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Bertrand Benjamin 2022-01-20 14:43:53 +01:00
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2nd/Evaluations/DM_2022-01-25/tpl_DM3.tex Normal file
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\documentclass[a4paper,10pt]{article}
\usepackage{myXsim}
\usepackage{pgfplots}
% \pgfplotsset{compat = newest}
% \usepgfplotslibrary{external}
% \tikzexternalize
% Title Page
\title{DM 3 \hfill \Var{Nom}}
\tribe{2nd6}
\date{À rendre pour mardi 25 janvier 2022}
\xsimsetup{
solution/print = false
}
\begin{document}
\maketitle
\begin{exercise}[subtitle={Calculs de fractions}, points=2]
Détailler les calculs suivants et donner le résultat sous la forme d'une fraction irréductible.
\begin{multicols}{3}
\begin{enumerate}[label={\Alph*=}]
%- set A = rdm.expression("{a} / {b} + {c}", ["a!=b", "b > 1"], global_config={"min_max":(1, 10)})
\item $\Var{A}$
%- set B = rdm.expression("{a} / {b} + {c} / {d}", ["a!=b", "b > 1"], global_config={"min_max":(-10, 10), "rejected":[-1, 0, 1]})
\item $\Var{B}$
%- set C = rdm.expression("{a} / {b} * {c} / {k*b}", ["a!=b", "c!=b", "b > 1"], global_config={"min_max":(1, 10)})
\item $\Var{C}$
\end{enumerate}
\end{multicols}
\end{exercise}
\begin{solution}
\begin{enumerate}[label={\Alph*=}]
\item $\Var{A.simplify().explain() | join('=')} = \Var{A.simplify().simplified}$
\item $\Var{B.simplify().explain() | join('=')} = \Var{B.simplify().simplified}$
\item $\Var{C.simplify().explain() | join('=')} = \Var{C.simplify().simplified}$
\end{enumerate}
\end{solution}
\begin{exercise}[subtitle={Développer}, points=2]
Développer puis réduire les expressions suivantes
\begin{multicols}{3}
\begin{enumerate}[label={\Alph*=}]
%- set A = rdm.expression("{a}x({c}x+{d}) - {b}x", [], global_config={"min_max":(1, 10)})
\item $\Var{A}$
%- set B = rdm.expression("({a}x+{b})({c}x+{d})", [], )
\item $\Var{B}$
%- set C = rdm.expression("({a}x+{b})^2", [], )
\item $\Var{C}$
\end{enumerate}
\end{multicols}
\end{exercise}
\begin{solution}
\begin{enumerate}
\item
\begin{align*}
A &= \Var{A.simplify().explain() | join('\\\\&=')}
\end{align*}
\item
\begin{align*}
B &= \Var{B.simplify().explain() | join('\\\\&=')}
\end{align*}
\item
\begin{align*}
C &= \Var{C.simplify().explain() | join('\\\\&=')}
\end{align*}
\end{enumerate}
\end{solution}
\begin{exercise}[subtitle={Inéquation et tableaux}, points=3]
Tracer le tableau de signe des fonctions suivantes en le démontrant à l'aide de la résolution d'une inéquation.
\begin{multicols}{2}
\begin{enumerate}
%- set f = rdm.expression("{a}x + {b}", global_config={"min_max":(1, 20)})
\item $f(x) = \Var{f}$
%- set g = rdm.expression("{a}x + {b}", global_config={"min_max":(1, 20)})
\item $g(x) = \Var{g}$
\end{enumerate}
\end{multicols}
\end{exercise}
\begin{solution}
\begin{enumerate}
\item
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} \\
\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
\begin{center}
\begin{tikzpicture}
\tkzTabInit[lgt=2,espcl=1]{$ t $/1,$ f(t) $/1}{, $\Var{racine}$ ,}
\tkzTabLine{, -, z, +, }
\end{tikzpicture}
\end{center}
\item
Pour déterminer les valeurs de $x$ pour lesquelles $g(x)$ est positive, il faut résoudre l'inéquation
%- set cst = -g[0]
%- set coef = g[1]
%- set racine = cst / coef
\begin{align*}
g(x) & \geq 0 \\
\Var{g} & \geq 0 \\
\Var{g + cst} &\geq \Var{0 + cst} \\
\frac{\Var{g + 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
\begin{center}
\begin{tikzpicture}
\tkzTabInit[lgt=2,espcl=1]{$ t $/1,$ g(t) $/1}{, $\Var{racine}$ ,}
\tkzTabLine{, -, z, +, }
\end{tikzpicture}
\end{center}
\end{enumerate}
\end{solution}
\begin{exercise}[subtitle={Vecteurs}, points=2]
\begin{enumerate}
\item Tracer les vecteurs $\vect{z} = \vect{u} + \vect{v}$ et $\vect{y} = 2\vect{u} - \vect{v}$ (le vecteur peur sortir du cadre)
%- set xa1, ya1 = rdm.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 = rdm.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{exercise}
\begin{solution}
\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{solution}
\end{document}