +14,21 @@ \end{enumerate} \end{exercise} +\begin{exercise}[subtitle={Automatismes}, points=5, step=2, type={automatismes}] + \begin{enumerate} + \item Un vélo coûte 300€. Calculer sont prix après une réduction de 40\%. + \vfill + \item Sur l'emballage d'un plat préparé de 350g, il est écrit 2\% de sel. Calculer la quantité de sel. + \vfill + \item Dans une entreprise de 350 personnes, 30 sont des cadres. Donner la proportion de cadre dans cette entreprise. + \vfill + \item Développer puis réduire l'expression: $6x^2 + x(x-1)$ + \vfill + \item Soit $f(x) = x^2 + 3x$. Calculer $f(-5)$. + \vfill + \end{enumerate} +\end{exercise} + \begin{exercise}[subtitle={Résultats d'une entreprise}, points=8, step=1, type={Exercise}] Soit $f$ la fonction définie sur $\intFF{0}{60}$ par $f(x) = -0,1x^2 + 6x - 50$. Cette fonction représente le résultat (en milion d'euros) que réalise une entrpirse pour la fabrication de $x$ milions de jouets. La représentation graphique $\mathcal{C}$ de la fonction $f$ représentée ci dessous. @@ -56,11 +71,52 @@ \item Déterminer pour quelle valeur de $x$ les bénéfices sont positifs. \end{enumerate} \end{enumerate} +\end{exercise} +\begin{exercise}[subtitle={Résultats d'une entreprise}, points=8, step=2, type={Exercise}] + Soit $f$ la fonction définie sur $\intFF{0}{60}$ par $f(x) = -0,1x^2 + 5,5x - 25$. Cette fonction représente le résultat (en million d'euros) que réalise une entreprise pour la fabrication de $x$ millions de jouets. La représentation graphique $\mathcal{C}$ de la fonction $f$ représentée ci dessous. + + \noindent + \begin{minipage}{0.55\textwidth} + \begin{enumerate} + \item Recherche graphique + \begin{enumerate} + \item Déterminer graphiquement le résultat maximal et le nombre de jouets fabriqués pour lequel ce maximum est atteint. + \item Résoudre graphiquement $f(x) > 35$. Interpréter votre réponse. + \end{enumerate} + \item Recherche par le calcul + \begin{enumerate} + \item Calculer $f'$ la dérivée de $f$. + \item Étudier le signe de $f'$ et en déduire les variations de $f$. + \item En déduire la valeur du maximum de $f$ ainsi que la valeur de $x$ pour laquel il est atteind. + \end{enumerate} + \end{enumerate} + \end{minipage} + \begin{minipage}{0.4\textwidth} + \begin{tikzpicture}[xscale=0.6, yscale=0.6] + \tkzInit[xmin=0,xmax=55,xstep=5, + ymin=-5,ymax=55,ystep=5] + \tkzGrid + \tkzGrid[sub, subxstep=1, subystep=1] + \tkzDrawX[label={Production}, below=-20pt] + \tkzLabelX + \tkzDrawY[label={Résultat}, left=-50pt] + \tkzLabelY + \tkzFct[domain=0:55,color=red,very thick]% + { -0.1*\x**2+5.5*\x-25 }; + \end{tikzpicture} + \end{minipage} + \begin{enumerate} + \setcounter{enumi}{2} + \item (*) Soit $g$ la fonction définie par $g(x)=-0,1x^2+7,5x-90$ qui représente les coûts de production en fonction de $x$. + \begin{enumerate} + \item Simplifier l'expression des bénéfices $b(x) = f(x) - g(x)$. + \item Déterminer pour quelle valeur de $x$ les bénéfices sont positifs. + \end{enumerate} + \end{enumerate} \end{exercise} \begin{exercise}[subtitle={Suites}, points=7, step=1, type={Exercise}] - \noindent \begin{minipage}{0.6\textwidth} \begin{enumerate} @@ -106,7 +162,54 @@ \item Écrire une programme python qui permettrait de calculer $v_{10}$. \end{enumerate} \end{enumerate} +\end{exercise} +\begin{exercise}[subtitle={Suites}, points=7, step=2, type={Exercise}] + \noindent + \begin{minipage}{0.65\textwidth} + \begin{enumerate} + \item On s'intéresse à une ruche qui n'est soumise ni au bruit ni à la pollution. Le graphique ci-contre représente l'évolution de la population en fonction des années. + + On note $n$ le numéro de l'année et $u_n$ le nombre d'abeilles à l'année $n$. + + \begin{enumerate} + \item Pourquoi peut-on estimer que la suite $(u_n)$ est arithmétique? Quelle est sa raison et son premier terme? + \item Quelle sera la population de cette ruche l'année 6? L'année 10? + \end{enumerate} + \end{enumerate} + \end{minipage} + \hfill + \begin{minipage}{0.3\textwidth} + \begin{tikzpicture}[baseline=(a.north), xscale=0.7, yscale=0.7] + \tkzInit[xmin=0,xmax=6,xstep=1, + ymin=6500,ymax=9000,ystep=500] + \tkzGrid + \tkzGrid[sub, subystep=100, subxstep=1] + \tkzDrawX[label={année}, below=-20pt] + \tkzLabelX + \tkzDrawY[label={Nombre}, left=-30pt] + \tkzLabelY + \global\edef\tkzFctLast{6600+x*450} + \foreach \va in {0, 1, ...,5}{% + \tkzDefPointByFct[draw](\va)% + } + \end{tikzpicture} + \end{minipage} + + \begin{enumerate} + \setcounter{enumi}{1} + \item On s'intéresse à une riche perturbée par la pollution et le bruit. Elle est composée initialement de \np{50000} abeilles dont la reine mais sa population diminue de 9\% par an. + \begin{enumerate} + \item Quelle est la population de cette ruche après un an de perturbation? + \item Expliquer pourquoi la population de cette ruche est multipliée par 0.91 chaque année. + \end{enumerate} + On modélise la population de cette ruche par la suite géométrique $(v_n)$ de premier terme $v_0 = \np{30000}$ et de raison $q = 0.91$ + \begin{enumerate} + \setcounter{enumii}{2} + \item Calculer $v_1$, $v_2$ et $v_3$. + \item Écrire une programme python qui permettrait de calculer $v_{10}$. + \end{enumerate} + \end{enumerate} \end{exercise}