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+++ b/TST/DS/DS_21_03_22/DS_21_03_22-2.tex @@ -0,0 +1,38 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} +\usepackage{moreverb} + +% Title Page +\title{DS 7 \hfill Sujet 2} +\tribe{TST} +\date{22 mars 2021} +\duree{1h} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + %type=Exercise, + tribe=2, +} + +\newcommand{\reponse}[1]{% + \begin{bclogo}[barre=none, logo=]{Réponse} + \vspace{#1} + \end{bclogo} +} + +\pagestyle{empty} + +\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_03_22/exercises.tex b/TST/DS/DS_21_03_22/exercises.tex new file mode 100644 index 0000000..e68db35 --- /dev/null +++ b/TST/DS/DS_21_03_22/exercises.tex @@ -0,0 +1,277 @@ +\collectexercises{banque} +\begin{exercise}[subtitle={Automatismes}, points=6, tribe={1}, type={automatismes}] + Les réponses aux questions suivantes devront être justifiées. + \begin{enumerate} + \item Calculer $P_A(B)$ et interpréter le résultat + + \begin{minipage}{0.6\linewidth} + \begin{tabular}{|*{4}{p{2cm}|}c|} + \hline + & Rénovation de moins de 5ans & entre 5 et 20ans & Plus de 20ans & Total \\ + \hline + Défauts de construction & 20 & 16 & 30 & 66\\ + \hline + Pas de défauts & 24 & 10 & 5 & 39\\ + \hline + Total & 44 & 26 & 35 & 105\\ + \hline + \end{tabular} + \end{minipage} + \hfill + \begin{minipage}{0.3\linewidth} + On note + \[ + A = \left\{ \mbox{Défauts de construction} \right\} + \] + \[ + B = \left\{ \mbox{Rénovation de moins de 5ans} \right\} \qquad + \] + \end{minipage} + + + \reponse{2cm} + + \item Soit $(u_n)$ une suite arithmétique de raison $r = 5$ et de premier terme $20$. Calculer la quantité suivante + \[ + \sum_{n=0}^{50} u_n = + \] + + \reponse{2cm} + + \item De janvier à juillet, une quantité a augmenté 15\%. Faire un schéma pour illustrer la situation et calculer le taux d'évolution mensuel moyen. + \reponse{2cm} + + \item Soit $(u_n)$ une suite arithmétique de raison 2 et de premier terme 10. Compléter le programme qui calcule la somme des 15 premiers termes. + \begin{center} + \begin{minipage}{0.4\linewidth} + + u = \cdots + + S = \cdots + + for i in range(\cdots): + + \hspace{1cm} u = \cdots + + \hspace{1cm} S = \cdots + + print(\cdots) + + \end{minipage} + \end{center} + + \end{enumerate} + \pagebreak +\end{exercise} + +\begin{exercise}[subtitle={Absentéismes}, points=6, tribe={1}, type={Exercise}] + On veut étudier l'absentéisme dans une entreprise. Pour cela, le responsable des ressources humaines estime qu'un employé est absent un jour avec une probabilité de 0.001. Pour cette étude, on supposera que l'absence d'un employé n'aura aucun lien avec l'absence de ses collègues. + + On s'intéresse à une équipe de 7 employés et on compte le nombre d'absents un jour précis. On note $X$ la variable aléatoire qui compte le nombre d'absent ce jour là. + \begin{enumerate} + \item Expliquer pourquoi on peut modéliser $X$ par une loi binomiale de paramètres 7 et 0.001. + \item On souhaite calculer des probabilités. + \begin{enumerate} + \item Compléter le triangle de Pascale ci-dessous (on note $n$ le nombre de répétitions et $k$ le nombre de succès). + \begin{center} + \begin{tabular}{|*{9}{p{0.8cm}|}} + \hline + n \backslash k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ + \hline + 0 & 1 & & & & & & & \\ + \hline + 1 & & & & & & & & \\ + \hline + 2 & & & & & & & & \\ + \hline + 3 & & & & & & & & \\ + \hline + 4 & & & & & & & & \\ + \hline + 5 & & & & & & & & \\ + \hline + 6 & & & & & & & & \\ + \hline + 7 & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 \\ + \hline + \end{tabular} + \end{center} + \item Calculer les probabilités suivantes + \[ + P(X = 2) \qquad \qquad P(X=4) \qquad P(X < 2) + \] + \end{enumerate} + \item Calculer l'espérance de $X$ puis interpréter le résultat dans le cadre de cette exercice. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Recyclage des déchets}, points=8, tribe={1}, type={Exercise}] + Une usine de recyclage ouvre en 2020. La première année, elle sera capable de recycler \np{60 000} tonnes de déchets par an. Les innovations du domaines permettent de prévoir une progression de 4\% de déchets recyclé par ans. + + On modélise la capacité de recyclage de l'usine par ans par la suite $(u_n)$ où $n$ décrit le nombre d'année depuis 2020. + + \begin{enumerate} + \item Expliquer pourquoi la suite $(u_n)$ est géométrique. Préciser les paramètres. + \item Calculer $u_1$, $u_2$ et $u_10$. + \item Exprimer $u_n$ en fonction de $n$. + \item Résoudre l'équation $\np{60000} \times 1.04^n \geq 120$. Comment peut-on interpréter ce résultat dans le cadre de l'exercice? + \item Quelle quantité de déchets seront recyclés entre 2020 et 2030? + \item Recopier puis compléter le programme