plein de trucs
authorDenise <dmaurice@phare.normalesup.org>
Fri, 18 Dec 2015 22:44:33 +0000 (23:44 +0100)
committerDenise <dmaurice@phare.normalesup.org>
Fri, 18 Dec 2015 22:44:33 +0000 (23:44 +0100)
95 files changed:
1ere/DST_4/dst4.tex
1ere/DST_4/dst4_corrige.pdf
1ere/DST_4/dst4_enonce.pdf
2nde/DM_4/dm4.tex [new file with mode: 0644]
2nde/DM_4/dm4_corrige.pdf [new file with mode: 0644]
2nde/DM_4/dm4_enonce.pdf [new file with mode: 0644]
2nde/chapitre5_geomespace/chap5.pdf
2nde/chapitre5_geomespace/chap5.tex
2nde/chapitre5_geomespace/chap5_page2.pdf [new file with mode: 0644]
2nde/chapitre5_geomespace/cube-ppconfondus.pdf [moved from 2nde/chapitre5_geomespace/ch4-cube-ppconfondus.pdf with 100% similarity]
2nde/chapitre5_geomespace/cube-ppconfondus.svg [moved from 2nde/chapitre5_geomespace/ch4-cube-ppconfondus.svg with 100% similarity]
2nde/chapitre5_geomespace/cube-ppparalleles.pdf [moved from 2nde/chapitre5_geomespace/ch4-cube-ppparalleles.pdf with 100% similarity]
2nde/chapitre5_geomespace/cube-ppparalleles.svg [moved from 2nde/chapitre5_geomespace/ch4-cube-ppparalleles.svg with 100% similarity]
2nde/chapitre5_geomespace/cube-ppsecants.pdf [moved from 2nde/chapitre5_geomespace/ch4-cube-ppsecants.pdf with 100% similarity]
2nde/chapitre5_geomespace/cube-ppsecants.svg [moved from 2nde/chapitre5_geomespace/ch4-cube-ppsecants.svg with 100% similarity]
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2nde/manuel_sesamath/2G2/MS2_2G2_chapitrecomplet.pdf [new file with mode: 0644]
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2nde/manuel_sesamath/2G2/corrections/corr-QCM-G2-48.tex [new file with mode: 0644]
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index e25d6ecff6cd87ac41220098afae5b5e16e47844..c4bfa98e51b10554e377350f19cde90d4df4d359 100644 (file)
@@ -9,48 +9,57 @@
 \nomprenom
 %\bigskip
 
-\chapoDST{
+\chapoDST{Second degré~:
+
+\bigskip
+
+
+Stats~:
+
+\bigskip
+
+Fonctions~:
+
+\bigskip
+
+Pourcentages
+
+\bigskip
 }
 
-\section{Équations du second degré (8)}
+\section{Second degré (8)}
+\emph{Note pour toutes les questions de cet exercice~: lorsque les résultats ne tombent pas juste, toujours donner la valeur exacte (des racines, ou autres)~: avec la racine carrée ou la fraction.}
 
 \begin{enumerate}
-        \item Pour chacune des équations suivantes, calculer le discriminant et indiquer (en cochant la case correspondante) combien elle admet de solutions~:\bareme{(2)}\\
-\begin{tabular}{|p{8cm}|p{8cm}|}
-\hline
-$x^2 + x + 15 = 0$ & $-3x^2 +5x -2 = 0$ \\
-$ \Delta = $ & $ \Delta = $ \\
-\quad & \quad \\
-\quad & \quad \\
-$\square$ Deux solutions distinctes & $\square$ Deux solutions distinctes \\
-$\square$ Une solution & $\square$ Une solution\\
-$\square$ Aucune solution & $\square$ Aucune solution \\
-\hline
-$7-x^2 = 0$ & $4x^2 -12x + 9 = 0$ \\
-$ \Delta = $ & $ \Delta = $ \\
-\quad & \quad \\
-\quad & \quad \\
-$\square$ Deux solutions distinctes & $\square$ Deux solutions distinctes \\
-$\square$ Une solution & $\square$ Une solution\\
-$\square$ Aucune solution & $\square$ Aucune solution \\
-\hline
-\end{tabular}
+\item Soit $f$ définie par $f(x) = x(2x+1) -x^2 -12$. 
+  \begin{enumerate}[a)]
+    \item Montrer qu'il s'agit bien d'une fonction polynôme du second degré, en mettant $f$ sous la forme $ax^2 + bx + c$ et en identifiant $a$, $b$ et $c$. \bareme{0,5}
+    \item Calculer ses éventuelles racines, puis mettre $f$ sous forme factorisée si c'est possible. \bareme{1}
+  \end{enumerate}
 
