192 lines
7.4 KiB
TeX
192 lines
7.4 KiB
TeX
|
\documentclass[a4paper,12pt]{article}
|
||
|
|
\usepackage[utf8x]{inputenc}
|
||
|
|
\usepackage[francais]{babel}
|
||
|
|
\usepackage[T1]{fontenc}
|
||
|
|
\usepackage{amssymb}
|
||
|
|
\usepackage{amsmath}
|
||
|
|
\usepackage{amsfonts}
|
||
|
|
\usepackage{subfig}
|
||
|
|
\usepackage{graphicx}
|
||
|
|
\usepackage{color}
|
||
|
|
\usepackage{gensymb}
|
||
|
|
\usepackage{ifthen, calc}
|
||
|
|
\usepackage{tabularx}
|
||
|
|
|
||
|
|
\newenvironment{solution}
|
||
|
|
{%
|
||
|
|
~\\
|
||
|
|
\newbox\tempbox%
|
||
|
|
\begin{lrbox}{\tempbox}\begin{minipage}{\linewidth}%
|
||
|
|
}{%
|
||
|
|
\end{minipage}\end{lrbox}%
|
||
|
|
\medskip%
|
||
|
|
\fbox{\usebox{\tempbox}}%
|
||
|
|
\medskip%
|
||
|
|
}
|
||
|
|
|
||
|
|
% Title Page
|
||
|
|
\title{DM 1}
|
||
|
|
\date{Novembre 2015}
|
||
|
|
% DS DSCorr DM DMCorr Corr
|
||
|
|
|
||
|
|
\begin{document}
|
||
|
|
|
||
|
|
\maketitle
|
||
|
|
|
||
|
|
Sujet numéro \Var{infos.num}
|
||
|
|
|
||
|
|
|
||
|
|
\section{Exercice}
|
||
|
|
\Block{set a,b,c = random_str("{a*d},{b*d},{c}", conditions = ["{a} != {b}", "{a} > 1", "{b}>1", "{c} > 1", "{d}>1"]).split(',')}
|
||
|
|
\Block{set total = int(a) + int(b) + int(c)}
|
||
|
|
Dans un sac, il y a \Var{a} bonbons à la menthe, \Var{b} bonbons à la fraise et \Var{c} au chocolat. On choisit un bonbon au hasard dans ce sac.
|
||
|
|
\begin{enumerate}
|
||
|
|
\item Calculer la probabilité de tirer un bonbon à la fraise.
|
||
|
|
\begin{solution}
|
||
|
|
$T($ tirer un bonbon à la fraise $) = \dfrac{\Var{a}}{\Var{total}}$
|
||
|
|
\end{solution}
|
||
|
|
\item Calculer la probabilité de tirer un bonbon qui n'est pas au chocolat.
|
||
|
|
\begin{solution}
|
||
|
|
\Block{set nonChoco = int(a) + int(b)}
|
||
|
|
$T($ tirer un bonbon à la fraise ou à la menthe $) = \dfrac{\Var{nonChoco}}{\Var{total}}$
|
||
|
|
\end{solution}
|
||
|
|
\item Calculer la probabilité de tirer un bonbon au réglisse.
|
||
|
|
\begin{solution}
|
||
|
|
$T($ tirer un bonbon au réglisse $) = \dfrac{0}{\Var{total}} = 0$
|
||
|
|
\end{solution}
|
||
|
|
\item Dans un autre sac, on place 25 bonbons à la menthe et 34 bonbons à la fraise. Lise préfère les bonbons à la menthe. Dans quel sac doit-elle tirer un bonbon pour avoir le plus de chance d'avoir un bonbon qu'elle préfère?
