DST 2nde, activité 2nde
authorDenise sur Lya <sekhmet@lya>
Fri, 27 May 2016 21:06:58 +0000 (23:06 +0200)
committerDenise sur Lya <sekhmet@lya>
Fri, 27 May 2016 21:06:58 +0000 (23:06 +0200)
12 files changed:
2nde/DST_8/cube.ggb [new file with mode: 0644]
2nde/DST_8/dst8.tex
2nde/DST_8/dst8_A_corrige.pdf [new file with mode: 0644]
2nde/DST_8/dst8_A_enonce.pdf [new file with mode: 0644]
2nde/DST_8/dst8_B_corrige.pdf [new file with mode: 0644]
2nde/DST_8/dst8_B_enonce.pdf [new file with mode: 0644]
2nde/DST_8/fond_graphe.pdf [new file with mode: 0644]
2nde/DST_8/fond_graphe.svg [new file with mode: 0644]
2nde/chapitre11_probas/activite_arbre.pdf [new file with mode: 0644]
2nde/chapitre11_probas/activite_arbre.tex [new file with mode: 0644]
2nde/chapitre11_probas/etudejeu.pdf [new file with mode: 0644]
2nde/chapitre11_probas/etudejeu.svg [new file with mode: 0644]

diff --git a/2nde/DST_8/cube.ggb b/2nde/DST_8/cube.ggb
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@@ -1,7 +1,7 @@
 \input{../../header.tex}
 
 \title{\vspace{-1.5cm}Devoir sur table numéro 8 -- classe de 2nde 6 \vspace{-1.5cm}}
-\date{\dateoucorr{27/05/2016} -- \version{A}{B}}
+\date{\dateoucorr{27/05/2016} -- Sujet \version{A}{B}}
 
 \begin{document}
 
@@ -17,6 +17,10 @@ Vecteurs
 
 \bigskip
 
+Fonctions 
+
+\bigskip
+
 Géométrie dans l'espace
 
 \bigskip
@@ -27,32 +31,35 @@ Géométrie dans l'espace
 \begin{enumerate}
 \item $A(x)$, $B(x)$, $C(x)$ sont trois expressions dont on donne les tableaux de signe :
 
-\begin{minipage}{0.3\textwidth}
+\begin{minipage}{0.4\textwidth}
 \variations
 x & \mI & & -1 & & \pI \\
 A(x) & \ga- & \z & \dr+ \\ 
 \fin
 \end{minipage}
-\begin{minipage}{0.3\textwidth}
+\begin{minipage}{0.4\textwidth}
+\variations
 x & \mI & & \frac{3}{2} & & \pI \\
 B(x) & \ga+ & \z & \dr- \\ 
 \fin
 \end{minipage}
-\begin{minipage}{0.3\textwidth}
+
+%\begin{minipage}{0.3\textwidth}
+\variations
 x & \mI & & 2 & & 3 & & \pI \\
-C(x) & \ga- & \z & + & \z \dr- \\ 
+C(x) & \ga- & \z & + & \z \dr- \\ 
 \fin
-\end{minipage}
+%\end{minipage}
 
 En utilisant ces tableaux de signe, dresser les tableaux de signe des expressions suivantes :
 \begin{enumerate}
 \item \version{$A(x)B(x)$}{$A(x)C(x)$} \bareme{0.75}
 \item \version{$B(x)C(x)$}{$A(x)B(x)$} \bareme{0.75}
-\item $\frac{\version{B(x)}{A(x)}}{C(x)}$ \bareme{1}
+\item $\frac{\version{A(x)}{B(x)}}{C(x)}$ \bareme{1}
 \end{enumerate}
 
 \item $f$ est la fonction définie par $f(x) = \version{(x-1)(-2x-1)}{(x-2)(-3x + 9)}$. Résoudre l'inéquation $f(x) <0$.\bareme{1,5}
-\item $g$ est la fonction définie par $g(x) = \version{\frac{2x-7}{-5x + 10}}{\frac{-5x+10}{2x-1}}$. Résoudre l'inéquation $g(x) \geq 0$, après avoir déterminé la ou les valeurs interdites. \bareme{1,5}
+\item $g$ est la fonction définie par $g(x) = \version{\frac{2x-7}{-5x + 10}}{\frac{-5x+10}{2x-1}}$. Résoudre l'inéquation $g(x) \geq 0$ et déterminer la ou les valeurs interdites. \bareme{1,5}
 
