From 8abb9b60175bba982d34f1f7ca15c3e3ea77019c Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Tue, 3 Nov 2020 11:11:32 +0100 Subject: [PATCH] Feat: compile corr DM1 TST1 --- TST/DM/2010_DM1/TST1/corr_01_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_02_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_03_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_04_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_05_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_06_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_07_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_08_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_09_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_10_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_11_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_12_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_13_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_14_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_15_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_16_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_17_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_18_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_19_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_20_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_21_2010_DM1.tex | 141 +++++++++++++++++++++ TST/DM/2010_DM1/TST1/corr_all_2010_DM1.pdf | Bin 0 -> 239321 bytes 22 files changed, 2961 insertions(+) create mode 100644 TST/DM/2010_DM1/TST1/corr_01_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_02_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_03_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_04_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_05_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_06_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_07_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_08_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_09_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_10_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_11_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_12_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_13_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_14_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_15_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_16_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_17_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_18_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_19_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_20_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_21_2010_DM1.tex create mode 100644 TST/DM/2010_DM1/TST1/corr_all_2010_DM1.pdf diff --git a/TST/DM/2010_DM1/TST1/corr_01_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_01_2010_DM1.tex new file mode 100644 index 0000000..a73f22b --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_01_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill AIOUAZ Ahmed} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{- 5}{5} - \dfrac{- 2}{5}$ + \item $B = \dfrac{7}{3} - \dfrac{8}{9}$ + + \item $C = \dfrac{- 4}{2} + \dfrac{2}{1}$ + \item $D = \dfrac{6}{10} + 9$ + + \item $E = \dfrac{- 7}{5} \times \dfrac{- 6}{4}$ + \item $F = \dfrac{- 10}{5} \times - 8$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{- 5}{5} - \dfrac{- 2}{5}=\dfrac{- 5}{5} + \dfrac{2}{5}=\dfrac{- 5 + 2}{5}=\dfrac{- 3}{5} + \] + \item + \[ + \dfrac{7}{3} - \dfrac{8}{9}=\dfrac{7}{3} - \dfrac{8}{9}=\dfrac{7 \times 3}{3 \times 3} - \dfrac{8}{9}=\dfrac{21}{9} - \dfrac{8}{9}=\dfrac{21 - 8}{9}=\dfrac{21 - 8}{9}=\dfrac{13}{9} + \] + \item + \[ + \dfrac{- 4}{2} + \dfrac{2}{1}=\dfrac{- 4}{2} + \dfrac{2 \times 2}{1 \times 2}=\dfrac{- 4}{2} + \dfrac{4}{2}=\dfrac{- 4 + 4}{2}=\dfrac{0}{2} + \] + \item + \[ + \dfrac{6}{10} + 9=\dfrac{6}{10} + \dfrac{9}{1}=\dfrac{6}{10} + \dfrac{9 \times 10}{1 \times 10}=\dfrac{6}{10} + \dfrac{90}{10}=\dfrac{6 + 90}{10}=\dfrac{96}{10} + \] + \item + \[ + \dfrac{- 7}{5} \times \dfrac{- 6}{4}=\dfrac{- 7 \times - 6}{5 \times 4}=\dfrac{42}{20} + \] + \item + \[ + \dfrac{- 10}{5} \times - 8=\dfrac{- 10 \times - 8}{5}=\dfrac{80}{5} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (- 1x + 10)(- 7x + 10)$ + \item $B = (8x - 1)(8x - 1)$ + + \item $C = (- 1x - 3)^{2}$ + \item $D = - 7 + x(- 3x - 6)$ + + \item $E = 2x^{2} + x(3x - 9)$ + \item $F = 4(x + 4)(x - 4)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (- 1x + 10)(- 7x + 10)\\&= - x \times - 7x - x \times 10 + 10 \times - 7x + 10 \times 10\\&= - 1 \times - 7 \times x^{1 + 1} + 10 \times - 1 \times x + 10 \times - 7 \times x + 100\\&= - 10x - 70x + 7x^{2} + 100\\&= (- 10 - 70) \times x + 7x^{2} + 100\\&= 7x^{2} - 80x + 100 + \end{align*} + \item + \begin{align*} + B &= (8x - 1)(8x - 1)\\&= 8x \times 8x + 8x \times - 1 - 1 \times 8x - 1 \times - 1\\&= 8 \times 8 \times x^{1 + 1} - 1 \times 8 \times x - 1 \times 8 \times x + 1\\&= - 8x - 8x + 64x^{2} + 1\\&= (- 8 - 8) \times x + 64x^{2} + 1\\&= 64x^{2} - 16x + 1 + \end{align*} + \item + \begin{align*} + C &= (- 1x - 3)^{2}\\&= (- x - 3)(- x - 3)\\&= - x \times - x - x \times - 3 - 3 \times - x - 3 \times - 3\\&= - 1 \times - 1 \times x^{1 + 1} - 3 \times - 1 \times x - 3 \times - 1 \times x + 9\\&= 3x + 3x + 1x^{2} + 9\\&= (3 + 3) \times x + x^{2} + 9\\&= x^{2} + 6x + 9 + \end{align*} + \item + \begin{align*} + D &= - 7 + x(- 3x - 6)\\&= - 7 + x \times - 3x + x \times - 6\\&= - 3x^{2} - 6x - 7 + \end{align*} + \item + \begin{align*} + E &= 2x^{2} + x(3x - 9)\\&= 2x^{2} + x \times 3x + x \times - 9\\&= 2x^{2} + 3x^{2} - 9x\\&= 2x^{2} + 3x^{2} - 9x\\&= (2 + 3) \times x^{2} - 9x\\&= 5x^{2} - 9x + \end{align*} + \item + \begin{align*} + F &= 4(x + 4)(x - 4)\\&= (4x + 4 \times 4)(x - 4)\\&= (4x + 16)(x - 4)\\&= 4x \times x + 4x \times - 4 + 16x + 16 \times - 4\\&= - 4 \times 4 \times x - 64 + 4x^{2} + 16x\\&= - 16x - 64 + 4x^{2} + 16x\\&= 4x^{2} - 16x + 16x - 64\\&= 4x^{2} + (- 16 + 16) \times x - 64\\&= 4x^{2} - 64 + 0x\\&= 4x^{2} - 64 