-        \item Résoudre les équations suivantes.\bareme{(1 par eq)}
+        \item Résoudre les équations suivantes (les détails sont demandés).\bareme{(1 par eq, 0,5 pour le delta négatif)}
 \begin{enumerate}[a)]
-\item  $x^2 + x -12 = 0$
-        \item  $2x^2 + x + 7 = 0$
-        \item $-4x^2 + 4x - 1= 0$ 
+  \item  $2x^2 + x + 7 = 0$
+  \item  $x^2 + x -1 = 0$
+  \item $-4x^2 + 4x - 1= 0$ 
+\end{enumerate}
+\item En utilisant un tableau de signe, résoudre les inéquations suivantes. \bareme{1,5 puis 1}
+\begin{enumerate}[a)]
+  \item$-x^2 -2x + 35 < 0$ 
+  \item $x^2 + 1 \geq 0$
 \end{enumerate}
-\item En utilisant un tableau de signe, résoudre l'inéquation $-x^2 -2x + 35 < 0$ \bareme{1,5}
-
 
-\item La somme des carrés de trois nombres positifs consécutifs est égale à $194$. Quels sont ces nombres~? \bareme{1,5}
+\item La somme des carrés de trois nombres positifs consécutifs est égale à $194$. Quels sont ces nombres~? \bareme{1}
 
 
 \end{enumerate}
 
-\section{Statistiques : problème}
+\section{Statistiques : problème (4)}
+\emph{Dans ce problème, arrondir les réponses au centième.}
+
+\bigskip
+
+
 Dans le parc informatique d'une entreprise, le responsable a compté le nombre de pannes survenues sur les machines durant un an. Il obtient les données suivantes
 
 \begin{tabular}{|c|c|c|c|c|c|c|c|}
@@ -62,27 +71,61 @@ Nombre de machines & 5 & 8 & 10 & 7 & 4 & 2 & 1 \\
 \end{tabular}
 
 \begin{enumerate}
-\item Calculer la médiane du nombre de pannes, et l'écart interquartile (avec justification). \bareme{1}
-\item Calculer le nombre moyen de pannes, ainsi que l'écart-type. \bareme{1}
-\item Le responsable décide de changer la machine qui avait eu 8 pannes, en en mettant une neuve à la place (qui, théoriquement, n'aura pas de panne l'année prochaine). Sans faire le calcul, indiquer comment va évoluer moyenne et écart-type l'année suivante, si les autres machines restent dans le même état. \bareme{1}
+\item Calculer la médiane du nombre de pannes, et l'écart interquartile (avec justification). \bareme{1,5} \reponse{$2$ et $2$}
+\item Calculer le nombre moyen de pannes, ainsi que l'écart-type. \bareme{1,5} \reponse{$\xb = 2,24$ et $\sigma = 1,67$}
+\item Le responsable décide de changer la machine qui avait eu 8 pannes, en en mettant une neuve à la place (qui, théoriquement, n'aura pas de panne l'année prochaine). Sans faire le calcul, indiquer comment va évoluer moyenne, écart-type, médiane et écart interquartile l'année suivante, si les autres machines restent dans le même état. \bareme{1}
 \end{enumerate}
 