 \end{enumerate}
 
@@ -60,34 +67,64 @@ En utilisant ces tableaux de signe, dresser les tableaux de signe des expression
 \begin{enumerate}
 \item Vecteurs et coordonnées :
 \begin{enumerate}
-\item Dans un repère, on a $\vu(3; -2)$ et $\vv(5; 2)$. Calculer les coordonnées du vecteur $2\vu_1 - \vu_2$.\bareme{0,5} % 1; -6 
-\item Montrer que ce vecteur est colinéaire au vecteur $\vw \left( -\frac{1}{2}; 3 \right)$.\bareme{0,5}
+\item Dans un repère, on a $\vu(3; -2)$ et $\vv(5; 2)$. Calculer les coordonnées du vecteur $2\vu - \vv$.\bareme{0,5} % 1; -6 
+\item Montrer que ce vecteur est colinéaire au vecteur $\vw \left( \version{-\frac{1}{3} ; 2}{-\frac{1}{2}; 3} \right)$.\bareme{0,5}
 %\item Calculer les coordonnées d'un vecteur $\w'$ tel que $\vu + \vv + \vw' = \ve{0}$. \bareme{1} 
 %\item $A(2; 1)$. Le point $B$ est tel que $\ve{AB} = \vu$. Trouver les coordonnées de $B$. \bareme{0,5} %(5; 1)
 %\item On a $C(-1; -1)$ et $D(-4; 1)$. Montrer que $ABCD$ est un parallélogramme. \bareme{1}
 \end{enumerate}
 \item Relation de Chasles : simplifier au maximum les sommes et produits suivants : \bareme{0,5 chq}
 \begin{enumerate}
-       \item $\ve{CH} + \ve{AB} +\ve{BC} = $
-       \item $\ve{MT} - 2\ve{MG} + 3\ve{QM} = $
-       \item 3($\ve{FK} - \ve{SP}) + 2(2\ve{KP} - \ve{FS}) = $
+       \item \version{$\ve{CH} + \ve{AB} +\ve{BC}$}{ $\ve{DH} + \ve{AB} + \ve{BD}$ }
+       \item \version{ $\ve{MT} - 2(\ve{MG} - \ve{TG}) $}{ $ \ve{MT} - 2(\ve{MA} - \ve{TA} )$ }
+%      \item 3($\ve{FK} - \ve{SP}) + 2(2\ve{KP} - \ve{FS})$
 \end{enumerate}
-\item On a $A$, $B$, $C$. Trouver les coordonnées d'un point $D$ tel que $ABCD$ soit un parallélogramme. \bareme{1}
-\item Les points $A(-1; -1)$, $B(1; -2)$ et $C(-4; 0,5) $ sont-ils alignés ? Justifier \bareme{1} %2; -1 et -3; +3/2
-\item $E(4; x)$. Trouver $x$ tel que $(AC) // (CE)$. \bareme{1}
+\item On a \version{$P(-1; 1)$, $A(2; 2)$, $R(6; 0)$, $L(3; -1)$}{ $P(-2; 2)$, $A(1; 3)$, $R(5; 1)$, $L(2; 0)$ }
+\begin{enumerate}
+  \item Montrer que $PARL$ est un parallélogramme. \bareme{1}
+  \item \version{$U(1, -3)$}{$U(0; -2)$}. Trouver les coordonnées de $M$ telles que $PLUM$ soit un parallélogramme. \bareme{1}
+  \item Que dire du quadrilatère $MARU$ ? Justifier. \bareme{1}
+  \item Les points $L$, $U$ et $R$ sont-ils alignés ? Justifier. \bareme{1}
+  \item Trouver les coordonnées d'un point $S$, de même abscisse que $L$, tel que $(LU) // (RS)$ \bareme{1}
+\end{enumerate}
+
+%\item $ABCD$ est un parallélogramme, et $EFCD$ est un parallélogramme. En utilisant des égalités de vecteurs, montrer que $ABFE$ est un parallélogramme. \bareme{1,5} % AB = DC, EF = DC. Donc AB = EF donc ABFE paraléllo.
+
+\end{enumerate}
+
+\section{Fonctions (3,5)}
+$f$ est une fonction dont on donne le tableau de variations :
+
+\begin{minipage}{9cm}
+\variations
+\version{x & -2 & & 3 & & 4 & & 7 \\}
+{x & -4 & & 1 & & 2 & & 4 \\}
+f(x) & \h{4} & \d & \b{-2} & \c & \h{0} & \d & \b{-1} \\
+\fin
+\end{minipage}
+\begin{minipage}{8cm}\includegraphics[width=8cm]{fond_graphe.pdf}\end{minipage}
+
+\begin{enumerate}
+\item Peut-on dire que $f$ est décroissante sur \version{$[-2; 7]$}{$[-4; 4]$} ? Justifier brièvement. \bareme{0.5}
+\item Quel est le maximum de $f$ sur son ensemble de définition, et où est-il atteint ? \bareme{0.5}
+\item Quel est le minimum de $f$ sur $\left[-1; \frac{\version{7}{3}}{2} \right]$, et où est-il atteint ? \bareme{0.5}
+\item On apprend que $f(0) = 0$. Étudier le signe de $f$ (sous forme de tableau ou avec des phrases). \bareme{1}
+\item Dessiner ci-dessus un graphique possible pour $f$. \bareme{1}
 