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = - 3x^{2} - 24x + 60$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = - 3 \times 1^{2} - 24 \times 1 + 60=- 3 \times 1 - 24 + 60=- 3 + 36=33 + \] + \[ + f(-1) = - 3 \times - 1^{2} - 24 \times - 1 + 60=- 3 \times 1 + 24 + 60=- 3 + 84=81 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_02_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_02_2010_DM1.tex new file mode 100644 index 0000000..dee783d --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_02_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill BAHBAH Zakaria} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{- 9}{7} - \dfrac{- 10}{7}$ + \item $B = \dfrac{6}{6} - \dfrac{- 5}{24}$ + + \item $C = \dfrac{- 10}{5} + \dfrac{7}{4}$ + \item $D = \dfrac{- 5}{9} - 10$ + + \item $E = \dfrac{6}{10} \times \dfrac{- 7}{9}$ + \item $F = \dfrac{- 6}{4} \times 8$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{- 9}{7} - \dfrac{- 10}{7}=\dfrac{- 9}{7} + \dfrac{10}{7}=\dfrac{- 9 + 10}{7}=\dfrac{1}{7} + \] + \item + \[ + \dfrac{6}{6} - \dfrac{- 5}{24}=\dfrac{6}{6} + \dfrac{5}{24}=\dfrac{6 \times 4}{6 \times 4} + \dfrac{5}{24}=\dfrac{24}{24} + \dfrac{5}{24}=\dfrac{24 + 5}{24}=\dfrac{29}{24} + \] + \item + \[ + \dfrac{- 10}{5} + \dfrac{7}{4}=\dfrac{- 10 \times 4}{5 \times 4} + \dfrac{7 \times 5}{4 \times 5}=\dfrac{- 40}{20} + \dfrac{35}{20}=\dfrac{- 40 + 35}{20}=\dfrac{- 5}{20} + \] + \item + \[ + \dfrac{- 5}{9} - 10=\dfrac{- 5}{9} + \dfrac{- 10}{1}=\dfrac{- 5}{9} + \dfrac{- 10 \times 9}{1 \times 9}=\dfrac{- 5}{9} + \dfrac{- 90}{9}=\dfrac{- 5 - 90}{9}=\dfrac{- 95}{9} + \] + \item + \[ + \dfrac{6}{10} \times \dfrac{- 7}{9}=\dfrac{6 \times - 7}{10 \times 9}=\dfrac{- 42}{90} + \] + \item + \[ + \dfrac{- 6}{4} \times 8=\dfrac{- 6 \times 8}{4}=\dfrac{- 48}{4} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (10x - 9)(- 4x - 9)$ + \item $B = (1x - 9)(1x - 9)$ + + \item $C = (9x - 10)^{2}$ + \item $D = - 1 + x(- 6x - 10)$ + + \item $E = - 10x^{2} + x(- 7x - 3)$ + \item $F = 7(x - 7)(x + 2)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (10x - 9)(- 4x - 9)\\&= 10x \times - 4x + 10x \times - 9 - 9 \times - 4x - 9 \times - 9\\&= 10 \times - 4 \times x^{1 + 1} - 9 \times 10 \times x - 9 \times - 4 \times x + 81\\&= - 90x + 36x - 40x^{2} + 81\\&= (- 90 + 36) \times x - 40x^{2} + 81\\&= - 40x^{2} - 54x + 81 + \end{align*} + \item + \begin{align*} + B &= (1x - 9)(1x - 9)\\&= x \times x + x \times - 9 - 9x - 9 \times - 9\\&= x^{2} + 81 + (- 9 - 9) \times x\\&= x^{2} - 18x + 81 + \end{align*} + \item + \begin{align*} + C &= (9x - 10)^{2}\\&= (9x - 10)(9x - 10)\\&= 9x \times 9x + 9x \times - 10 - 10 \times 9x - 10 \times - 10\\&= 9 \times 9 \times x^{1 + 1} - 10 \times 9 \times x - 10 \times 9 \times x + 100\\&= - 90x - 90x + 81x^{2} + 100\\&= (- 90 - 90) \times x + 81x^{2} + 100\\&= 81x^{2} - 180x + 100 + \end{align*} + \item + \begin{align*} + D &= - 1 + x(- 6x - 10)\\&= - 1 + x \times - 6x + x \times - 10\\&= - 6x^{2} - 10x - 1 + \end{align*} + \item + \begin{align*} + E &= - 10x^{2} + x(- 7x - 3)\\&= - 10x^{2} + x \times - 7x + x \times - 3\\&= - 10x^{2} - 7x^{2} - 3x\\&= - 10x^{2} - 7x^{2} - 3x\\&= (- 10 - 7) \times x^{2} - 3x\\&= - 17x^{2} - 3x + \end{align*} + \item + \begin{align*} + F &= 7(x - 7)(x + 2)\\&= (7x + 7 \times - 7)(x + 2)\\&= (7x - 49)(x + 2)\\&= 7x \times x + 7x \times 2 - 49x - 49 \times 2\\&= 2 \times 7 \times x - 98 + 7x^{2} - 49x\\&= 14x - 98 + 7x^{2} - 49x\\&= 7x^{2} + 14x - 49x - 98\\&= 7x^{2} + (14 - 49) \times x - 98\\&= 7x^{2} - 35x - 98 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = - 7x^{2} - 98x - 280$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = - 7 \times 1^{2} - 98 \times 1 - 280=- 7 \times 1 - 98 - 280=- 7 - 378=- 385 + \] + \[ + f(-1) = - 7 \times - 1^{2} - 98 \times - 1 - 280=- 7 \times 1 + 98 - 280=- 7 - 182=- 189 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_03_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_03_2010_DM1.tex new file mode 100644 index 0000000..e6f1353 --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_03_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill BALLOFFET Kenza} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{- 10}{3} - \dfrac{8}{3}$ + \item $B = \dfrac{3}{3} - \dfrac{2}{15}$ + + \item $C = \dfrac{1}{4} + \dfrac{- 3}{3}$ + \item $D = \dfrac{6}{10} + 10$ + + \item $E = \dfrac{- 7}{2} \times \dfrac{5}{1}$ + \item $F = \dfrac{2}{4} \times 1$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{- 10}{3} - \dfrac{8}{3}=\dfrac{- 10}{3} - \dfrac{8}{3}=\dfrac{- 10 - 8}{3}=\dfrac{- 10 - 8}{3}=\dfrac{- 18}{3} + \] + \item + \[ + \dfrac{3}{3} - \dfrac{2}{15}=\dfrac{3}{3} - \dfrac{2}{15}=\dfrac{3 \times 5}{3 \times 5} - \dfrac{2}{15}=\dfrac{15}{15} - \dfrac{2}{15}=\dfrac{15 - 2}{15}=\dfrac{15 - 2}{15}=\dfrac{13}{15} + \] + \item + \[ + \dfrac{1}{4} + \dfrac{- 3}{3}=\dfrac{1 \times 3}{4 \times 3} + \dfrac{- 3 \times 4}{3 \times 4}=\dfrac{3}{12} + \dfrac{- 12}{12}=\dfrac{3 - 12}{12}=\dfrac{- 9}{12} + \] + \item + \[ + \dfrac{6}{10} + 10=\dfrac{6}{10} + \dfrac{10}{1}=\dfrac{6}{10} + \dfrac{10 \times 10}{1 \times 10}=\dfrac{6}{10} + \dfrac{100}{10}=\dfrac{6 + 100}{10}=\dfrac{106}{10} + \] + \item + \[ + \dfrac{- 7}{2} \times \dfrac{5}{1}=\dfrac{- 7 \times 5}{2 \times 1}=\dfrac{- 35}{2} + \] + \item + \[ + \dfrac{2}{4} \times 1=\dfrac{2}{4} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (7x - 8)(- 3x - 8)$ + \item $B = (5x + 7)(- 9x + 7)$ + + \item $C = (- 10x - 9)^{2}$ + \item $D = 1 + x(8x + 2)$ + + \item $E = 10x^{2} + x(- 6x - 4)$ + \item $F = 1(x - 7)(x + 5)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (7x - 8)(- 3x - 8)\\&= 7x \times - 3x + 7x \times - 8 - 8 \times - 3x - 8 \times - 8\\&= 7 \times - 3 \times x^{1 + 1} - 8 \times 7 \times x - 8 \times - 3 \times x + 64\\&= - 56x + 24x - 21x^{2} + 64\\&= (- 56 + 24) \times x - 21x^{2} + 64\\&= - 21x^{2} - 32x + 64 + \end{align*} + \item + \begin{align*} + B &= (5x + 7)(- 9x + 7)\\&= 5x \times - 9x + 5x \times 7 + 7 \times - 9x + 7 \times 7\\&= 5 \times - 9 \times x^{1 + 