|
||
|
|
\begin{solution}
|
||
|
|
\Block{if (int(a)/total) > (25/34)}
|
||
|
|
Elle prefera tirer dans le premier sac car
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\frac{\Var{a}{\Var{total}} & > & \frac{25}{34}
|
||
|
|
\end{eqnarray*}
|
||
|
|
\Block{else}
|
||
|
|
Elle prefera tirer dans le deuxième sac car
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\frac{\Var{a}}{\Var{total}} & < & \frac{25}{34}
|
||
|
|
\end{eqnarray*}
|
||
|
|
\Block{endif}
|
||
|
|
|
||
|
|
\end{solution}
|
||
|
|
\end{enumerate}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
\section{Exercice}
|
||
|
|
\begin{enumerate}
|
||
|
|
\item Compléter les pointillés pour qu'il y est bien égalité.
|
||
|
|
\hspace{-1cm}
|
||
|
|
\begin{center}
|
||
|
|
\Block{set a,b,c = random_str("{a},{b},{b*c}", conditions = ["{a} != {b}", "{a} > 1", "{b}>1","{c}>1"]).split(',')}%
|
||
|
|
$\dfrac{\Var{a}}{\Var{b}} = \dfrac{\ldots}{\Var{c}}$
|
||
|
|
\hfill
|
||
|
|
\Block{set a,b,c = random_str("{a},{b},{b*c}", conditions = ["{a} != {b}", "{a} > 1", "{b}>1","{c}>1"]).split(',')}%
|
||
|
|
$\dfrac{\Var{a}}{\Var{b}} = \dfrac{\ldots}{\Var{c}}$
|
||
|
|
\hfill
|
||
|
|
\Block{set a,b,c = random_str("{a},{b},{b*c}", conditions = ["{a} != {b}", "{a} > 1", "{b}>1","{c}>1"]).split(',')}%
|
||
|
|
$\dfrac{\cdots}{\Var{c}} = \dfrac{\Var{a}}{\Var{b}}$
|
||
|
|
\hfill
|
||
|
|
\Block{set a,b,c = random_str("{a},{b},{a*c}", conditions = ["{a} != {b}", "{a} > 1", "{b}>1","{c}>1"]).split(',')}%
|
||
|
|
$\dfrac{\Var{a}}{\Var{b}} = \dfrac{\Var{c}}{\cdots}$
|
||
|
|
\end{center}
|
||
|
|
|
||
|
|
|
||
|
|
\item Faire les calculs suivants en détaillant les étapes (penser à simplifier les fractions quand c'est possible).
|
||
|
|
\begin{enumerate}
|
||
|
|
\Block{set e = Expression.random("{a} / {b} + {c} / {b}", ["{b} > 1", "{a} > 0", "{c} > 0"])}
|
||
|
|
\item $A = \Var{e}$
|
||
|
|
\begin{solution}
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\Var{e.simplify().explain() | calculus(name = "A")}
|
||
|
|
\end{eqnarray*}
|
||
|
|
\end{solution}
|
||
|
|
\Block{set e = Expression.random("{a} / {b} + {c} / {b}", ["{b} > 1"])}
|
||
|
|
\item $B = \Var{e}$
|
||
|
|
\begin{solution}
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\Var{e.simplify().explain() | calculus(name = "B")}
|
||
|
|
\end{eqnarray*}
|
||
|
|
\end{solution}
|
||
|
|
\Block{set e = Expression.random("{a} / {b} + {c} / {d*b}", ["{b} > 1", "{c} > 1", "{d} > 1"])}
|
||
|
|
\item $C = \Var{e}$
|
||
|
|
\begin{solution}
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\Var{e.simplify().explain() | calculus(name = "C")}
|
||
|
|
\end{eqnarray*}
|
||
|
|
\end{solution}
|
||
|
|