-\item $ABCD$ est un parallélogramme, et $EFCD$ est un parallélogramme. En utilisant des égalités de vecteurs, montrer que $ABFE$ est un parallélogramme. \bareme{1,5} % AB = DC, EF = DC. Donc AB = EF donc ABFE paraléllo.
 
-\end{itemize}
+\end{enumerate}
+
 
-\section{Géométrie}
-\begin{minipage}{6cm}\includegraphics[width=6cm]{cubex.pdf}\end{minipage}
-\begin{minipage}{10cm}
-$ABCDEFGH$ est un cube de côté \ucm{\version{5}{4}}. On place les points $A'$, $C'$ et $F'$ sur les arêtes $[BA]$, $[BC]$ et $[BF]$, de telle sorte que $[BA'] = [BC'] = [BF'] = \ucm{x}$ (voir la figure). On \og coupe \fg et on enlève le petit tétraèdre $BA'C'F'$, et on s'intéresse à l'\souligne{aire} de la figure restante, qu'on notera $A(x)$.
+\section{Géométrie dans l'espace (4)}
+\begin{minipage}{4.5cm}\includegraphics[width=4cm]{cubex.pdf}\end{minipage}
+\begin{minipage}{12cm}
+$ABCDEFGH$ est un cube de côté \ucm{\version{5}{4}}. On place les points $A'$, $C'$ et $F'$ sur les arêtes $[BA]$, $[BC]$ et $[BF]$, de telle sorte que $[BA'] = [BC'] = [BF'] = \ucm{x}$ (voir la figure). On \og coupe \fg et on enlève le petit tétraèdre $BA'C'F'$, et on s'intéresse à l'\souligne{aire} de la figure restante ($AA'C'CDF'EFGH$), qu'on notera $A(x)$.
 \end{minipage}
-\begin{itemize}
+\begin{enumerate}
 \item Dans quel intervalle peut varier $x$ ? \bareme{0,5}
-\item Calculer l'aire lorsque $x=0$, et $x=\version{5}{4}$. Aide : vous pouvez utiliser (sans avoir besoin de la démontrer), la propriété suivante : l'aire d'un triangle équilatéral de côté $c$ est $c^2 \sqrt{3}{4}$. \bareme{1,5}
+\item Calculer l'aire lorsque $x=0$, et $x=\version{5}{4}$. 
+
+\emph{Aide : vous pouvez utiliser la propriété suivante : l'aire d'un triangle équilatéral de côté $c$ est $ \frac{c^2\sqrt{3}}{4}$.} \bareme{1,5}
 \reponse{
 La surface de la figure est composée de 3 carrés \og entiers \fg, trois triangles rectangles isocèles, plus le triangle $A'C'F'$.
 
@@ -96,26 +133,40 @@ L'aire des trois carrés est égale à \[ A_1 = 3 \times \version{5}{4}^2\]
 L'aire des trois carrés coupés est égale à \[ A_2 = \frac{b \times h}{2} = \frac{\version{5}{4}^2}{2} = \]
 
 L'aire du triangle équilatéral est égale à \[ A_3 = A'C' \frac{\sqrt 3}{4}\]
-Dans le triangle rectangle $A'C'B$ rectangle en $B$, d'après le théorème de Pytja
+Dans le triangle rectangle $A'C'B$ rectangle en $B$, d'après le théorème de Pythagore
 }
 \item Montrer que 
-\[ A(x) = \]
-(vous pouvez encore utiliser la formule de l'aire du triangle équilatéral)
+\[ A(x) = \version{150}{96} - \frac{3x^2}{2} + \frac{x^2\sqrt 3}{2}\]
+(vous pouvez encore utiliser la formule de l'aire du triangle équilatéral) \bareme{1,5}
 \reponse{
 La surface de la figure est composée de 3 carrés \og entiers \fg, trois carrés \og coupés \fg (un triangle rectangle isocèle en moins), plus le triangle $A'C'F'$.
 