1} + 7 \times 5 \times x + 7 \times - 9 \times x + 49\\&= 35x - 63x - 45x^{2} + 49\\&= (35 - 63) \times x - 45x^{2} + 49\\&= - 45x^{2} - 28x + 49 + \end{align*} + \item + \begin{align*} + C &= (- 10x - 9)^{2}\\&= (- 10x - 9)(- 10x - 9)\\&= - 10x \times - 10x - 10x \times - 9 - 9 \times - 10x - 9 \times - 9\\&= - 10 \times - 10 \times x^{1 + 1} - 9 \times - 10 \times x - 9 \times - 10 \times x + 81\\&= 90x + 90x + 100x^{2} + 81\\&= (90 + 90) \times x + 100x^{2} + 81\\&= 100x^{2} + 180x + 81 + \end{align*} + \item + \begin{align*} + D &= 1 + x(8x + 2)\\&= 1 + x \times 8x + x \times 2\\&= 8x^{2} + 2x + 1 + \end{align*} + \item + \begin{align*} + E &= 10x^{2} + x(- 6x - 4)\\&= 10x^{2} + x \times - 6x + x \times - 4\\&= 10x^{2} - 6x^{2} - 4x\\&= 10x^{2} - 6x^{2} - 4x\\&= (10 - 6) \times x^{2} - 4x\\&= 4x^{2} - 4x + \end{align*} + \item + \begin{align*} + F &= 1(x - 7)(x + 5)\\&= (x - 7)(x + 5)\\&= x \times x + x \times 5 - 7x - 7 \times 5\\&= x^{2} - 35 + (5 - 7) \times x\\&= x^{2} - 2x - 35 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = - 9x^{2} + 54x - 72$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = - 9 \times 1^{2} + 54 \times 1 - 72=- 9 \times 1 + 54 - 72=- 9 - 18=- 27 + \] + \[ + f(-1) = - 9 \times - 1^{2} + 54 \times - 1 - 72=- 9 \times 1 - 54 - 72=- 9 - 126=- 135 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_04_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_04_2010_DM1.tex new file mode 100644 index 0000000..ccf3d3f --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_04_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill BENHATTAL Chakir} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{5}{3} - \dfrac{5}{3}$ + \item $B = \dfrac{6}{7} - \dfrac{6}{14}$ + + \item $C = \dfrac{- 7}{8} + \dfrac{- 1}{7}$ + \item $D = \dfrac{- 8}{2} + 10$ + + \item $E = \dfrac{- 2}{9} \times \dfrac{- 9}{8}$ + \item $F = \dfrac{- 3}{3} \times - 10$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{5}{3} - \dfrac{5}{3}=\dfrac{5}{3} - \dfrac{5}{3}=\dfrac{5 - 5}{3}=\dfrac{5 - 5}{3}=\dfrac{0}{3} + \] + \item + \[ + \dfrac{6}{7} - \dfrac{6}{14}=\dfrac{6}{7} - \dfrac{6}{14}=\dfrac{6 \times 2}{7 \times 2} - \dfrac{6}{14}=\dfrac{12}{14} - \dfrac{6}{14}=\dfrac{12 - 6}{14}=\dfrac{12 - 6}{14}=\dfrac{6}{14} + \] + \item + \[ + \dfrac{- 7}{8} + \dfrac{- 1}{7}=\dfrac{- 7 \times 7}{8 \times 7} + \dfrac{- 1 \times 8}{7 \times 8}=\dfrac{- 49}{56} + \dfrac{- 8}{56}=\dfrac{- 49 - 8}{56}=\dfrac{- 57}{56} + \] + \item + \[ + \dfrac{- 8}{2} + 10=\dfrac{- 8}{2} + \dfrac{10}{1}=\dfrac{- 8}{2} + \dfrac{10 \times 2}{1 \times 2}=\dfrac{- 8}{2} + \dfrac{20}{2}=\dfrac{- 8 + 20}{2}=\dfrac{12}{2} + \] + \item + \[ + \dfrac{- 2}{9} \times \dfrac{- 9}{8}=\dfrac{- 2 \times - 9}{9 \times 8}=\dfrac{18}{72} + \] + \item + \[ + \dfrac{- 3}{3} \times - 10=\dfrac{- 3 \times - 10}{3}=\dfrac{30}{3} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (- 10x + 2)(9x + 2)$ + \item $B = (2x - 7)(- 9x - 7)$ + + \item $C = (4x - 8)^{2}$ + \item $D = - 3 + x(9x - 8)$ + + \item $E = - 4x^{2} + x(6x - 5)$ + \item $F = - 10(x + 5)(x + 9)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (- 10x + 2)(9x + 2)\\&= - 10x \times 9x - 10x \times 2 + 2 \times 9x + 2 \times 2\\&= - 10 \times 9 \times x^{1 + 1} + 2 \times - 10 \times x + 2 \times 9 \times x + 4\\&= - 20x + 18x - 90x^{2} + 4\\&= (- 20 + 18) \times x - 90x^{2} + 4\\&= - 90x^{2} - 2x + 4 + \end{align*} + \item + \begin{align*} + B &= (2x - 7)(- 9x - 7)\\&= 2x \times - 9x + 2x \times - 7 - 7 \times - 9x - 7 \times - 7\\&= 2 \times - 9 \times x^{1 + 1} - 7 \times 2 \times x - 7 \times - 9 \times x + 49\\&= - 14x + 63x - 18x^{2} + 49\\&= (- 14 + 63) \times x - 18x^{2} + 49\\&= - 18x^{2} + 49x + 49 + \end{align*} + \item + \begin{align*} + C &= (4x - 8)^{2}\\&= (4x - 8)(4x - 8)\\&= 4x \times 4x + 4x \times - 8 - 8 \times 4x - 8 \times - 8\\&= 4 \times 4 \times x^{1 + 1} - 8 \times 4 \times x - 8 \times 4 \times x + 64\\&= - 32x - 32x + 16x^{2} + 64\\&= (- 32 - 32) \times x + 16x^{2} + 64\\&= 16x^{2} - 64x + 64 + \end{align*} + \item + \begin{align*} + D &= - 3 + x(9x - 8)\\&= - 3 + x \times 9x + x \times - 8\\&= 9x^{2} - 8x - 3 + \end{align*} + \item + \begin{align*} + E &= - 4x^{2} + x(6x - 5)\\&= - 4x^{2} + x \times 6x + x \times - 5\\&= - 4x^{2} + 6x^{2} - 5x\\&= - 4x^{2} + 6x^{2} - 5x\\&= (- 4 + 6) \times x^{2} - 5x\\&= 2x^{2} - 5x + \end{align*} + \item + \begin{align*} + F &= - 10(x + 5)(x + 9)\\&= (- 10x - 10 \times 5)(x + 9)\\&= (- 10x - 50)(x + 9)\\&= - 10x \times x - 10x \times 9 - 50x - 50 \times 9\\&= 9 \times - 10 \times x - 450 - 10x^{2} - 50x\\&= - 90x - 450 - 10x^{2} - 50x\\&= - 10x^{2} - 90x - 50x - 450\\&= - 10x^{2} + (- 90 - 50) \times x - 450\\&= - 10x^{2} - 140x - 450 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = - 2x^{2} + 8x + 24$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = - 2 \times 1^{2} + 8 \times 1 + 24=- 2 \times 1 + 8 + 24=- 2 + 32=30 + \] + \[ + f(-1) = - 2 \times - 1^{2} + 8 \times - 1 + 24=- 2 \times 1 - 8 + 24=- 2 + 16=14 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_05_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_05_2010_DM1.tex new file mode 100644 index 0000000..06eff6a --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_05_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill CLAIN Avinash} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{4}{4} - \dfrac{- 9}{4}$ + \item $B = \dfrac{2}{6} - \dfrac{1}{60}$ + + \item $C = \dfrac{4}{6} + \dfrac{- 3}{5}$ + \item $D = \dfrac{- 8}{7} - 8$ + + \item $E = \dfrac{- 1}{3} \times \dfrac{- 4}{2}$ + \item $F = \dfrac{2}{6} \times 8$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{4}{4} - \dfrac{- 9}{4}=\dfrac{4}{4} + \dfrac{9}{4}=\dfrac{4 + 9}{4}=\dfrac{13}{4} + \] + \item + \[ + \dfrac{2}{6} - \dfrac{1}{60}=\dfrac{2}{6} - \dfrac{1}{60}=\dfrac{2 \times 10}{6 \times 10} - \dfrac{1}{60}=\dfrac{20}{60} - \dfrac{1}{60}=\dfrac{20 - 1}{60}=\dfrac{20 - 1}{60}=\dfrac{19}{60} + \] + \item + \[ + \dfrac{4}{6} + \dfrac{- 3}{5}=\dfrac{4 \times 5}{6 \times 5} + \dfrac{- 3 \times 6}{5 \times 6}=\dfrac{20}{30} + \dfrac{- 