229 lines
10 KiB
TeX
229 lines
10 KiB
TeX
\documentclass[a4paper,12pt]{article}
|
|
\usepackage{myXsim}
|
|
\usepackage{pgfplots}
|
|
\usetikzlibrary{decorations.markings}
|
|
\pgfplotsset{compat=1.18}
|
|
|
|
\title{ DM1 \hfill \Var{ subject.Nom }}
|
|
\tribe{2nd}
|
|
\date{A rendre pour le 2 décembre 2022}
|
|
\duree{}
|
|
|
|
\xsimsetup{
|
|
solution/print = false
|
|
}
|
|
|
|
|
|
\pagestyle{empty}
|
|
|
|
\begin{document}
|
|
\maketitle
|
|
|
|
Le barème est donné à titre indicatif, il pourra être modifié.
|
|
|
|
\begin{exercise}[subtitle={Calculs avec des fractions}, points=5]
|
|
Détailler les calculs suivants et donner le résultat sous la forme d'une fraction irréductible.
|
|
\Block{
|
|
set fractions = {
|
|
"A": random_expression("{a} / {b} + {c} / {d}", ["a!=b", "c!=d", "b > 1", "d > 1"], global_config={"min_max": (0, 10)}),
|
|
"B": random_expression("{a} / {b} + {c}", ["a!=b", "b > 1"], global_config={"min_max": (1, 10)}),
|
|
"C": random_expression("{a} / {b} * {c}", ["a!=b", "b > 1"], global_config={"min_max": (1, 10)}),
|
|
"D": random_expression("{a} / {b} * {c} / {b}", ["a!=b", "c!=b", "b > 1"], global_config={"min_max": (1, 10)}),
|
|
"E": random_expression("{a} / {b} / ({c} / {b})", ["a!=b", "c!=b", "b > 1"], global_config={"min_max": (1, 10)}),
|
|
}
|
|
}
|
|
\begin{multicols}{3}
|
|
\begin{enumerate}
|
|
%- for (l, e) in fractions.items()
|
|
\item $\Var{l} = \Var{e}$
|
|
%- endfor
|
|
\end{enumerate}
|
|
\end{multicols}
|
|
\end{exercise}
|
|
|
|
\begin{solution}
|
|
\begin{multicols}{3}
|
|
\begin{enumerate}
|
|
%- for (l, e) in fractions.items()
|
|
\item
|
|
\begin{align*}
|
|
\Var{l} & = \Var{e.simplify().explain() | join(' \\\\ & = ')} \\
|
|
& = \Var{e.simplify().simplified}
|
|
\end{align*}
|
|
%- endfor
|
|
\end{enumerate}
|
|
\end{multicols}
|
|
\end{solution}
|
|
|
|
\begin{exercise}[subtitle={Inéquation et tableaux}, points=3]
|
|
Tracer le tableau de signe de la fonction suivante en le démontrant à l'aide de la résolution d'une inéquation.
|
|
%- set f = random_expression("{a}x + {b}", global_config={"min_max":(-20, 20), "rejected":[0, 1]})
|
|
$$f(x) = \Var{f}$$
|
|
\end{exercise}
|
|
|
|
\begin{solution}
|
|
Pour déterminer les valeurs de $x$ pour lesquelles $f(x)$ est positive, il faut résoudre l'inéquation
|
|
|
|
%- set cst = -f[0]
|
|
%- set coef = f[1]
|
|
%- set racine = cst / coef
|
|
\begin{align*}
|
|
f(x) & \geq 0 \\
|
|
\Var{f} & \geq 0 \\
|
|
\Var{f + cst} &\geq \Var{0 + cst} \\
|
|
%- if coef > 0
|
|
\frac{\Var{f + cst}}{\Var{coef}} &\geq \frac{\Var{cst}}{\Var{coef}} \\
|
|
x &\geq \Var{racine.simplify()} \\
|
|
\end{align*}
|
|
|
|
Donc $f(x)$ est positif quand $x$ est supérieur à $\Var{racine}$. On en déduit le tableau de signe
|
|
%- else
|
|
\frac{\Var{f + cst}}{\Var{coef}} &\leq \frac{\Var{cst}}{\Var{coef}} \\
|
|
x &\leq \Var{racine.simplify()} \\
|
|
\end{align*}
|
|
|
|
Donc $f(x)$ est positif quand $x$ est inférieur à $\Var{racine}$. On en déduit le tableau de signe
|
|
%- endif
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}
|
|
\tkzTabInit[lgt=2,espcl=1]{$ t $/1,$ f(t) $/1}{, $\Var{racine}$ ,}
|
|
%- if coef > 0
|
|
\tkzTabLine{, -, z, +, }
|
|
%- else
|
|
