From 907a28ffdceb19258b5315afc8889b72037e674a Mon Sep 17 00:00:00 2001 From: simejo Date: Thu, 23 Sep 2021 21:35:47 +0200 Subject: [PATCH] added --- algebre_1_td_1.tex | 210 +++++++++++++++++++++++++ analyse_1_td_1.tex | 59 +++++++ mecanique_du_point_td_1.tex | 268 ++++++++++++++++++++++++++++++++ pdf/algebre_1_td_1.pdf | Bin 0 -> 92298 bytes pdf/analyse_1_td_1.pdf | Bin 39704 -> 76528 bytes pdf/mecanique_du_point_td_1.pdf | Bin 0 -> 83798 bytes 6 files changed, 537 insertions(+) create mode 100644 algebre_1_td_1.tex create mode 100644 mecanique_du_point_td_1.tex create mode 100644 pdf/algebre_1_td_1.pdf create mode 100644 pdf/mecanique_du_point_td_1.pdf diff --git a/algebre_1_td_1.tex b/algebre_1_td_1.tex new file mode 100644 index 0000000..6fad136 --- /dev/null +++ b/algebre_1_td_1.tex @@ -0,0 +1,210 @@ +\documentclass{article} + +\usepackage{mathtools} +\usepackage{amssymb} + +\title{Algèbre 1 - TD 1 \\ NON CORRIGÉ} +\author{Timéo Pochin} + +\begin{document} + \maketitle + + \section*{Exercice 1} + + \subsection*{1)} + \begin{align} + & \neg ( P \land \neg Q ) \nonumber \\ + \iff\quad & \neg P \lor Q \nonumber + \end{align} + + \subsection*{2)} + \begin{align} + & \neg\big(P\land (Q\land R)\big) \nonumber \\ + \iff\quad & \neg ( P \land Q \land R ) \nonumber \\ + \iff\quad & \neg P \lor \neg Q \lor \neg R \nonumber + \end{align} + + \subsection*{3)} + \begin{align} + & \neg\big(P\lor (Q\land R)\big) \nonumber \\ + \iff\quad & \neg P \land \neg (Q \land R) \nonumber \\ + \iff\quad & \neg P \land (\neg Q \lor \neg R) \nonumber + \end{align} + + \subsection*{4)} + \begin{align} + & \neg\big((P\land Q)\Rightarrow(R\Rightarrow S)\big) \nonumber \\ + \iff\quad & P \land Q \land \neg (R \Rightarrow S) \nonumber \\ + \iff\quad & P \land Q \land R \land \neg S \nonumber + \end{align} + + \section*{Exercice 2} + + \subsection*{} + \begin{tabular}{ |c|c|c||c|c|c| } + \hline + $P$ & $Q$ & $R$ & $\neg P$ & $\neg Q$ & $\neg P \Leftrightarrow \neg Q$ \\ \hline + \hline + F & F & F & V & V & V \\ \hline + F & F & V & V & V & V \\ \hline + V & V & F & F & F & V \\ \hline + V & V & V & F & F & V \\ \hline + \end{tabular} + + \subsection*{} + \begin{tabular}{ |c|c|c||c|c|c| } + \hline + $P$ & $Q$ & $R$ & $P\Rightarrow R$ & $Q\Rightarrow R$ & $(P\Rightarrow R)\Leftrightarrow(Q\Rightarrow R)$ \\ \hline + \hline + F & F & F & V & V & V \\ \hline + F & F & V & V & V & V \\ \hline + V & V & F & F & F & V \\ \hline + V & V & V & V & V & V \\ \hline + \end{tabular} + + \subsection*{} + \begin{tabular}{ |c|c|c||c|c|c| } + \hline + $P$ & $Q$ & $R$ & $R\Rightarrow P$ & $R\Rightarrow Q$ & $(R\Rightarrow P)\Leftrightarrow(R\Rightarrow Q)$ \\ \hline + \hline + F & F & F & V & V & V \\ \hline + F & F & V & F & F & V \\ \hline + V & V & F & V & V & V \\ \hline + V & V & V & V & V & V \\ \hline + \end{tabular} + + \subsection*{} + \begin{tabular}{ |c|c|c||c|c|c| } + \hline + $P$ & $Q$ & $R$ & $P\land R$ & $Q\land R$ & $(P\land R)\Leftrightarrow(Q\land R)$ \\ \hline + \hline + F & F & F & F & F & V \\ \hline + F & F & V & F & F & V \\ \hline + V & V & F & F & F & V \\ \hline + V & V & V & V & V & V \\ \hline + \end{tabular} + + \subsection*{} + \begin{tabular}{ |c|c|c||c|c|c| } + \hline + $P$ & $Q$ & $R$ & $P\lor R$ & $Q\lor R$ & $(P\lor R)\Leftrightarrow(Q\lor R)$ \\ \hline + \hline + F & F & F & F & F & V \\ \hline + F & F & V & V & V & V \\ \hline + V & V & F & V & V & V \\ \hline + V & V & V & V & V & V \\ \hline + \end{tabular} + + \section*{Exercice 3} + + \subsection*{1)} + \begin{tabular}{ |c|c||c|c|c|c|c|c| } + \hline + $P$ & $Q$ & $P\lor Q$ & $\neg(P\lor Q)$ & $\neg P$ & $\neg Q$ & $\neg P\land\neg Q$ & $\neg(P\lor Q)\Leftrightarrow\neg P\land\neg Q$ \\ \hline + \hline + F & F & F & V & V & V & V & V \\ \hline + F & V & V & F & V & F & F & V \\ \hline + V & F & V & F & F & V & F & V \\ \hline + V & V & V & F & F & F & F & V \\ \hline + \end{tabular} + + \subsection*{2)} + \begin{tabular}{ |c|c|c||c|c|c|c|c| } + \hline + $P$ & $Q$ & $R$ & $P\Rightarrow Q$ & $Q\Rightarrow R$ & $(P\Rightarrow Q)\land(Q\Rightarrow R)$ & $P\Rightarrow R$ & $\big((P\Rightarrow Q)\land(Q\Rightarrow R)\big)\Rightarrow(P\Rightarrow R)$ \\ \hline + \hline + %P Q R + F & F & F & V & V & V & V & V \\ \hline + F & F & V & V & V & V & V & V \\ \hline + F & V & F & V & F & F & V & V \\ \hline + F & V & V & V & V & V & V & V \\ \hline + V & F & F & F & V & F & F & V \\ \hline + V & F & V & F & V & F & V & V \\ \hline + V & V & F & V & F & F & F & V \\ \hline + V & V & V & V & V & V & V & V \\ \hline + \end{tabular} + + \section*{Exercice 4} + + \subsection*{1)} + \begin{align} + & (P\land Q)\Rightarrow P \nonumber \\ + \iff\quad & \neg(P\land Q)\lor P \nonumber \\ + \iff\quad & \neg P\lor\neg Q\lor P \nonumber \\ + \iff\quad & \text{Vrai}\lor\neg Q \nonumber \\ + \iff\quad & \text{Vrai} \nonumber + \end{align} + + \subsection*{2)} + \begin{align} + & \bigg(\big(P\Rightarrow(Q\Rightarrow R)\big)\Leftrightarrow\big((P\land Q)\Rightarrow R\big)\bigg) \nonumber \\ + \iff\quad & \bigg(\big(\neg P\lor(Q\Rightarrow R)\Leftrightarrow\neg(P\land Q)\lor R\big)\bigg) \nonumber \\ + \iff\quad & (\neg P\lor\neg Q\lor R\Leftrightarrow\neg P\lor\neg Q\lor R) \nonumber \\ + \iff\quad & \text{Vrai} \nonumber + \end{align} + + \subsection*{3)} + \begin{align} + & \bigg(\big((P\land Q)\Rightarrow R\big)\Leftrightarrow\big(\neg P\lor(Q\Rightarrow R)\big)\bigg) \nonumber \\ + \iff\quad & \bigg(\big(\neg(P\land Q)\lor R\big)\Leftrightarrow(\neg P\lor\neg Q\lor R)\bigg) \nonumber \\ + \iff\quad & \big((\neg P\lor\neg Q\lor R)\Leftrightarrow(\neg P\lor\neg Q\lor R)\big) \nonumber \\ + \iff\quad & \text{Vrai} \nonumber + \end{align} + + \section*{Exercice 5} + + \subsection*{1)} + Faux + + \subsection*{2)} + Vrai + + \subsection*{3)} + Faux + + \section*{Exercice 6} + + \subsection*{1)} + Il existe un triangle rectangle qui ne possède pas un angle droit. + + \subsection*{2)} + Il existe une maison où au moins un enfant n’aime pas