18}{30}=\dfrac{20 - 18}{30}=\dfrac{2}{30} + \] + \item + \[ + \dfrac{- 8}{7} - 8=\dfrac{- 8}{7} + \dfrac{- 8}{1}=\dfrac{- 8}{7} + \dfrac{- 8 \times 7}{1 \times 7}=\dfrac{- 8}{7} + \dfrac{- 56}{7}=\dfrac{- 8 - 56}{7}=\dfrac{- 64}{7} + \] + \item + \[ + \dfrac{- 1}{3} \times \dfrac{- 4}{2}=\dfrac{- 1 \times - 4}{3 \times 2}=\dfrac{4}{6} + \] + \item + \[ + \dfrac{2}{6} \times 8=\dfrac{2 \times 8}{6}=\dfrac{16}{6} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (9x - 7)(3x - 7)$ + \item $B = (- 10x - 7)(9x - 7)$ + + \item $C = (3x - 7)^{2}$ + \item $D = 9 + x(- 1x - 4)$ + + \item $E = - 2x^{2} + x(10x + 6)$ + \item $F = 2(x - 9)(x + 7)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (9x - 7)(3x - 7)\\&= 9x \times 3x + 9x \times - 7 - 7 \times 3x - 7 \times - 7\\&= 9 \times 3 \times x^{1 + 1} - 7 \times 9 \times x - 7 \times 3 \times x + 49\\&= - 63x - 21x + 27x^{2} + 49\\&= (- 63 - 21) \times x + 27x^{2} + 49\\&= 27x^{2} - 84x + 49 + \end{align*} + \item + \begin{align*} + B &= (- 10x - 7)(9x - 7)\\&= - 10x \times 9x - 10x \times - 7 - 7 \times 9x - 7 \times - 7\\&= - 10 \times 9 \times x^{1 + 1} - 7 \times - 10 \times x - 7 \times 9 \times x + 49\\&= 70x - 63x - 90x^{2} + 49\\&= (70 - 63) \times x - 90x^{2} + 49\\&= - 90x^{2} + 7x + 49 + \end{align*} + \item + \begin{align*} + C &= (3x - 7)^{2}\\&= (3x - 7)(3x - 7)\\&= 3x \times 3x + 3x \times - 7 - 7 \times 3x - 7 \times - 7\\&= 3 \times 3 \times x^{1 + 1} - 7 \times 3 \times x - 7 \times 3 \times x + 49\\&= - 21x - 21x + 9x^{2} + 49\\&= (- 21 - 21) \times x + 9x^{2} + 49\\&= 9x^{2} - 42x + 49 + \end{align*} + \item + \begin{align*} + D &= 9 + x(- 1x - 4)\\&= 9 + x \times - x + x \times - 4\\&= - x^{2} - 4x + 9 + \end{align*} + \item + \begin{align*} + E &= - 2x^{2} + x(10x + 6)\\&= - 2x^{2} + x \times 10x + x \times 6\\&= - 2x^{2} + 10x^{2} + 6x\\&= - 2x^{2} + 10x^{2} + 6x\\&= (- 2 + 10) \times x^{2} + 6x\\&= 8x^{2} + 6x + \end{align*} + \item + \begin{align*} + F &= 2(x - 9)(x + 7)\\&= (2x + 2 \times - 9)(x + 7)\\&= (2x - 18)(x + 7)\\&= 2x \times x + 2x \times 7 - 18x - 18 \times 7\\&= 7 \times 2 \times x - 126 + 2x^{2} - 18x\\&= 14x - 126 + 2x^{2} - 18x\\&= 2x^{2} + 14x - 18x - 126\\&= 2x^{2} + (14 - 18) \times x - 126\\&= 2x^{2} - 4x - 126 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = 6x^{2} + 18x - 168$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = 6 \times 1^{2} + 18 \times 1 - 168=6 \times 1 + 18 - 168=6 - 150=- 144 + \] + \[ + f(-1) = 6 \times - 1^{2} + 18 \times - 1 - 168=6 \times 1 - 18 - 168=6 - 186=- 180 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_06_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_06_2010_DM1.tex new file mode 100644 index 0000000..09a81c6 --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_06_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill COLASSI Alexis} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{- 9}{7} - \dfrac{- 4}{7}$ + \item $B = \dfrac{- 10}{8} - \dfrac{8}{80}$ + + \item $C = \dfrac{9}{8} + \dfrac{- 8}{7}$ + \item $D = \dfrac{- 2}{3} + 1$ + + \item $E = \dfrac{8}{3} \times \dfrac{- 1}{2}$ + \item $F = \dfrac{1}{8} \times 9$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{- 9}{7} - \dfrac{- 4}{7}=\dfrac{- 9}{7} + \dfrac{4}{7}=\dfrac{- 9 + 4}{7}=\dfrac{- 5}{7} + \] + \item + \[ + \dfrac{- 10}{8} - \dfrac{8}{80}=\dfrac{- 10}{8} - \dfrac{8}{80}=\dfrac{- 10 \times 10}{8 \times 10} - \dfrac{8}{80}=\dfrac{- 100}{80} - \dfrac{8}{80}=\dfrac{- 100 - 8}{80}=\dfrac{- 100 - 8}{80}=\dfrac{- 108}{80} + \] + \item + \[ + \dfrac{9}{8} + \dfrac{- 8}{7}=\dfrac{9 \times 7}{8 \times 7} + \dfrac{- 8 \times 8}{7 \times 8}=\dfrac{63}{56} + \dfrac{- 64}{56}=\dfrac{63 - 64}{56}=\dfrac{- 1}{56} + \] + \item + \[ + \dfrac{- 2}{3} + 1=\dfrac{- 2}{3} + \dfrac{1}{1}=\dfrac{- 2}{3} + \dfrac{1 \times 3}{1 \times 3}=\dfrac{- 2}{3} + \dfrac{3}{3}=\dfrac{- 2 + 3}{3}=\dfrac{1}{3} + \] + \item + \[ + \dfrac{8}{3} \times \dfrac{- 1}{2}=\dfrac{8 \times - 1}{3 \times 2}=\dfrac{- 8}{6} + \] + \item + \[ + \dfrac{1}{8} \times 9=\dfrac{1 \times 9}{8}=\dfrac{9}{8} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (8x + 8)(4x + 8)$ + \item $B = (7x + 4)(5x + 4)$ + + \item $C = (- 5x - 5)^{2}$ + \item $D = 5 + x(5x - 5)$ + + \item $E = 8x^{2} + x(- 4x + 8)$ + \item $F = 4(x + 5)(x - 5)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (8x + 8)(4x + 8)\\&= 8x \times 4x + 8x \times 8 + 8 \times 4x + 8 \times 8\\&= 8 \times 4 \times x^{1 + 1} + 8 \times 8 \times x + 8 \times 4 \times x + 64\\&= 64x + 32x + 32x^{2} + 64\\&= (64 + 32) \times x + 32x^{2} + 64\\&= 32x^{2} + 96x + 64 + \end{align*} + \item + \begin{align*} + B &= (7x + 4)(5x + 4)\\&= 7x \times 5x + 7x \times 4 + 4 \times 5x + 4 \times 4\\&= 7 \times 5 \times x^{1 + 1} + 4 \times 7 \times x + 4 \times 5 \times x + 16\\&= 28x + 20x + 35x^{2} + 16\\&= (28 + 20) \times x + 35x^{2} + 16\\&= 35x^{2} + 48x + 16 + \end{align*} + \item + \begin{align*} + C &= (- 5x - 5)^{2}\\&= (- 5x - 5)(- 5x - 5)\\&= - 5x \times - 5x - 5x \times - 5 - 5 \times - 5x - 5 \times - 5\\&= - 5 \times - 5 \times x^{1 + 1} - 5 \times - 5 \times x - 5 \times - 5 \times x + 25\\&= 25x + 25x + 25x^{2} + 25\\&= (25 + 25) \times x + 25x^{2} + 25\\&= 25x^{2} + 50x + 25 + \end{align*} + \item + \begin{align*} + D &= 5 + x(5x - 5)\\&= 5 + x \times 5x + x \times - 5\\&= 5x^{2} - 5x + 5 + \end{align*} + \item + \begin{align*} + E &= 8x^{2} + x(- 4x + 8)\\&= 8x^{2} + x \times - 4x + x \times 8\\&= 8x^{2} - 4x^{2} + 8x\\&= 8x^{2} - 4x^{2} + 8x\\&= (8 - 4) \times x^{2} + 8x\\&= 4x^{2} + 8x + \end{align*} + \item + \begin{align*} + F &= 4(x + 5)(x - 5)\\&= (4x + 4 \times 5)(x - 5)\\&= (4x + 20)(x - 5)\\&= 4x \times x + 4x \times - 5 + 20x + 20 \times - 5\\&= - 5 \times 4 \times x - 100 + 4x^{2} + 20x\\&= - 20x - 100 + 4x^{2} + 20x\\&= 4x^{2} - 20x + 20x - 100\\&= 4x^{2} + (- 20 + 20) \times x - 100\\&= 4x^{2} - 100 + 0x\\&= 4x^{2} - 100 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = 9x^{2} + 81x + 72$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = 9 \times 1^{2} + 81 \times 1 + 72=9 \times 