\Block{set e = Expression.random("{a} / {b} + {c} / {d*b}", ["{b} > 1", "{d} > 1"])}
|
||
|
|
\item $D = \Var{e}$
|
||
|
|
\begin{solution}
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\Var{e.simplify().explain() | calculus(name = "D")}
|
||
|
|
\end{eqnarray*}
|
||
|
|
\end{solution}
|
||
|
|
\Block{set e = Expression.random("{a} / {b} * {c}", ["{b} > 1", "{a} > 0", "{c} > 1", "{c} != {b}"])}
|
||
|
|
\item $E = \Var{e}$
|
||
|
|
\begin{solution}
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\Var{e.simplify().explain() | calculus(name = "E")}
|
||
|
|
\end{eqnarray*}
|
||
|
|
\end{solution}
|
||
|
|
\Block{set e = Expression.random("{a} / {b} * {c} / {d}", ["{b} > 1", "{a} > 0", "{c} > 0", "{d} > 1"])}
|
||
|
|
\item $F = \Var{e}$
|
||
|
|
\begin{solution}
|
||
|
|
\begin{eqnarray*}
|
||
|
|
\Var{e.simplify().explain() | calculus(name = "F")}
|
||
|
|
\end{eqnarray*}
|
||
|
|
\end{solution}
|
||
|
|
\end{enumerate}
|
||
|
|
|
||
|
|
\end{enumerate}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
\section{Exercice}
|
||
|
|
\Block{set AO, OD, CD, OB = random_str("{a},{b},{c},{d}", ["{a} < {b}", "{c} != {d}"], 1, 20).split(',')}
|
||
|
|
Dans la figure suivante, $(AB)$ et $(CD)$ sont parallèles, $AO = \Var{AO}$, $OD = \Var{OD}$, $CD = \Var{CD}$ et $OB = \Var{OB}$.
|
||
|
|
|
||
|
|
\Block{set fig = random_str("{a}", [], 1, 2)}
|
||
|
|
|
||
|
|
\includegraphics[scale=0.4]{thales\Var{fig}}
|
||
|
|
|
||
|
|
Calculer les longueurs $OC$ et $AB$.
|
||
|
|
|
||
|
|
\begin{solution}
|
||
|
|
On sait que
|
||
|
|
\begin{itemize}
|
||
|
|
\item $(AB)$ et $(CD)$ sont parallèles
|
||
|
|
\item $A$,$O$ et $D$ sont alignés
|
||
|
|
\item $B$,$O$ et $C$ sont alignés
|
||
|
|
\end{itemize}
|
||
|
|
Donc d'après le théorème de Thalès
|
||
|
|
|
||
|
|
\begin{tabular}{|c|*{3}{c|}}
|
||
|
|
\hline
|
||
|
|
Triangle $OAB$ & $AO = \Var{AO}$ & $OB = \Var{OB}$ & $AB $ \\
|
||
|
|
\hline
|
||
|
|
Triangle $OCD$ & $DO = \Var{OD}$ & $OC $ & $CD = \Var{CD}$ \\
|
||
|
|
\hline
|
||
|
|
\end{tabular}
|
||
|
|
est un tableau de proportionnalité.
|
||
|
|
|
||
|
|
On en déduit que
|
||
|
|
\begin{eqnarray*}
|
||
|
|
OC & = & \frac{DO \times OB}{AO} = \frac{\Var{OD} \times \Var{OB}}{\Var{AO}} = \Var{int(OD)*int(OB)/int(AO) | round(2)}
|
||
|
|
\end{eqnarray*}
|
||
|
|
Et que
|
||
|
|
\begin{eqnarray*}
|
||
|
|
AB & = & \frac{CD \times AO}{DO} = \frac{\Var{CD} \times \Var{AO}}{\Var{OD}} = \Var{int(CD)*int(AO)/int(OD) |round(2)}
|
||
|
|
\end{eqnarray*}
|
||
|
|
|
||
|
|
\end{solution}
|
||
|
|
|
||
|
|
|
||
|
|
\end{document}
|
||
|
|
|
||
|
|
%%% Local Variables:
|
||
|
|
%%% mode: latex
|
||
|
|
%%% TeX-master: "master"
|
||
|
|
%%% End:
|
||
|
|
|