-\section{Statistiques, logique}
-On considère une série statistique à 3 valeurs~:
+%\section{Statistiques, logique}
+%On considère une série statistique à 3 valeurs~:
 
-\begin{tabular}{|c|c|c|c|}
-\hline
-Valeur & $x_1$ & $x_2$ & $x_3$ \\
-\hline
-Effectif & $n_1$ & $n_2$ & $n_3$ \\
-\hline
-\end{tabular}
+%\begin{tabular}{|c|c|c|c|}
+%\hline
+%Valeur & $x_1$ & $x_2$ & $x_3$ \\
+%\hline
+%Effectif & $n_1$ & $n_2$ & $n_3$ \\
+%\hline
+%\end{tabular}
+
+% Dire si les affirmations suivantes sont vraies ou fausses, et justifier.
+%\begin{enumerate}
+%\item Si toutes les valeurs de la série augmentent de 1, alors la moyenne augmente de 1.
+%\item Si toutes les valeurs de la série augmentent de 1, alors l'écart-type augmente de 1.
+%\item Si toutes les valeurs de la série sont multipliées par 3, alors la moyenne est multipliée par 3.
+%\item Si toutes les valeurs de la série sont multipliées par 3, alors l'écart-type est multiplié par 3.
+%\end{enumerate}
+
+\section{Pourcentages et taux (4)}
+\emph{Dans ce problème, arrondir tous les pourcentages au centième de \%.}
+
+\bigskip
 