1 + 81 + 72=9 + 153=162 + \] + \[ + f(-1) = 9 \times - 1^{2} + 81 \times - 1 + 72=9 \times 1 - 81 + 72=9 - 9=0 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_07_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_07_2010_DM1.tex new file mode 100644 index 0000000..4db635a --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_07_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill COUBAT Alexis} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{- 1}{2} - \dfrac{- 3}{2}$ + \item $B = \dfrac{- 5}{5} - \dfrac{- 7}{30}$ + + \item $C = \dfrac{- 1}{4} + \dfrac{3}{3}$ + \item $D = \dfrac{7}{10} - 7$ + + \item $E = \dfrac{- 8}{2} \times \dfrac{- 4}{1}$ + \item $F = \dfrac{1}{7} \times 5$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{- 1}{2} - \dfrac{- 3}{2}=\dfrac{- 1}{2} + \dfrac{3}{2}=\dfrac{- 1 + 3}{2}=\dfrac{2}{2} + \] + \item + \[ + \dfrac{- 5}{5} - \dfrac{- 7}{30}=\dfrac{- 5}{5} + \dfrac{7}{30}=\dfrac{- 5 \times 6}{5 \times 6} + \dfrac{7}{30}=\dfrac{- 30}{30} + \dfrac{7}{30}=\dfrac{- 30 + 7}{30}=\dfrac{- 23}{30} + \] + \item + \[ + \dfrac{- 1}{4} + \dfrac{3}{3}=\dfrac{- 1 \times 3}{4 \times 3} + \dfrac{3 \times 4}{3 \times 4}=\dfrac{- 3}{12} + \dfrac{12}{12}=\dfrac{- 3 + 12}{12}=\dfrac{9}{12} + \] + \item + \[ + \dfrac{7}{10} - 7=\dfrac{7}{10} + \dfrac{- 7}{1}=\dfrac{7}{10} + \dfrac{- 7 \times 10}{1 \times 10}=\dfrac{7}{10} + \dfrac{- 70}{10}=\dfrac{7 - 70}{10}=\dfrac{- 63}{10} + \] + \item + \[ + \dfrac{- 8}{2} \times \dfrac{- 4}{1}=\dfrac{- 8 \times - 4}{2 \times 1}=\dfrac{32}{2} + \] + \item + \[ + \dfrac{1}{7} \times 5=\dfrac{1 \times 5}{7}=\dfrac{5}{7} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (- 2x - 6)(- 4x - 6)$ + \item $B = (- 10x - 3)(- 5x - 3)$ + + \item $C = (10x - 7)^{2}$ + \item $D = - 4 + x(10x - 3)$ + + \item $E = - 3x^{2} + x(- 1x - 5)$ + \item $F = - 3(x + 2)(x + 8)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (- 2x - 6)(- 4x - 6)\\&= - 2x \times - 4x - 2x \times - 6 - 6 \times - 4x - 6 \times - 6\\&= - 2 \times - 4 \times x^{1 + 1} - 6 \times - 2 \times x - 6 \times - 4 \times x + 36\\&= 12x + 24x + 8x^{2} + 36\\&= (12 + 24) \times x + 8x^{2} + 36\\&= 8x^{2} + 36x + 36 + \end{align*} + \item + \begin{align*} + B &= (- 10x - 3)(- 5x - 3)\\&= - 10x \times - 5x - 10x \times - 3 - 3 \times - 5x - 3 \times - 3\\&= - 10 \times - 5 \times x^{1 + 1} - 3 \times - 10 \times x - 3 \times - 5 \times x + 9\\&= 30x + 15x + 50x^{2} + 9\\&= (30 + 15) \times x + 50x^{2} + 9\\&= 50x^{2} + 45x + 9 + \end{align*} + \item + \begin{align*} + C &= (10x - 7)^{2}\\&= (10x - 7)(10x - 7)\\&= 10x \times 10x + 10x \times - 7 - 7 \times 10x - 7 \times - 7\\&= 10 \times 10 \times x^{1 + 1} - 7 \times 10 \times x - 7 \times 10 \times x + 49\\&= - 70x - 70x + 100x^{2} + 49\\&= (- 70 - 70) \times x + 100x^{2} + 49\\&= 100x^{2} - 140x + 49 + \end{align*} + \item + \begin{align*} + D &= - 4 + x(10x - 3)\\&= - 4 + x \times 10x + x \times - 3\\&= 10x^{2} - 3x - 4 + \end{align*} + \item + \begin{align*} + E &= - 3x^{2} + x(- 1x - 5)\\&= - 3x^{2} + x \times - x + x \times - 5\\&= - 3x^{2} - x^{2} - 5x\\&= - 3x^{2} - x^{2} - 5x\\&= (- 3 - 1) \times x^{2} - 5x\\&= - 4x^{2} - 5x + \end{align*} + \item + \begin{align*} + F &= - 3(x + 2)(x + 8)\\&= (- 3x - 3 \times 2)(x + 8)\\&= (- 3x - 6)(x + 8)\\&= - 3x \times x - 3x \times 8 - 6x - 6 \times 8\\&= 8 \times - 3 \times x - 48 - 3x^{2} - 6x\\&= - 24x - 48 - 3x^{2} - 6x\\&= - 3x^{2} - 24x - 6x - 48\\&= - 3x^{2} + (- 24 - 6) \times x - 48\\&= - 3x^{2} - 30x - 48 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = - 6x^{2} + 72x - 162$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = - 6 \times 1^{2} + 72 \times 1 - 162=- 6 \times 1 + 72 - 162=- 6 - 90=- 96 + \] + \[ + f(-1) = - 6 \times - 1^{2} + 72 \times - 1 - 162=- 6 \times 1 - 72 - 162=- 6 - 234=- 240 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_08_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_08_2010_DM1.tex new file mode 100644 index 0000000..1267cc7 --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_08_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill COULLON Anis} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{- 10}{10} - \dfrac{- 1}{10}$ + \item $B = \dfrac{8}{8} - \dfrac{5}{16}$ + + \item $C = \dfrac{- 2}{9} + \dfrac{8}{8}$ + \item $D = \dfrac{10}{6} - 8$ + + \item $E = \dfrac{- 5}{8} \times \dfrac{- 4}{7}$ + \item $F = \dfrac{5}{7} \times 6$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{- 10}{10} - \dfrac{- 1}{10}=\dfrac{- 10}{10} + \dfrac{1}{10}=\dfrac{- 10 + 1}{10}=\dfrac{- 9}{10} + \] + \item + \[ + \dfrac{8}{8} - \dfrac{5}{16}=\dfrac{8}{8} - \dfrac{5}{16}=\dfrac{8 \times 2}{8 \times 2} - \dfrac{5}{16}=\dfrac{16}{16} - \dfrac{5}{16}=\dfrac{16 - 5}{16}=\dfrac{16 - 5}{16}=\dfrac{11}{16} + \] + \item + \[ + \dfrac{- 2}{9} + \dfrac{8}{8}=\dfrac{- 2 \times 8}{9 \times 8} + \dfrac{8 \times 9}{8 \times 9}=\dfrac{- 16}{72} + \dfrac{72}{72}=\dfrac{- 16 + 72}{72}=\dfrac{56}{72} + \] + \item + \[ + \dfrac{10}{6} - 8=\dfrac{10}{6} + \dfrac{- 8}{1}=\dfrac{10}{6} + \dfrac{- 8 \times 6}{1 \times 6}=\dfrac{10}{6} + \dfrac{- 48}{6}=\dfrac{10 - 48}{6}=\dfrac{- 38}{6} + \] + \item + \[ + \dfrac{- 5}{8} \times \dfrac{- 4}{7}=\dfrac{- 5 \times - 4}{8 \times 7}=\dfrac{20}{56} + \] + \item + \[ + \dfrac{5}{7} \times 6=\dfrac{5 \times 6}{7}=\dfrac{30}{7} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (5x - 3)(4x - 3)$ + \item $B = (3x + 5)(- 9x + 5)$ + + \item $C = (8x - 5)^{2}$ + \item $D = - 3 + x(5x + 7)$ + + \item $E = 10x^{2} + x(- 1x - 7)$ + \item $F = - 6(x + 2)(x + 9)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (5x - 3)(4x - 3)\\&= 5x \times 4x + 5x \times - 3 - 3 \times 4x - 3 \times - 3\\&= 5 \times 4 \times x^{1 + 1} - 3 \times 5 \times x - 3 \times 4 \times x + 9\\&= - 15x - 12x + 20x^{2} + 9\\&= (- 15 - 12) \times x + 20x^{2} + 9\\&= 20x^{2} - 27x + 9 + \end{align*} + \item + \begin{align*} + B &= (3x + 5)(- 9x + 5)\\&= 3x \times - 9x + 3x \times 5 + 5 \times - 9x + 5 \times 5\\&= 3 \times - 9 \times x^{1 + 1} + 5 \times 3 \times x + 5 \times - 9 \times x + 25\\&= 