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\documentclass[a5paper,10pt]{article}
\usepackage{myXsim}
\usepackage{tasks}
% Title Page
\title{DS 4 }
\tribe{TST}
\date{11 décembre 2020}
\duree{1h}
\DeclareExerciseCollection{banque}
\xsimsetup{
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type=Exercise,
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\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"
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\documentclass[a4paper,10pt, landscape, twocolumn]{article}
\usepackage{myXsim}
% Title Page
\title{DS 4}
\tribe{TST}
\date{11 décembre 2020}
\duree{15 minutes}
\pagestyle{empty}
\DeclareExerciseCollection{banque}
\xsimsetup{
%tribe=TST1,
type=automatismes,
}
\begin{document}
\input{exercises.tex}
\maketitle
\medskip
{\Large Nom - Prénom: \dotfill}
\medskip
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\printcollection{banque}
\vfill
\pagebreak
\maketitle
\medskip
{\Large Nom - Prénom: \dotfill}
\medskip
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\printcollection{banque}
\end{document}
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TST/DS/DS_20_12_11/exercises.tex
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\collectexercises{banque}
\begin{exercise}[subtitle={Automatismes}, points=6, tribe={TST1}, type={automatismes}]
\begin{enumerate}
\item Un article de mode est vendu en solde 135\euro. Les vendeurs expliquent qu'il est soldé à 20\%. Quel était son prix avant les soldes?
\vfill
\item Simplifier le calcul suivant
\[
\frac{5^4 \times 5^{-3}}{5^2 \times 5^6} &=
\]
\vfill
\item Soit $X$ la variable aléatoire dont la loi de probabilité est résumé pas le tableau suivant
\begin{center}
\begin{tabular}{|c|*{3}{p{2cm}|}}
\hline
$x_i$ & -3 & 2 & 5 \\
\hline
$p_i$ & 0.4 & 0.1 & 0.5 \\
\hline
\end{tabular}
\end{center}
Calculer la valeur de $P(X > 0)$.
\vfill
\item Calculer la moyenne géométrique de $a=3$ et $b=9$.
\vfill
\item Soit $(d)$ la droite d'équation $y = 4x + 1$. Est-ce que le point $A(3; 11)$ appartient à la droite?
\vfill
\end{enumerate}
\end{exercise}
\begin{exercise}[subtitle={Fonctions polynômes}, points=8, tribe={TST1}, type={Exercise}]
Soit $f$ la fonction définie sur $\intFF{0}{5}$ par
\[
f(x) = x^3 - 7,5x^2 + 18x - 11,5
\]
\begin{enumerate}
\item Calculer $f(1)$ puis interpréter.
\item
\begin{enumerate}
\item Calculer $f'(x)$ la dérivée de $f(x)$.
\item Démontrer que $x=3$ et $x=2$ sont des racines de $f'(x)$.
\end{enumerate}
Dans la suite, on supposera que l'on a $f'(x) = 3(x-3)(x-2)$.
\begin{enumerate}
\setcounter{enumii}{2}
\item Tracer le tableau de signe de $f'(x)$ et en déduire les variations de $f$ sur $\intFF{0}{5}$.
\end{enumerate}
\item (Vrai/faux) Les phrases suivantes sont-elles vraies ou fausses. Vous justifierez vos réponses.
\begin{enumerate}
\item $f$ est croissante sur l'intervalle $\intFF{0}{2}$ et décroissante sur l'intervalle $\intFF{2}{3}$.
\item $f$ est une fonction polynôme de degré 2.
\item $f(x)$ a pour maximum 2,5.
\end{enumerate}
\end{enumerate}
\end{exercise}
\begin{exercise}[subtitle={Placements}, points=7, tribe={TST1}, type={Exercise}]
\begin{enumerate}
\item Bob a placé, en 2015, \np{2000}\euro sur un compte avec un rendement de 2,5\% par ans.
\begin{enumerate}
\item Quel sera le solde du compte en 2016, et 2020?
\item Si on modélise le solde du compte par la suite $(u_n)$. Quelle est la nature de la suite? Préciser les paramètres.
\item Exprimer $u_n$ en fonction de $n$.
\end{enumerate}
\item Sa copine Sarah a fait un placement similaire qui a débuté en 2015. On note $v_n$ la suite qui modélise le solde de son compte. On supposera qu'elle est géométrique et on a les valeurs suivantes
\[
u_1 = 2500 \qquad \qquad u_3 = 2700
\]
\begin{enumerate}
\item Calculer la moyenne géométrique de $u_1$ et de $u_3$ pour démontrer que la valeur de $u_2$ est d'environ \np{2598}.
\item Démontrer que la raison de la suite est $q=1,39$.
\item En déduire le rendement du placement de Sarah.
\item Quelle était la somme que Sarah a placé sur son compte en 2015?
\end{enumerate}
\end{enumerate}
\end{exercise}
\collectexercisesstop{banque}