- Dire si les affirmations suivantes sont vraies ou fausses, et justifier.
+Le patron de l'entreprise de papier toilette ToutDoux fait le point depuis la fin de l'année 2012. Fin 2013, il a constaté une augmentation de 10 \% des ventes par rapport à fin 2012. Fin 2014, une augmentation de 6 \%.
 \begin{enumerate}
-\item Si toutes les valeurs de la série augmentent de 1, alors la moyenne augmente de 1.
-\item Si toutes les valeurs de la série augmentent de 1, alors l'écart-type augmente de 1.
-\item Si toutes les valeurs de la série sont multipliées par 3, alors la moyenne est multipliée par 3.
-\item Si toutes les valeurs de la série sont multipliées par 3, alors l'écart-type est multiplié par 3.
+\item Calculer le taux d'évolution global entre fin 2012 et fin 2014.\bareme{1}
+\item L'année 2015, l'entreprise a subit une rude concurrence, et les ventes ont alors chuté de 16 \%. Le patron, philosophe, s'est dit alors \og Ce n'est pas grave, nous sommes juste revenus aux ventes de début 2013 \fg. A-t-il raison~? Justifier par un calcul approprié. \bareme{1,5}
+\reponse{perte de $2,06 \%$ en vrai.}
+\item Le patron a alors eu une idée révolutionnaire, celle de vendre du papier toilette imprimé avec des petits chatons roses. Et fin 2015, il a constaté que le taux d'évolution total des ventes (depuis fin 2012 donc) était de $12,6~\%$. Quel a été le taux d'évolution entre fin 2014 et fin 2015 pour obtenir ce résultat~? \bareme{1,5}
 \end{enumerate}
+
+\section{Fonctions de référence (4)}
+
+\begin{enumerate}
+\item $f$ est la fonction cube. Donner son tableau de variations et son tableau de signe. \bareme{1}
+\item $g$ est la fonction définie par $g(x) = 5x$. Quel est son sens de variation~? \bareme{0,5}
+\item En utilisant la représentation graphique de ces deux fonctions avec la calculatrice, résoudre graphiquement l'équation $f(x) = g(x)$ (arrondi au dixième). \bareme{1}
+\item Résolution algébrique~: montrer que l'équation $f(x) = g(x)$ est équivalente à \bareme{0,5}
+\[
+x(x^2-5) = 0
+\]
+\item Résoudre cette équation de manière exacte.\bareme{1}
+\end{enumerate}
+
+\section{Bonus~: taux mystère}
+%Une évolution d'un certain taux, suivie d'une évolution d'un taux trois fois plus élevé donnent, au final, un taux d'évolution global de $50$ \%. Quel est ce taux d'évolution~?
+\og Cette année, le taux d'évolution de notre chiffre d'affaire est le triple de celui de l'année précédente. Nous obtenons donc une augmentation globale, sur ces deux années, de 50 \%. \fg
+
+Trouver le taux d'évolution sur la première année.
+
+
 \end{document}
index caefe14575ddee3fbe8953277574a5c767f28ffe..b9a3463c667d34090b2d279db3fc767b9339a8b6 100644 (file)
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diff --git a/2nde/DM_4/dm4.tex b/2nde/DM_4/dm4.tex
new file mode 100644 (file)
index 0000000..9de0284
--- /dev/null
@@ -0,0 +1,52 @@
+\input{../../header.tex}
+
+%\corrigetrue
+
+%\newcommand\boldmath{}
+
+\title{\vspace{-2cm}Classe de 2nde 6 -- Devoir Maison numéro 4}
+\date{\vspace{-1.5cm}\dateoucorr{Pour le 7 janvier}}
+
+\begin{document}
+\maketitle
+
+\nomprenom
+
+\chapoDM
+
+\section*{Une formule pour la médiane}
+
+ Dans un vieux livre, Mongi trouve la formule donnant la longueur des médianes d'un triangle en fonction des longueurs de ces trois côtés.
+ $$L^2=\dfrac{1}{4}\left( 2a^2+2b^2 - c^2\right)$$
+ \begin{center}
+  \begin{tikzpicture}[scale=0.8]
+   \draw (0,0)--(6,0)--(2,3)--cycle;
+   \draw (2,3)--(3,0);
+   \draw (0,0) node[left] {$A$};
+   \draw (6,0) node[right] {$B$};
+   \draw (2,3) node[above] {$C$};
+   \draw (4,1.5) node[above right] {{\boldmath $a$}};
+   \draw (1,1.5) node[above left] {{\boldmath $b$}};
+   \draw (3,0) node[below] {{\boldmath $c$}};
+   \draw (3,0) node[above right] {$I$};
+   \draw (1.5,0) node[rotate=90] {{\boldmath $\approx$}};
+   \draw (4.5,0) node[rotate=90] {{\boldmath $\approx$}};
+   \draw (2.5,1.5) node[right] {{\boldmath $L$}};
+  \end{tikzpicture}
+ \end{center}
+
+Mongi est curieux de la preuve de cette formule.  On se place dans un repère orthonormé d'origine $A$où $(AB)$ est l'axe des abscisses.
+\begin{enumerate}
+ \item Quelles sont les coordonnées de $A$ et de $B$?
+ \item Quelles sont les coordonnées du milieu $I$ de $[AB]$?
+ \item On note $(x_C;y_C)$, les coordonnées du point $C$.
+ Vérifier que : $$L^2=IC^2=x_C^2+y_C^2-cx_C+\dfrac{c^2}{4}$$
+ \item Calculer  $a^2$ et $b^2$ en fonction de $x_C$, $y_C$ et de $c$. Donner les résultats sous forme développée.
+ \item \'Etablir la formule du livre de Mongi. 
+
+\emph{Indication~: il existe plusieurs façons de s'y prendre, mais on peut par exemple partir du membre de droite de la formule ($\dfrac{1}{4}\left( 2a^2+2b^2 - c^2\right)$), et remplacer $a^2$ et $b^2$ par leurs expressions trouvées à la question précédente pour arriver enfin à l'expression de $IC^2$ établie à la question 3.}
+\end{enumerate}
+
+
+
+\end{document}
diff --git a/2nde/DM_4/dm4_corrige.pdf b/2nde/DM_4/dm4_corrige.pdf
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diff --git a/2nde/DM_4/dm4_enonce.pdf b/2nde/DM_4/dm4_enonce.pdf
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index 0ec6a0c03a8a17c1e34516444fc1ef307e481949..178d75fda46c75fe958af8b0d66686337248865c 100644 (file)
@@ -75,7 +75,7 @@ Pour déterminer un plan dans l'espace, il faut (et il suffit de) donner~:
 \begin{minipage}{4cm}
 \includegraphics[width=4cm]{cubedebase_abcdefgh.pdf}
 \end{minipage}
-\begin{minipage}{10cm}
+\begin{minipage}{12cm}
 Exemple~: \lignepoint
 \lignepoint
 \lignepoint
@@ -85,10 +85,47 @@ Exemple~: \lignepoint
 Dans chaque plan de l'espace, on peut appliquer les théorèmes de la géométrie plane (Pythagore, Thalès, \ldots).
 \end{prop}
 