 L'aire des trois carrés est égale à \[ A_1 = 3 \times \version{5}{4}^2\]
 
-L'aire des trois carrés coupés est égale à \[ A_2 = \version{5}{4}^2 - \frac{b \times h}{2} = \version{5}{4}^2  - \frac{x^2}{2} \]
+L'aire des trois carrés coupés est égale à \[ A_2 = 3 \version{5}{4}^2 - 3 \frac{b \times h}{2} = 3 \version{5}{4}^2  - 3 \frac{x^2}{2} \]
 
-L'aire du triangle équilatéral est égale à \[ A_3 = A'C' \frac{\sqrt 3}{4}\]
-Dans le triangle rectangle $A'C'B$ rectangle en $B$, d'après le théorème de Pytja
+L'aire du triangle équilatéral est égale à \[ A_3 = A'C'^2 \frac{\sqrt 3}{4}\]
+Dans le triangle rectangle $A'C'B$ rectangle en $B$, d'après le théorème de Pythagore, $A'C'^2 = A'B^2 + C'B^2 = 2x^2$. D'où $A'C' = x \sqrt{2}$
+Donc \[ A_3 = 2x^2 \frac{\sqrt{3}}{4} = x^2 \frac{\sqrt{3}}{2} \]
+
+D'où \[ A(x) = 6 \version{5}{4}^2 - \frac{x^2}{2} (3 - \sqrt 3)\]
 
 }
-\item 
-\end{itemize}
+\item À l'aide de la calculatrice, dresser le tableau de variations de $A$ sur son ensemble de définition. \bareme{0,5}
+\end{enumerate}
+
+
+\section*{Bonus}
+%\emph{Dans cette question, une réponse sans un minimum de justification ne vaudra pas de bonus ! Par contre, une réponse argumentée, même fausse, ou un raisonnement partiel pourra en valoir...}
+
+
+Vous êtes face à deux coffres. Chacun peut contenir un trésor ou un piège mortel (mais il se peut très bien que les deux coffres contiennent un trésor, ou que les deux coffres contiennent un piège). Les inscriptions sur les coffres sont soit toutes les deux vraies, soit toutes les deux fausses.
 
+Le premier coffre a pour inscription~: «~Il y a un piège ici, ou un trésor dans l'autre coffre~».
 
+Le second coffre a pour inscription~: «~Il y a un trésor dans l'autre coffre~».
 