suivant pour qu'il calcule le nombre d'année qu'il faudra attendre avant que la capacité de recyclage dépasse \np{100 000} tonnes. + \begin{center} + \begin{minipage}{0.4\linewidth} + + n = 1 + + u = \cdots + + while u \cdots: + + \hspace{1cm} n = n + 1 + + \hspace{1cm} u = \cdots + + print(n) + + \end{minipage} + \end{center} + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Automatismes}, points=6, tribe={2}, type={automatismes}] + Les réponses aux questions suivantes devront être justifiées. + \begin{enumerate} + \item Calculer $P_B(A)$ et interpréter le résultat + + \begin{minipage}{0.6\linewidth} + \begin{tabular}{|*{4}{p{2cm}|}c|} + \hline + & Rénovation de moins de 5ans & entre 5 et 20ans & Plus de 20ans & Total \\ + \hline + Défauts de construction & 20 & 16 & 30 & 66\\ + \hline + Pas de défauts & 24 & 10 & 5 & 39\\ + \hline + Total & 44 & 26 & 35 & 105\\ + \hline + \end{tabular} + \end{minipage} + \hfill + \begin{minipage}{0.3\linewidth} + On note + \[ + A = \left\{ \mbox{Défauts de construction} \right\} + \] + \[ + B = \left\{ \mbox{Rénovation de moins de 5ans} \right\} \qquad + \] + \end{minipage} + + + \reponse{2cm} + + \item Soit $(u_n)$ une suite arithmétique de raison $r = 3$ et de premier terme $50$. Calculer la quantité suivante + \[ + \sum_{n=0}^{30} u_n = + \] + + \reponse{2cm} + + \item De janvier à juillet, une quantité a augmenté 11\%. Faire un schéma pour illustrer la situation et calculer le taux d'évolution mensuel moyen. + \reponse{2cm} + + \item Soit $(u_n)$ une suite géométrique de raison 2 et de premier terme 10. Compléter le programme qui calcule la somme des 20 premiers termes. + \begin{center} + \begin{minipage}{0.4\linewidth} + + u = \cdots + + S = \cdots + + for i in range(\cdots): + + \hspace{1cm} u = \cdots + + \hspace{1cm} S = \cdots + + print(\cdots) + + \end{minipage} + \end{center} + \end{enumerate} + \pagebreak +\end{exercise} + +\begin{exercise}[subtitle={Absentéismes}, points=6, tribe={2}, type={Exercise}] + On veut étudier l'absentéisme dans une entreprise. Pour cela, le responsable des ressources humaines estime qu'un employé est absent un jour avec une probabilité de 0.005. Pour cette étude, on supposera que l'absence d'un employé n'aura aucun lien avec l'absence de ses collègues. + + On s'intéresse à une équipe de 7 employés et on compte le nombre d'absents un jour précis. On note $X$ la variable aléatoire qui compte le nombre d'absent ce jour là. + \begin{enumerate} + \item Expliquer pourquoi on peut modéliser $X$ par une loi binomiale de paramètres 7 et 0.005. + \item On souhaite calculer des probabilités. + \begin{enumerate} + \item Compléter le triangle de Pascale ci-dessous (on note $n$ le nombre de répétitions et $k$ le nombre de succès). + \begin{center} + \begin{tabular}{|*{9}{p{0.8cm}|}} + \hline + n \backslash k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ + \hline + 0 & 1 & & & & & & & \\ + \hline + 1 & & & & & & & & \\ + \hline + 2 & & & & & & & & \\ + \hline + 3 & & & & & & & & \\ + \hline + 4 & & & & & & & & \\ + \hline + 5 & & & & & & & & \\ + \hline + 6 & & & & & & & & \\ + \hline + 7 & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 \\ + \hline + \end{tabular} + \end{center} + \item Calculer les probabilités suivantes + \[ + P(X = 3) \qquad \qquad P(X=6) \qquad P(X < 2) + \] + \end{enumerate} + \item Calculer l'espérance de $X$ puis interpréter le résultat dans le cadre de cette exercice. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Recyclage des déchets}, points=8, tribe={2}, type={Exercise}] + Une usine de recyclage ouvre en 2020. La première année, elle sera capable de recycler \np{60 000} tonnes de déchets par an. Les innovations du domaines permettent de prévoir une progression de 1\% de déchets recyclé par ans. + + On modélise la capacité de recyclage de l'usine par ans par la suite $(u_n)$ où $n$ décrit le nombre d'année depuis 2020. + + \begin{enumerate} + \item Expliquer pourquoi la suite $(u_n)$ est géométrique. Préciser les paramètres. + \item Calculer $u_1$, $u_2$ et $u_10$. + \item Exprimer $u_n$ en fonction de $n$. + \item Résoudre l'équation $\np{60000} \times 1.01^n \geq 120$. Comment peut-on interpréter ce résultat dans le cadre de l'exercice? + \item Quelle quantité de déchets seront recyclés entre 2020 et 2030? + \item Recopier puis compléter le programme suivant pour qu'il calcule le nombre d'année qu'il faudra attendre avant que la capacité de recyclage dépasse \np{100 000} tonnes. + \begin{center} + \begin{minipage}{0.4\linewidth} + + n = 1 + + u = \cdots + + while u \cdots: + + \hspace{1cm} n = n + 1 + + \hspace{1cm} u = \cdots + + print(n) + + \end{minipage} + \end{center} + \end{enumerate} +\end{exercise} + +\collectexercisesstop{banque}