\tkzTabLine{, +, z, -, }
|
|
%- endif
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\end{solution}
|
|
|
|
\begin{exercise}[subtitle={Vecteurs}, points=3]
|
|
\begin{multicols}{2}
|
|
\begin{enumerate}
|
|
\item Tracer les vecteurs $\vect{z} = \vect{u} + \vect{v}$ et $\vect{y} = 2\vect{u} - \vect{v}$ (le vecteur peut sortir du cadre)
|
|
%- set xa1, ya1 = random_list(["x", "y"], global_config={"min_max": (-5, 5), "rejected":[-2, -1, 0, 1, 2]})
|
|
%- set xa2, ya2 = -xa1, ya1
|
|
|
|
%- set aminx = min(0, xa1, xa2, xa1+xa2, 2*xa1-xa2)
|
|
%- set amaxx = max(0, xa1, xa2, xa1+xa2, 2*xa1-xa2)
|
|
%- set aminy = min(0, ya1, ya2, ya1+ya2, 2*ya1-ya2)
|
|
%- set amaxy = max(0, ya1, ya2, ya1+ya2, 2*ya1-ya2)
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.4]
|
|
\draw (\Var{aminx-1}, \Var{aminy-1}) rectangle (\Var{amaxx+1}, \Var{amaxy+1});
|
|
\draw[very thick, ->] (0, 0) -- node [midway, sloped, above] {$\vect{u}$} (\Var{xa1}, \Var{ya1});
|
|
\draw[very thick, -> ] (0, 0) -- node [midway, sloped, above] {$\vect{v}$} (\Var{xa2}, \Var{ya2});
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\item Tracer la force résultat de la somme des 3 forces exercées sur le point $0$ représenté ci-dessous.
|
|
%- set x1, y1 = random_list(["x", "y"], global_config={"min_max": (-5, 5), "rejected":[-2, -1, 0, 1, 2]})
|
|
%- set x2, y2 = -x1, y1
|
|
%- set x3, y3 = x2, 0
|
|
|
|
%- set minx = min(0, x1, x2, x3, x1+x2, x2+x3, x1+x3, x1+x2+x3 )
|
|
%- set maxx = max(0, x1, x2, x3, x1+x2, x2+x3, x1+x3, x1+x2+x3 )
|
|
%- set miny = min(0, y1, y2, y3, y1+y2, y2+y3, y1+y3, y1+y2+y3 )
|
|
%- set maxy = max(0, y1, y2, y3, y1+y2, y2+y3, y1+y3, y1+y2+y3 )
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.4]
|
|
\draw (\Var{minx-1}, \Var{miny-1}) rectangle (\Var{maxx+1}, \Var{maxy+1});
|
|
\draw[very thick, ->] (0, 0) -- node [midway, sloped, above] {$\vect{F_1}$} (\Var{x1}, \Var{y1});
|
|
\draw[very thick, -> ] (0, 0) -- node [midway, sloped, above] {$\vect{F_2}$} (\Var{x2}, \Var{y2});
|
|
\draw[very thick, -> ] (0, 0) -- node [midway, sloped, above] {$\vect{F_3}$} (\Var{x3}, \Var{y3});
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\end{enumerate}
|
|
\end{multicols}
|
|
\end{exercise}
|
|
|
|
\begin{solution}
|
|
\begin{multicols}{2}
|
|
\begin{enumerate}
|
|
\item
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.4]
|
|
\draw (\Var{aminx-1}, \Var{aminy-1}) rectangle (\Var{amaxx+1}, \Var{amaxy+1});
|
|
\draw[very thick, ->] (0, 0) -- node [midway, sloped, above] {$\vect{u}$} (\Var{xa1}, \Var{ya1});
|
|
\draw[very thick, -> ] (0, 0) -- node [midway, sloped, above] {$\vect{v}$} (\Var{xa2}, \Var{ya2});
|
|
|
|
|
|
\draw[very thick, ->, color=blue] (0, 0)
|
|
-- ++ (\Var{xa1}, \Var{ya1}) node [midway, sloped, above] {$\vect{u}$}
|
|
-- ++ (\Var{xa2}, \Var{ya2}) node [midway, sloped, above] {$\vect{v}$} ;
|
|
\draw[very thick, ->, color=blue] (0, 0) -- node [midway, sloped, left] {$\vect{z}$} (\Var{xa2+xa1}, \Var{ya2+ya1});
|
|
\draw[very thick, ->, color=green] (0, 0)
|
|
-- ++ (\Var{xa1}, \Var{ya1}) node [midway, sloped, above] {$\vect{u}$}
|
|
-- ++ (\Var{xa1}, \Var{ya1}) node [midway, sloped, above] {$\vect{u}$}
|
|