+\subsection{Positions relatives de deux plans}
+
+\begin{rgl}
+Deux plans de l'espace sont soit \souligne{sécants}, soit \souligne{parallèles}.
+
+\vspace{0.5cm}
+
+\noindent \begin{tabular}{p{5.5cm}||p{5.5cm}p{5.5cm}}
+\lignepoint & \multicolumn{2}{c}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots } \\
+\includegraphics[width=5cm]{cube-ppsecants.pdf} & \includegraphics[width=4.5cm]{cube-ppparalleles.pdf}
+& \includegraphics[width=4.5cm]{cube-ppconfondus.pdf}\\
+\vspace{1cm}\lignepoint \lignepoint &  \vspace{1cm} \lignepoint \lignepoint & \vspace{1cm} \lignepoint \lignepoint
+\end{tabular}
+\end{rgl}
+
+Exemples~:
+
+\begin{minipage}{4cm}
+\includegraphics[width=4cm]{cubedebase_abcdefgh.pdf}
+\end{minipage}
+\begin{minipage}{12cm}
+Quelle est la position relative des plans $(ABF)$ et $(DCG)$~?
+\lignepoint
+
+\bigskip
+
+Quelle est la position relative des plans $(ABC)$ et $(CGH)$~?
+\lignepoint
+\lignepoint
+
+\bigskip
+
+Déterminer l'intersection des plans  $(ABC)$ et $(CGH)$.
+\lignepoint
+\lignepoint
+\end{minipage}
+
 \subsection{Positions relatives de deux droites}
 
 \begin{rgl}
-Deux droites de l'espace sont soit \souligne{coplanaires}, soit \souligne{non coplanaires}.
+Deux droites de l'espace sont soit \souligne{coplanaires} (c'est-à-dire qu'elles appartiennent à un même plan), soit \souligne{non coplanaires}.
 
 \smallskip
 
@@ -114,18 +151,7 @@ $d$ et $\Pl$ ont un point d'intersection $B$. & $d$ et $\Pl$ sont strictement pa
 \end{tabular}
 \end{rgl}
 
-\subsection{Positions relatives de deux plans}
-
-\begin{rgl}
-Deux plans de l'espace sont soit sécants, soit parallèles.
 
-\noindent \begin{tabular}{p{5.5cm}||p{5cm}p{5cm}}
-Sécants & \multicolumn{2}{c}{Parallèles} \\
-\includegraphics[width=4.5cm]{ch4-cube-ppsecants.pdf} & \includegraphics[width=4cm]{ch4-cube-ppparalleles.pdf}
-& \includegraphics[width=4cm]{ch4-cube-ppconfondus.pdf}\\
-$\Pl$ et $\Pl'$ ont une droite d'intersection, $d$. & $\Pl$ et $\Pl'$ sont strictement parallèles (pas de point d'intersection). & $\Pl$ et $\Pl'$ sont confondus.
-\end{tabular}
-\end{rgl}
 
 \section{Parallélisme dans l'espace}
 
diff --git a/2nde/chapitre5_geomespace/chap5_page2.pdf b/2nde/chapitre5_geomespace/chap5_page2.pdf
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