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+\DeclareExerciseCollection{banque} +\xsimsetup{ + %type=Exercise, +} + +\newcommand{\reponse}[1]{% + \begin{bclogo}[barre=none, logo=]{Réponse} + \vspace{#1} + \end{bclogo} +} + +\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_21_01_13/exercises.tex b/TST/DS/DS_21_01_13/exercises.tex new file mode 100644 index 0000000..c5d59b4 --- /dev/null +++ b/TST/DS/DS_21_01_13/exercises.tex @@ -0,0 +1,115 @@ +\collectexercises{banque} +\begin{exercise}[subtitle={Automatismes}, points=6, tribe={}, type={automatismes}] + \begin{enumerate} + \item Simplifier le calcul suivant + \begin{bclogo}[barre=none, logo=]{Réponse} + $\ds \frac{2\times10^{6} \times 10^{-3}\times 3}{5\times 10^2 \times 10^6} =$ + \vspace{2cm} + \end{bclogo} + + \item Une quantité est augmenté de 15\%. Quel taux d'évolution doit-on appliqué pour la faire revenir à sa valeur initiale? + \reponse{2.5cm} + + \item En 2010, la chiffre d'affaire d'une entreprise était de \np{15 000}. Chaque année, il a progressé de 8\%. Quel est le taux d'évolution global entre 2010 et 2020? + \reponse{2.5cm} + + \item En 2015, j'achète une voiture \np{10000}\euro. En 2019, elle a perdu 50\% de sa valeur. Quel a été la perte annuelle moyenne? + \reponse{2.5cm} + + \item Convertir $89,45m^2$ en $cm^2$ + \reponse{2cm} + + \pagebreak + + \item Convertir 3,8h en heure et minutes. + \reponse{2cm} + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Équation et logarithme}, points=4.5, tribe={}, type={Exercise}] + Résoudre par un calcul les équations et inéquations suivantes + \begin{multicols}{3} + \begin{enumerate} + \item $10^x = 150$ + \item $10^{-2x + 4} \leq 5$ + \item $2\times 10^x = 100$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{exercise}[subtitle={Production en transition}, points=9.5, tribe={}, type={Exercise}] + \noindent + Une usine qui fabrique un produit A, décide de fabriquer un nouveau produit B afin d'augmenter son chiffre d'affaires. La quantité, exprimée en tonnes, fabriquée par jour par l'usine est modélisée par : + \begin{multicols}{2} + \begin{itemize} + \item la fonction $f$ définie sur [0~;~14] par + + \[f(x) = \np{2000}\times 0.81^{x}\] + + pour le produit A ; + \item la fonction $g$ définie sur [0~;~14] par + + \[g (x)= 15x^2 + 50 x\] + + pour le produit B + \end{itemize} + \end{multicols} + Où $x$ est la durée écoulée depuis le lancement du nouveau produit B exprimée en mois. + + \noindent + \begin{minipage}{0.5\textwidth} + \textbf{Partie A} + + Leurs courbes représentatives respectives $\mathcal{C}_f$ et $\mathcal{C}_g$ sont données ci-contre. + + Par lecture graphique, sans justification et avec la précision permise par le graphique : + + + \begin{enumerate} + \item Quelle est la quantité de produit A au lancement du produit B? + \item Quelle est la quantité de produit B produite 9 mois après le lancement? + \item Déterminer la durée nécessaire pour que la quantité de produit B dépasse celle du produit A. + \item L'usine ne peut pas fabriquer une quantité journalière de produit B supérieure à \np{3000}~tonnes. + + Au bout de combien de mois cette quantité journalière sera atteinte? + \end{enumerate} + \end{minipage} + \begin{minipage}{0.5\textwidth} + \begin{tikzpicture}[xscale=0.6, yscale=1] + \tkzInit[xmin=0,xmax=14,xstep=1, + ymin=0,ymax=3500,ystep=500] + \tkzGrid + \tkzGrid[sub, subystep=100, subxstep=1] + \tkzDrawX[label={\textit{Temps (en mois)}},below=10pt] + \tkzLabelX + \tkzDrawY[label={\textit{Production (en tonnes)}}, right=10pt] + \tkzLabelY + + \tkzFct[domain=0:14,color=red,very thick]{2000*0.81**(\x)} + \tkzDefPointByFct(1) + \tkzText[above right,text=red](tkzPointResult){${\mathcal{C}}_f$} + + \tkzFct[domain=0:14,color=black,very thick]{15*\x**2 + 50*\x} + \tkzDefPointByFct(1) + \tkzText[above right,text=black](tkzPointResult){${\mathcal{C}}_g$} + + %\tkzFct[domain=0:14,color=green,very thick]{15*\x**2 + 50*\x + 2000*0.81**\x} + \end{tikzpicture} + \end{minipage} + + \noindent + \textbf{Partie B} + + \noindent + Vos réponses aux questions suivantes ne pourront pas être justifiées à l'aide du graphique. + + \noindent + \begin{enumerate} + \item Calculer et interpréter $f(4)$ et $g(4)$. + \item Quelle sera la production de produit A 9 mois après le lancement? + \item À partir de la formule de $f(x)$ justifier que la fonction est décroissante. + \item Combien de temps faut-il attendre pour que la production de produit A soit inférieur à 400tonnes. + \item (*) Combien de temps faut-il attendre pour que la production totale soit supérieur à 2100 tonnes? + \end{enumerate} +\end{exercise} +\collectexercisesstop{banque}