-- ++ (-\Var{xa2}, -\Var{ya2}) node [midway, sloped, above] {$-\vect{v}$} ;
|
|
\draw[very thick, ->, color=green] (0, 0) -- node [midway, sloped, above] {$\vect{y}$} (\Var{2*xa1 - xa2}, \Var{2*ya1 - ya2});
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
\item
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.4]
|
|
\draw (\Var{minx-1}, \Var{miny-1}) rectangle (\Var{maxx+1}, \Var{maxy+1});
|
|
\draw[very thick, ->] (0, 0) -- node [midway, sloped, above] {$\vect{F_1}$} (\Var{x1}, \Var{y1});
|
|
\draw[very thick, -> ] (0, 0) -- node [midway, sloped, above] {$\vect{F_2}$} (\Var{x2}, \Var{y2});
|
|
\draw[very thick, -> ] (0, 0) -- node [midway, sloped, above] {$\vect{F_3}$} (\Var{x3}, \Var{y3});
|
|
|
|
\draw[very thick, ->, color=blue] (0, 0)
|
|
--++ (\Var{x1}, \Var{y1}) node [midway, sloped, above] {$\vect{F_1}$}
|
|
--++ (\Var{x2}, \Var{y2}) node [midway, sloped, above] {$\vect{F_2}$}
|
|
--++ (\Var{x3}, \Var{y3}) node [midway, sloped, above] {$\vect{F_3}$}
|
|
;
|
|
\draw[very thick, ->, color=blue] (0, 0) -- (\Var{x1+x2+x3}, \Var{y1+y2+y3}) node [midway, sloped, above] {$\vect{F_1}+\vect{F_2}+\vect{F_3}$};
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\end{enumerate}
|
|
\end{multicols}
|
|
\end{solution}
|
|
|
|
\begin{exercise}[subtitle={Statistiques}, points=5]
|
|
%- set center = random.randint(30, 50)
|
|
%- set qty = random.randint(20, 40)
|
|
%- set dataset = stat.Dataset.random(qty, rd_args=(center, 1.5), nbr_format=int)
|
|
Ci-dessous la taille des poissons péchés lors du dernier challenge PêcheParty.
|
|
\begin{center}
|
|
\Var{dataset.tabular_latex(ceil(qty/15))}
|
|
\end{center}
|
|
\begin{enumerate}
|
|
\item Décrire la série statistique et donner l'effectif total.
|
|
\item Calculer la moyenne, les quartiles, l'écart interquartile et la médiane de cette série statistique.
|
|
\item Quelle est la valeur de l'écart-type de cette série statistique?
|
|
\end{enumerate}
|
|
\end{exercise}
|
|
|
|
\begin{solution}
|
|
Dans cette correction les étapes de construction des indicateurs ne sont pas détaillés.
|
|
|
|
Tableau des effectifs
|
|
%- set wdataset = stat.WeightedDataset(dataset)
|
|
\begin{center}
|
|
\Var{wdataset.tabular_latex()}
|
|
\end{center}
|
|
La population sont les poissons péchés lors du dernier challenge PêcheParty. Les individus sont les poissons. Le caractère est la taille des poissons.
|
|
\begin{multicols}{2}
|
|
\begin{itemize}
|
|
\item Effectif total : $\Var{dataset.effectif_total()}$
|
|
\item Premier quartile $ Q_1 = \Var{dataset.quartile(1)}$ (position $\Var{dataset.posi_quartile(1)}$)
|
|
\item Médiane $ Me = \Var{dataset.quartile(2)}$ (position $\Var{dataset.posi_quartile(2)}$)
|
|
\item Troisième quartile $ Q_3 = \Var{dataset.quartile(3)}$ (position $\Var{dataset.posi_quartile(3)}$)
|
|
\item interquartile: $Q_3 - Q_1 = \Var{dataset.quartile(3)} - \Var{dataset.quartile(1)} = \Var{dataset.quartile(3) - dataset.quartile(1) }$
|
|
|
|
\item Moyenne: $\overline{x} = \Var{dataset.mean()}$
|
|
\item Écart-type: $\sigma = \Var{dataset.sd()}$
|
|
\end{itemize}
|
|
\end{multicols}
|
|
\end{solution}
|
|
|
|
|
|
\end{document}
|
|
|
|
%%% Local Variables:
|
|
%%% mode: latex
|
|
%%% TeX-master: "master"
|
|
%%% End:
|