15x - 45x - 27x^{2} + 25\\&= (15 - 45) \times x - 27x^{2} + 25\\&= - 27x^{2} - 30x + 25 + \end{align*} + \item + \begin{align*} + C &= (8x - 5)^{2}\\&= (8x - 5)(8x - 5)\\&= 8x \times 8x + 8x \times - 5 - 5 \times 8x - 5 \times - 5\\&= 8 \times 8 \times x^{1 + 1} - 5 \times 8 \times x - 5 \times 8 \times x + 25\\&= - 40x - 40x + 64x^{2} + 25\\&= (- 40 - 40) \times x + 64x^{2} + 25\\&= 64x^{2} - 80x + 25 + \end{align*} + \item + \begin{align*} + D &= - 3 + x(5x + 7)\\&= - 3 + x \times 5x + x \times 7\\&= 5x^{2} + 7x - 3 + \end{align*} + \item + \begin{align*} + E &= 10x^{2} + x(- 1x - 7)\\&= 10x^{2} + x \times - x + x \times - 7\\&= 10x^{2} - x^{2} - 7x\\&= 10x^{2} - x^{2} - 7x\\&= (10 - 1) \times x^{2} - 7x\\&= 9x^{2} - 7x + \end{align*} + \item + \begin{align*} + F &= - 6(x + 2)(x + 9)\\&= (- 6x - 6 \times 2)(x + 9)\\&= (- 6x - 12)(x + 9)\\&= - 6x \times x - 6x \times 9 - 12x - 12 \times 9\\&= 9 \times - 6 \times x - 108 - 6x^{2} - 12x\\&= - 54x - 108 - 6x^{2} - 12x\\&= - 6x^{2} - 54x - 12x - 108\\&= - 6x^{2} + (- 54 - 12) \times x - 108\\&= - 6x^{2} - 66x - 108 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = - 7x^{2} - 70x - 175$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = - 7 \times 1^{2} - 70 \times 1 - 175=- 7 \times 1 - 70 - 175=- 7 - 245=- 252 + \] + \[ + f(-1) = - 7 \times - 1^{2} - 70 \times - 1 - 175=- 7 \times 1 + 70 - 175=- 7 - 105=- 112 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_09_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_09_2010_DM1.tex new file mode 100644 index 0000000..f616520 --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_09_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill DINGER Sölen} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{9}{3} - \dfrac{- 4}{3}$ + \item $B = \dfrac{8}{8} - \dfrac{- 9}{16}$ + + \item $C = \dfrac{- 9}{10} + \dfrac{6}{9}$ + \item $D = \dfrac{- 4}{8} - 6$ + + \item $E = \dfrac{7}{5} \times \dfrac{9}{4}$ + \item $F = \dfrac{10}{3} \times 5$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{9}{3} - \dfrac{- 4}{3}=\dfrac{9}{3} + \dfrac{4}{3}=\dfrac{9 + 4}{3}=\dfrac{13}{3} + \] + \item + \[ + \dfrac{8}{8} - \dfrac{- 9}{16}=\dfrac{8}{8} + \dfrac{9}{16}=\dfrac{8 \times 2}{8 \times 2} + \dfrac{9}{16}=\dfrac{16}{16} + \dfrac{9}{16}=\dfrac{16 + 9}{16}=\dfrac{25}{16} + \] + \item + \[ + \dfrac{- 9}{10} + \dfrac{6}{9}=\dfrac{- 9 \times 9}{10 \times 9} + \dfrac{6 \times 10}{9 \times 10}=\dfrac{- 81}{90} + \dfrac{60}{90}=\dfrac{- 81 + 60}{90}=\dfrac{- 21}{90} + \] + \item + \[ + \dfrac{- 4}{8} - 6=\dfrac{- 4}{8} + \dfrac{- 6}{1}=\dfrac{- 4}{8} + \dfrac{- 6 \times 8}{1 \times 8}=\dfrac{- 4}{8} + \dfrac{- 48}{8}=\dfrac{- 4 - 48}{8}=\dfrac{- 52}{8} + \] + \item + \[ + \dfrac{7}{5} \times \dfrac{9}{4}=\dfrac{7 \times 9}{5 \times 4}=\dfrac{63}{20} + \] + \item + \[ + \dfrac{10}{3} \times 5=\dfrac{10 \times 5}{3}=\dfrac{50}{3} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (4x + 7)(- 4x + 7)$ + \item $B = (- 8x + 1)(6x + 1)$ + + \item $C = (9x + 3)^{2}$ + \item $D = - 4 + x(5x - 10)$ + + \item $E = - 3x^{2} + x(- 2x - 5)$ + \item $F = 9(x - 4)(x + 8)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (4x + 7)(- 4x + 7)\\&= 4x \times - 4x + 4x \times 7 + 7 \times - 4x + 7 \times 7\\&= 4 \times - 4 \times x^{1 + 1} + 7 \times 4 \times x + 7 \times - 4 \times x + 49\\&= 28x - 28x - 16x^{2} + 49\\&= (28 - 28) \times x - 16x^{2} + 49\\&= 0x - 16x^{2} + 49\\&= - 16x^{2} + 49 + \end{align*} + \item + \begin{align*} + B &= (- 8x + 1)(6x + 1)\\&= - 8x \times 6x - 8x \times 1 + 1 \times 6x + 1 \times 1\\&= - 8 \times 6 \times x^{1 + 1} - 8x + 6x + 1\\&= - 48x^{2} - 8x + 6x + 1\\&= - 48x^{2} + (- 8 + 6) \times x + 1\\&= - 48x^{2} - 2x + 1 + \end{align*} + \item + \begin{align*} + C &= (9x + 3)^{2}\\&= (9x + 3)(9x + 3)\\&= 9x \times 9x + 9x \times 3 + 3 \times 9x + 3 \times 3\\&= 9 \times 9 \times x^{1 + 1} + 3 \times 9 \times x + 3 \times 9 \times x + 9\\&= 27x + 27x + 81x^{2} + 9\\&= (27 + 27) \times x + 81x^{2} + 9\\&= 81x^{2} + 54x + 9 + \end{align*} + \item + \begin{align*} + D &= - 4 + x(5x - 10)\\&= - 4 + x \times 5x + x \times - 10\\&= 5x^{2} - 10x - 4 + \end{align*} + \item + \begin{align*} + E &= - 3x^{2} + x(- 2x - 5)\\&= - 3x^{2} + x \times - 2x + x \times - 5\\&= - 3x^{2} - 2x^{2} - 5x\\&= - 3x^{2} - 2x^{2} - 5x\\&= (- 3 - 2) \times x^{2} - 5x\\&= - 5x^{2} - 5x + \end{align*} + \item + \begin{align*} + F &= 9(x - 4)(x + 8)\\&= (9x + 9 \times - 4)(x + 8)\\&= (9x - 36)(x + 8)\\&= 9x \times x + 9x \times 8 - 36x - 36 \times 8\\&= 8 \times 9 \times x - 288 + 9x^{2} - 36x\\&= 72x - 288 + 9x^{2} - 36x\\&= 9x^{2} + 72x - 36x - 288\\&= 9x^{2} + (72 - 36) \times x - 288\\&= 9x^{2} + 36x - 288 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = 2x^{2} + 2x - 40$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = 2 \times 1^{2} + 2 \times 1 - 40=2 \times 1 + 2 - 40=2 - 38=- 36 + \] + \[ + f(-1) = 2 \times - 1^{2} + 2 \times - 1 - 40=2 \times 1 - 2 - 40=2 - 42=- 40 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_10_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_10_2010_DM1.tex new file mode 100644 index 0000000..85d4c91 --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_10_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill EYRAUD Cynthia} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{6}{8} - \dfrac{- 10}{8}$ + \item $B = \dfrac{- 1}{2} - \dfrac{- 2}{8}$ + + \item $C = \dfrac{- 1}{5} + \dfrac{- 8}{4}$ + \item $D = \dfrac{- 3}{9} + 7$ + + \item $E = \dfrac{- 10}{9} \times \dfrac{7}{8}$ + \item $F = \dfrac{3}{3} \times - 6$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{6}{8} - \dfrac{- 10}{8}=\dfrac{6}{8} + \dfrac{10}{8}=\dfrac{6 + 10}{8}=\dfrac{16}{8} + \] + \item + \[ + \dfrac{- 1}{2} - \dfrac{- 2}{8}=\dfrac{- 1}{2} + \dfrac{2}{8}=\dfrac{- 1 \times 4}{2 \times 4} + \dfrac{2}{8}=\dfrac{- 4}{8} + \dfrac{2}{8}=\dfrac{- 4 + 2}{8}=\dfrac{- 2}{8} + \] + \item + \[ + \dfrac{- 1}{5} + \dfrac{- 8}{4}=\dfrac{- 1 \times 4}{5 \times 4} + \dfrac{- 8 \times 5}{4 \times 5}=\dfrac{- 4}{20} + \dfrac{- 40}{20}=\dfrac{- 4 - 40}{20}=\dfrac{- 