ses deux parents. + + \subsection*{3)} + Il existe un entier $x$ tel que, pour tout entier $y$, pour tout entier $z$, la relation $z\geq x+1$ implique la relation $z\geq y$. + + \subsection*{4)} + \[ + \exists\epsilon>0,\quad \forall\alpha>0,\quad |5x-7|\geq\epsilon\Rightarrow|x-\frac{7}{5}|\geq\alpha + \] + + \section*{Exercice 7} + + \subsection*{1)} + \[ + \forall n\in\mathbb{R}^+,\quad \exists a\in\{x\in\mathbb{R}\mid-n-4) \nonumber \\ + \iff\quad & (x<5)\land(x>-3) \nonumber \\ + \iff\quad & x=\left]-3,5\right[ \nonumber + \end{align} \subsection*{d)} + \begin{align} + & |x-2|\geq 3 \nonumber \\ + \iff\quad & (x-2\geq 3)\lor(x-2\leq-3) \nonumber \\ + \iff\quad & (x\geq 5)\lor(x\leq-1) \nonumber \\ + \iff\quad & x=\left]-\infty,-1\right]\ \cup\ \left[5,+\infty\right[ \nonumber + \end{align} \subsection*{e)} + \begin{align} + & |x-2|\leq|x| \nonumber \\ + \iff\quad & (x-2)^2\leq x^2 \nonumber \\ + \iff\quad & x^2-4x+4\leq x^2 \nonumber \\ + \iff\quad & 4\leq 4x \nonumber \\ + \iff\quad & 1\leq x \nonumber \\ + \iff\quad & x=\left[1,+\infty\right[ \nonumber + \end{align} \subsection*{f)} \subsection*{g)} + \begin{align} + & \sqrt{x+1}<2 \nonumber \\ + \iff\quad & (x+1<4)\land(x+1\geq0) \nonumber \\ + \iff\quad & (x<3)\land(x\geq-1) \nonumber \\ + \iff\quad & x=\left[-1,3\right[ \nonumber + \end{align} \subsection*{h)} + \begin{align} + & x^2+1\leq 3 \nonumber \\ + \iff\quad & x^2\leq 2 \nonumber \\ + \iff\quad & (x\leq\sqrt{2})\land(x\geq-\sqrt{2}) \nonumber \\ + \iff\quad & x=\left[-\sqrt{2},\sqrt{2}\right] \nonumber + \end{align} \subsection*{i)} + \begin{align} + & x^2+3x<4 \nonumber \\ + \iff\quad & x^2+3x-4<0 \nonumber \\ + \iff\quad & (x-1)(x+4)<0 \nonumber \\ + \iff\quad & (x<1)\land(x>-4) \nonumber \\ + \iff\quad & x=\left]-4,1\right[ \nonumber + \end{align} \subsection*{j)} + \begin{align} + & x^3-3x^2+2x\geq 0 \nonumber \\ + \iff\quad & x(x^2-3x+2)\geq 0 \nonumber \\ + \iff\quad & x(x-1)(x-2)\geq 0 \nonumber \\ + \iff\quad & \big((x\geq 0)\land(x\leq 1)\big)\lor(x\geq 2) \nonumber \\ + \iff\quad & x=\left[0,1\right]\cup\left[2,+\infty\right[ \nonumber + \end{align} \subsection*{k)} diff --git a/mecanique_du_point_td_1.tex b/mecanique_du_point_td_1.tex new file mode 100644 index 0000000..44e3fc5 --- /dev/null +++ b/mecanique_du_point_td_1.tex @@ -0,0 +1,268 @@ +\documentclass{article} + +\usepackage{mathtools} +\usepackage{amssymb} + +\title{Mécanique du point - TD 1 \\ NON CORRIGÉ} +\author{Timéo Pochin} + +\begin{document} + \maketitle + + \section*{Exercice 1} + + \subsection*{a)} + \begin{align} + \vec{u}\land\vec{v} + \nonumber + &= + \begin{pmatrix} + 1 \\ + 2 \\ + 3 + \end{pmatrix} + \land + \begin{pmatrix} + 4 \\ + 5 \\ + 6 + \end{pmatrix} + \\ \nonumber + &= + \begin{pmatrix} + 2\cdot 6-5\cdot 3 \\ + 4\cdot 3-1\cdot 6 \\ + 1\cdot 5-4\cdot 2 + \end{pmatrix} + \\ \nonumber + &= + \begin{pmatrix} + -3 \\ + 6 \\ + -3 + \end{pmatrix} + \\ \nonumber + &=-3\vec{e_x}+6\vec{e_y}-3\vec{e_z} + \end{align} + + \subsection*{b)} + \begin{align} + \vec{v}\land\vec{u} + \nonumber + &= + \begin{pmatrix} + 4 \\ + 5 \\ + 6 + \end{pmatrix} + \land + \begin{pmatrix} + 1 \\ + 2 \\ + 3 + \end{pmatrix} + \\ \nonumber + &= + \begin{pmatrix} + 5\cdot 3-2\cdot 6 \\ + 1\cdot 6-4\cdot 3 \\ + 4\cdot 2-1\cdot 5 + \end{pmatrix} + \\ \nonumber + &= + \begin{pmatrix} + 3 \\ + -6 \\ + 3 + \end{pmatrix} + \\ \nonumber + &=3\vec{e_x}-6\vec{e_y}+3\vec{e_z} + \end{align} + + \subsection*{c)} + \begin{align} + \vec{u}\cdot\vec{v} + \nonumber + &= + \begin{pmatrix} + 1 \\ + 2 \\ + 3 + \end{pmatrix} + \cdot + \begin{pmatrix} + 4 \\ + 5 \\ + 6 + \end{pmatrix} + \\ \nonumber + &=1\cdot 4+2\cdot 5+3\cdot 6\\ \nonumber + &=32 + \end{align} + + \subsection*{d)} + \begin{align} + \vec{u}\cdot(\vec{u}\land\vec{v}) + \nonumber + &= + \begin{pmatrix} + 1 \\ + 2 \\ + 3 + \end{pmatrix} + \cdot + \begin{pmatrix} + -3 \\ + 6 \\ + -3 + \end{pmatrix} + \\ \nonumber + &=1\cdot-3+2\cdot 6+3\cdot -3\\ \nonumber + &=0 + \end{align} + + \subsection*{e)} + \begin{align} + \|\vec{u}\| + \nonumber + &=\sqrt{1^2+2^2+3^2} + \\ \nonumber + &=\sqrt{14} + \end{align} + + \subsection*{f)} + \begin{align} + \|\vec{u}+\vec{v}\| + \nonumber + &= + \left\| + \begin{pmatrix} + 1 \\ + 2 \\ + 3 + \end{pmatrix} + + + \begin{pmatrix} + 4 \\ + 5 \\ + 6 + \end{pmatrix} + \right\| + \\ \nonumber + &= + \left\| + \begin{pmatrix} + 5 \\ + 7 \\ + 9 + \end{pmatrix} + \right\| + \\ \nonumber + &=\sqrt{5^2+7^2+9^2} + \\ \nonumber + &=\sqrt{155} + \end{align} + + \subsection*{g)} + \begin{align} + (\vec{u}\land\vec{v})\land\vec{w} + \nonumber + &= + \begin{pmatrix} + -3 \\ + 6 \\ + -3 + \end{pmatrix} + \land + \begin{pmatrix} + 7 \\ + 8 \\ + 9 + \end{pmatrix} + \\ \nonumber + &= + \begin{pmatrix} + 6\cdot 9-8\cdot-3 \\ + 7\cdot-3-9\cdot-3 \\ + -3\cdot 8-7\cdot 6 + \end{pmatrix} + \\ \nonumber + &= + \begin{pmatrix} + 78 \\ + 6 \\ + -66 + \end{pmatrix} + \\ \nonumber + &=78\vec{e_x}+6\vec{e_y}-66\vec{e_z} + \end{align} + + \subsection*{h)} + \begin{align} + \vec{u}\land(\vec{v}\land\vec{w}) + \nonumber + &= + \begin{pmatrix} + 1 \\ + 2 \\ + 3 + \end{pmatrix} + \land + \left( + \begin{pmatrix} + 4 \\ + 5 \\ + 6 + \end{pmatrix} + \land + \begin{pmatrix} + 7 \\ + 8 \\ + 9 + \end{pmatrix} + \right) + \\ \nonumber + &= + \begin{pmatrix} + 1 \\ + 2 \\ + 3 + \end{pmatrix} + \land + \begin{pmatrix} + 5\cdot 9-8\cdot 6 \\ + 7\cdot 6-4\cdot 9 \\ + 4\cdot 8-7\cdot 5 + \end{pmatrix} + \\ \nonumber + &= + \begin{pmatrix} + 1 \\ + 2 \\ + 3 + \end{pmatrix} + \land + \begin{pmatrix} + -3 \\ + 6 \\ + -3 + \end{pmatrix} + \\ \nonumber + &= + \begin{pmatrix} + 2\cdot-3-6\cdot 3 \\ + -3\cdot 3-1\cdot-3 \\ + 1\cdot 6--3\cdot 3 + \end{pmatrix} + \\ \nonumber + &= + \begin{pmatrix} + -24 \\ + -6 \\ + 12 + \end{pmatrix} + \\ \nonumber + &= -24\vec{e_x}-6\vec{e_y}+12\vec{e_z} + \end{align} + +\end{document} diff --git a/pdf/algebre_1_td_1.pdf b/pdf/algebre_1_td_1.pdf new file mode 100644 index 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