+\smallskip
 
+Faut-il ouvrir un coffre (lequel) ? Les deux ? Aucun ?
 \end{document}
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diff --git a/2nde/chapitre11_probas/activite_arbre.pdf b/2nde/chapitre11_probas/activite_arbre.pdf
new file mode 100644 (file)
index 0000000..00e372c
Binary files /dev/null and b/2nde/chapitre11_probas/activite_arbre.pdf differ
diff --git a/2nde/chapitre11_probas/activite_arbre.tex b/2nde/chapitre11_probas/activite_arbre.tex
new file mode 100644 (file)
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--- /dev/null
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+\input{../../header_a5.tex}
+
+\begin{document}
+\section*{Jeu de rôle sur calculatrice : étude}
+
+On rappelle les règles du jeu sur calculatrice : on doit choisir entre trois caractéristiques : force, agilité, santé, et la somme de ces caractéristiques doit être égale à 10. 
+
+Lors du combat, une attaque est résolue de la façon suivante : l'attaquant a une probabilité égale à $0,5 + 0,05 \times \text{agilité}$ de toucher son ennemi. S'il le touche, il inflige sa force en dégâts (s'il le rate, il ne lui inflige évidemment pas de dégâts).
+
+\exo{Premier cas}
+On suppose que le joueur a choisi 3 points de santé, il lui reste donc 7 points à répartir entre agilité et force. On s'intéresse à l'efficacité des attaques. Le joueur a choisi 2 points en force et 5 en agilité.
+\begin{enumerate}
+\item On considère une attaque. L'univers de l'expérience est donc $\Omega = \{ T, R \}$ (pour \og touche \fg et \og rate \fg). Quelle est la loi de probabilité associée à cette expérience ?
+
+\newcommand\bla{2cm}
+\begin{tabular}{|c|p{\bla}|p{\bla}|}
+\hline
+Issue & \vspace{0.3cm} & \\
+\hline
+Proba & \vspace{0.3cm} & \\
+\hline
+\end{tabular}
+
+\item On considère maintenant trois attaques successives.
+\begin{enumerate}
+  \item L'arbre ci-dessous représente les différentes issues possibles.
+
+\includegraphics[width=0.6\textwidth]{etudejeu.pdf}
+
+  On notera les issues sous cette forme : \og $RTR$ \fg signifie \og l'attaque 1 rate, l'attaque 2 touche, l'attaque 3 rate \fg par exemple. Décrire l'univers de cette expérience \\
+$\Omega = \dotfill $
+
+\item Cet arbre est \souligne{pondéré} : chaque branche doit indiquer la probabilité de \og choisir \fg cette branche. Compléter l'arbre en indiquant les probabilités en question.
+\item Lorsqu'on a affaire à un arbre pondéré, la probabilité d'une issue (comme $RTR$) est le produit des probabilités sur le chemin menant à cette issue. 
+ À l'aide de l'arbre, déterminer la loi de probabilité de cette expérience :
+
+\newcommand\bli{1cm}
+\hspace{-2cm}\begin{tabular}{|c|p{\bli}|p{\bli}|p{\bli}|p{\bli}|p{\bli}|p{\bli}|p{\bli}|p{\bli}|}
+\hline
+Issue & \vspace{0.3cm} & & & & & & & \\
+\hline
+Proba & \vspace{0.3cm} & & & & & & & \\
+\hline
+\end{tabular}
+
+
+    \item On suppose que l'ennemi a 6 points de vie. Décrire en terme d'ensemble l'événement $K$ : \og On réussit à tuer l'ennemi en trois coups (ou moins).\fg
+\vspace{1cm}
+  \item Calculer sa probabilité.
+\vspace{1cm}
+
+\end{enumerate}
+\end{enumerate}
+
+\exo{Deuxième cas}
+On considère cette fois que le joueur a choisi 5 points en force et 2 en agilité. Reprendre les questions de l'exercice précédent avec ces nouvelles données.
+
+\begin{enumerate}
+\item Loi de probabilité pour une attaque : \vspace{2cm}
+\item \begin{enumerate} \item Pour trois attaques : arbre pondéré :\newpage
+~
+
+\vspace{4cm}
+\item Loi de probabilité :
+
+\newcommand\bli{1cm}
+\hspace{-2cm}\begin{tabular}{|c|p{\bli}|p{\bli}|p{\bli}|p{\bli}|p{\bli}|p{\bli}|p{\bli}|p{\bli}|}
+\hline
+Issue & \vspace{0.3cm} & & & & & & & \\
+\hline
+Proba & \vspace{0.3cm} & & & & & & & \\
+\hline
+\end{tabular}
+\item $K$ : \og L'ennemi est tué en trois coups (ou moins) \fg. \vspace{1cm}
+\item $p(K)$ : 
+\vspace{1cm}
+\end{enumerate}
+\item Comparer les probabilités obtenues. Quelle semble être la meilleure stratégie ?
+\vspace{2cm}
+\end{enumerate}
+
+\exo{Pour aller plus loin : étude des dégâts moyens}
+On suppose toujours que le joueur a 7 points à répartir entre agilité et force. On note $x$ l'agilité choisie.
+\begin{enumerate}
+\item Donner, en fonction de $x$ la loi de probabilité pour une seule attaque.
+
+\vspace{2cm}
+\item Si on effectue $N$ attaques (avec $N$ suffisamment grand pour pouvoir appliquer la loi des grands nombres), combien d'entre elles toucheraient (en fonction de $x$ et de $N$) ?
+\vspace{2cm}
+\item En déduire les dégâts totaux infligés pour les $N$ attaques, puis les dégâts moyens pour une attaque.
+\vspace{3cm}
+\item À l'aide de la calculatrice, chercher quelle valeur de $x$ optimise les dégâts moyens.
+
+\vspace{2cm}
+
+\item On suppose qu'au bout d'un moment, le personnage monte en niveau et gagne de nouveaux points à répartir. Reprendre l'étude avec 15 points à répartir (vous n'êtes pas obligé(e) de refaire toutes les étapes).
+\end{enumerate}
+
+
+\end{document}
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