44}{20} + \] + \item + \[ + \dfrac{- 3}{9} + 7=\dfrac{- 3}{9} + \dfrac{7}{1}=\dfrac{- 3}{9} + \dfrac{7 \times 9}{1 \times 9}=\dfrac{- 3}{9} + \dfrac{63}{9}=\dfrac{- 3 + 63}{9}=\dfrac{60}{9} + \] + \item + \[ + \dfrac{- 10}{9} \times \dfrac{7}{8}=\dfrac{- 10 \times 7}{9 \times 8}=\dfrac{- 70}{72} + \] + \item + \[ + \dfrac{3}{3} \times - 6=\dfrac{3 \times - 6}{3}=\dfrac{- 18}{3} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (7x - 6)(- 2x - 6)$ + \item $B = (6x + 8)(10x + 8)$ + + \item $C = (5x + 5)^{2}$ + \item $D = - 1 + x(- 4x + 3)$ + + \item $E = - 4x^{2} + x(2x - 3)$ + \item $F = - 7(x - 8)(x + 7)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (7x - 6)(- 2x - 6)\\&= 7x \times - 2x + 7x \times - 6 - 6 \times - 2x - 6 \times - 6\\&= 7 \times - 2 \times x^{1 + 1} - 6 \times 7 \times x - 6 \times - 2 \times x + 36\\&= - 42x + 12x - 14x^{2} + 36\\&= (- 42 + 12) \times x - 14x^{2} + 36\\&= - 14x^{2} - 30x + 36 + \end{align*} + \item + \begin{align*} + B &= (6x + 8)(10x + 8)\\&= 6x \times 10x + 6x \times 8 + 8 \times 10x + 8 \times 8\\&= 6 \times 10 \times x^{1 + 1} + 8 \times 6 \times x + 8 \times 10 \times x + 64\\&= 48x + 80x + 60x^{2} + 64\\&= (48 + 80) \times x + 60x^{2} + 64\\&= 60x^{2} + 128x + 64 + \end{align*} + \item + \begin{align*} + C &= (5x + 5)^{2}\\&= (5x + 5)(5x + 5)\\&= 5x \times 5x + 5x \times 5 + 5 \times 5x + 5 \times 5\\&= 5 \times 5 \times x^{1 + 1} + 5 \times 5 \times x + 5 \times 5 \times x + 25\\&= 25x + 25x + 25x^{2} + 25\\&= (25 + 25) \times x + 25x^{2} + 25\\&= 25x^{2} + 50x + 25 + \end{align*} + \item + \begin{align*} + D &= - 1 + x(- 4x + 3)\\&= - 1 + x \times - 4x + x \times 3\\&= - 4x^{2} + 3x - 1 + \end{align*} + \item + \begin{align*} + E &= - 4x^{2} + x(2x - 3)\\&= - 4x^{2} + x \times 2x + x \times - 3\\&= - 4x^{2} + 2x^{2} - 3x\\&= - 4x^{2} + 2x^{2} - 3x\\&= (- 4 + 2) \times x^{2} - 3x\\&= - 2x^{2} - 3x + \end{align*} + \item + \begin{align*} + F &= - 7(x - 8)(x + 7)\\&= (- 7x - 7 \times - 8)(x + 7)\\&= (- 7x + 56)(x + 7)\\&= - 7x \times x - 7x \times 7 + 56x + 56 \times 7\\&= 7 \times - 7 \times x + 392 - 7x^{2} + 56x\\&= - 49x + 392 - 7x^{2} + 56x\\&= - 7x^{2} - 49x + 56x + 392\\&= - 7x^{2} + (- 49 + 56) \times x + 392\\&= - 7x^{2} + 7x + 392 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = 7x^{2} + 28x - 315$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = 7 \times 1^{2} + 28 \times 1 - 315=7 \times 1 + 28 - 315=7 - 287=- 280 + \] + \[ + f(-1) = 7 \times - 1^{2} + 28 \times - 1 - 315=7 \times 1 - 28 - 315=7 - 343=- 336 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_11_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_11_2010_DM1.tex new file mode 100644 index 0000000..4e04c0d --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_11_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill FERREIRA Léo} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{4}{7} - \dfrac{7}{7}$ + \item $B = \dfrac{9}{10} - \dfrac{6}{20}$ + + \item $C = \dfrac{10}{3} + \dfrac{- 1}{2}$ + \item $D = \dfrac{1}{5} + 3$ + + \item $E = \dfrac{- 1}{4} \times \dfrac{- 7}{3}$ + \item $F = \dfrac{3}{7} \times 9$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{4}{7} - \dfrac{7}{7}=\dfrac{4}{7} - \dfrac{7}{7}=\dfrac{4 - 7}{7}=\dfrac{4 - 7}{7}=\dfrac{- 3}{7} + \] + \item + \[ + \dfrac{9}{10} - \dfrac{6}{20}=\dfrac{9}{10} - \dfrac{6}{20}=\dfrac{9 \times 2}{10 \times 2} - \dfrac{6}{20}=\dfrac{18}{20} - \dfrac{6}{20}=\dfrac{18 - 6}{20}=\dfrac{18 - 6}{20}=\dfrac{12}{20} + \] + \item + \[ + \dfrac{10}{3} + \dfrac{- 1}{2}=\dfrac{10 \times 2}{3 \times 2} + \dfrac{- 1 \times 3}{2 \times 3}=\dfrac{20}{6} + \dfrac{- 3}{6}=\dfrac{20 - 3}{6}=\dfrac{17}{6} + \] + \item + \[ + \dfrac{1}{5} + 3=\dfrac{1}{5} + \dfrac{3}{1}=\dfrac{1}{5} + \dfrac{3 \times 5}{1 \times 5}=\dfrac{1}{5} + \dfrac{15}{5}=\dfrac{1 + 15}{5}=\dfrac{16}{5} + \] + \item + \[ + \dfrac{- 1}{4} \times \dfrac{- 7}{3}=\dfrac{- 1 \times - 7}{4 \times 3}=\dfrac{7}{12} + \] + \item + \[ + \dfrac{3}{7} \times 9=\dfrac{3 \times 9}{7}=\dfrac{27}{7} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (- 8x - 7)(- 1x - 7)$ + \item $B = (5x - 1)(- 2x - 1)$ + + \item $C = (- 3x + 7)^{2}$ + \item $D = - 4 + x(1x + 1)$ + + \item $E = - 10x^{2} + x(5x - 1)$ + \item $F = 10(x + 5)(x - 10)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (- 8x - 7)(- 1x - 7)\\&= - 8x \times - x - 8x \times - 7 - 7 \times - x - 7 \times - 7\\&= - 8 \times - 1 \times x^{1 + 1} - 7 \times - 8 \times x - 7 \times - 1 \times x + 49\\&= 56x + 7x + 8x^{2} + 49\\&= (56 + 7) \times x + 8x^{2} + 49\\&= 8x^{2} + 63x + 49 + \end{align*} + \item + \begin{align*} + B &= (5x - 1)(- 2x - 1)\\&= 5x \times - 2x + 5x \times - 1 - 1 \times - 2x - 1 \times - 1\\&= 5 \times - 2 \times x^{1 + 1} - 1 \times 5 \times x - 1 \times - 2 \times x + 1\\&= - 5x + 2x - 10x^{2} + 1\\&= (- 5 + 2) \times x - 10x^{2} + 1\\&= - 10x^{2} - 3x + 1 + \end{align*} + \item + \begin{align*} + C &= (- 3x + 7)^{2}\\&= (- 3x + 7)(- 3x + 7)\\&= - 3x \times - 3x - 3x \times 7 + 7 \times - 3x + 7 \times 7\\&= - 3 \times - 3 \times x^{1 + 1} + 7 \times - 3 \times x + 7 \times - 3 \times x + 49\\&= - 21x - 21x + 9x^{2} + 49\\&= (- 21 - 21) \times x + 9x^{2} + 49\\&= 9x^{2} - 42x + 49 + \end{align*} + \item + \begin{align*} + D &= - 4 + x(1x + 1)\\&= - 4 + x \times x + x \times 1\\&= x^{2} + x - 4 + \end{align*} + \item + \begin{align*} + E &= - 10x^{2} + x(5x - 1)\\&= - 10x^{2} + x \times 5x + x \times - 1\\&= - 10x^{2} + 5x^{2} - x\\&= - 10x^{2} + 5x^{2} - x\\&= (- 10 + 5) \times x^{2} - x\\&= - 5x^{2} - x + \end{align*} + \item + \begin{align*} + F &= 10(x + 5)(x - 10)\\&= (10x + 10 \times 5)(x - 10)\\&= (10x + 50)(x - 10)\\&= 10x \times x + 10x \times - 10 + 50x + 50 \times - 10\\&= - 10 \times 10 \times x - 500 + 10x^{2} + 50x\\&= - 100x - 500 + 10x^{2} + 50x\\&= 10x^{2} - 100x + 50x - 500\\&= 10x^{2} + (- 100 + 50) \times x - 500\\&= 10x^{2} - 50x - 500 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = - x^{2} - 5x + 14$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = - 1 \times 1^{2} - 5 \times 1 + 14=- 1 \times 1 - 5 + 14=- 1 + 9=8 + \] + \[ + f(-1) = - 1 \times - 1^{2} - 5 \times - 1 + 14=- 1 \times 1 + 5 + 14=- 1 + 19=18 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_12_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_12_2010_DM1.tex new file mode 100644 index 0000000..caf355f --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_12_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill FILALI Zakaria} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{- 2}{6} - \dfrac{2}{6}$ + \item $B = \dfrac{9}{4} - \dfrac{2}{20}$ + + \item $C = \dfrac{- 4}{8} + \dfrac{- 6}{7}$ + \item $D = \dfrac{- 7}{6} - 1$ + + \item $E = \dfrac{5}{2} \times \dfrac{5}{1}$ + \item $F = \dfrac{1}{9} \times 4$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{- 2}{6} - \dfrac{2}{6}=\dfrac{- 2}{6} - \dfrac{2}{6}=\dfrac{- 2 - 2}{6}=\dfrac{- 2 - 2}{6}=\dfrac{- 4}{6} + \] + \item + \[ + \dfrac{9}{4} - \dfrac{2}{20}=\dfrac{9}{4} - \dfrac{2}{20}=\dfrac{9 \times 5}{4 \times 5} - \dfrac{2}{20}=\dfrac{45}{20} - \dfrac{2}{20}=\dfrac{45 - 2}{20}=\dfrac{45 - 2}{20}=\dfrac{43}{20} + \] + \item + \[ + \dfrac{- 4}{8} + \dfrac{- 6}{7}=\dfrac{- 4 \times 7}{8 \times 7} + \dfrac{- 6 \times 8}{7 \times 8}=\dfrac{- 28}{56} + \dfrac{- 48}{56}=\dfrac{- 28 - 48}{56}=\dfrac{- 76}{56} + \] + \item + \[ + \dfrac{- 7}{6} - 1=\dfrac{- 7}{6} + \dfrac{- 1}{1}=\dfrac{- 7}{6} + \dfrac{- 1 \times 6}{1 \times 6}=\dfrac{- 7}{6} + \dfrac{- 6}{6}=\dfrac{- 7 - 6}{6}=\dfrac{- 13}{6} + \] + \item + \[ + \dfrac{5}{2} \times \dfrac{5}{1}=\dfrac{5 \times 5}{2 \times 1}=\dfrac{25}{2} + \] + \item + \[ + \dfrac{1}{9} \times 4=\dfrac{1 \times 4}{9}=\dfrac{4}{9} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (8x + 2)(- 8x + 2)$ + \item $B = (- 5x - 8)(9x - 8)$ + + \item $C = (- 1x - 1)^{2}$ + \item $D = - 7 + x(7x + 7)$ + + \item $E = 5x^{2} + x(- 6x + 8)$ + \item $F = - 10(x + 5)(x - 10)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (8x + 2)(- 8x + 2)\\&= 8x \times - 8x + 8x \times 2 + 2 \times - 8x + 2 \times 2\\&= 8 \times - 8 \times x^{1 + 1} + 2 \times 8 \times x + 2 \times - 8 \times x + 4\\&= 16x - 16x - 64x^{2} + 4\\&= (16 - 16) \times x - 64x^{2} + 4\\&= 0x - 64x^{2} + 4\\&= - 64x^{2} + 4 + \end{align*} + \item + \begin{align*} + B &= (- 5x - 8)(9x - 8)\\&= - 5x \times 9x - 5x \times - 8 - 8 \times 9x - 8 \times - 8\\&= - 5 \times 9 \times x^{1 + 1} - 8 \times - 5 \times x - 8 \times 9 \times x + 64\\&= 40x - 72x - 45x^{2} + 64\\&= (40 - 72) \times x - 45x^{2} + 64\\&= - 45x^{2} - 32x + 64 + \end{align*} + \item + \begin{align*} + C &= (- 1x - 1)^{2}\\&= (- x - 1)(- x - 1)\\&= - x \times - x - x \times - 1 - 1 \times - x - 1 \times - 1\\&= - 1 \times - 1 \times x^{1 + 1} - 1 \times - 1 \times x - 1 \times - 1 \times x + 1\\&= x^{2} + 2x + 1 + \end{align*} + \item + \begin{align*} + D &= - 7 + x(7x + 7)\\&= - 7 + x \times 7x + x \times 7\\&= 7x^{2} + 7x - 7 + \end{align*} + \item + \begin{align*} + E &= 5x^{2} + x(- 6x + 8)\\&= 5x^{2} + x \times - 6x + x \times 8\\&= 5x^{2} - 6x^{2} + 8x\\&= 5x^{2} - 6x^{2} + 8x\\&= (5 - 6) \times x^{2} + 8x\\&= - x^{2} + 8x + \end{align*} + \item + \begin{align*} + F &= - 10(x + 5)(x - 10)\\&= (- 10x - 10 \times 5)(x - 10)\\&= (- 10x - 50)(x - 10)\\&= - 10x \times x - 10x \times - 10 - 50x - 50 \times - 10\\&= - 10 \times - 10 \times x + 500 - 10x^{2} - 50x\\&= 100x + 500 - 10x^{2} - 50x\\&= - 10x^{2} + 100x - 50x + 500\\&= - 10x^{2} + (100 - 50) \times x + 500\\&= - 10x^{2} + 50x + 500 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = - 3x^{2} - 45x - 168$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = - 3 \times 1^{2} - 45 \times 1 - 168=- 3 \times 1 - 45 - 168=- 3 - 213=- 216 + \] + \[ + f(-1) = - 3 \times - 1^{2} - 45 \times - 1 - 168=- 3 \times 1 + 45 - 168=- 3 - 123=- 126 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_13_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_13_2010_DM1.tex new file mode 100644 index 0000000..f31d0bf --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_13_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill FOIGNY Romain} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{- 6}{8} - \dfrac{2}{8}$ + \item $B = \dfrac{- 4}{6} - \dfrac{10}{60}$ + + \item $C = \dfrac{8}{10} + \dfrac{- 3}{9}$ + \item $D = \dfrac{6}{7} - 1$ + + \item $E = \dfrac{7}{4} \times \dfrac{6}{3}$ + \item $F = \dfrac{- 3}{5} \times - 8$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{- 6}{8} - \dfrac{2}{8}=\dfrac{- 6}{8} - \dfrac{2}{8}=\dfrac{- 6 - 2}{8}=\dfrac{- 6 - 2}{8}=\dfrac{- 8}{8} + \] + \item + \[ + \dfrac{- 4}{6} - \dfrac{10}{60}=\dfrac{- 4}{6} - \dfrac{10}{60}=\dfrac{- 4 \times 10}{6 \times 10} - \dfrac{10}{60}=\dfrac{- 40}{60} - \dfrac{10}{60}=\dfrac{- 40 - 10}{60}=\dfrac{- 40 - 10}{60}=\dfrac{- 50}{60} + \] + \item + \[ + \dfrac{8}{10} + \dfrac{- 3}{9}=\dfrac{8 \times 9}{10 \times 9} + \dfrac{- 3 \times 10}{9 \times 10}=\dfrac{72}{90} + \dfrac{- 30}{90}=\dfrac{72 - 30}{90}=\dfrac{42}{90} + \] + \item + \[ + \dfrac{6}{7} - 1=\dfrac{6}{7} + \dfrac{- 1}{1}=\dfrac{6}{7} + \dfrac{- 1 \times 7}{1 \times 7}=\dfrac{6}{7} + \dfrac{- 7}{7}=\dfrac{6 - 7}{7}=\dfrac{- 1}{7} + \] + \item + \[ + \dfrac{7}{4} \times \dfrac{6}{3}=\dfrac{7 \times 6}{4 \times 3}=\dfrac{42}{12} + \] + \item + \[ + \dfrac{- 3}{5} \times - 8=\dfrac{- 3 \times - 8}{5}=\dfrac{24}{5} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (5x + 10)(- 1x + 10)$ + \item $B = (- 7x + 9)(5x + 9)$ + + \item $C = (- 7x - 10)^{2}$ + \item $D = - 9 + x(- 1x - 2)$ + + \item $E = - 4x^{2} + x(- 5x + 5)$ + \item $F = 5(x + 10)(x + 9)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (5x + 10)(- 1x + 10)\\&= 5x \times - x + 5x \times 10 + 10 \times - x + 10 \times 10\\&= 5 \times - 1 \times x^{1 + 1} + 10 \times 5 \times x + 10 \times - 1 \times x + 100\\&= 50x - 10x - 5x^{2} + 100\\&= (50 - 10) \times x - 5x^{2} + 100\\&= - 5x^{2} + 40x + 100 + \end{align*} + \item + \begin{align*} + B &= (- 7x + 9)(5x + 9)\\&= - 7x \times 5x - 7x \times 9 + 9 \times 5x + 9 \times 9\\&= - 7 \times 5 \times x^{1 + 1} + 9 \times - 7 \times x + 9 \times 5 \times x + 81\\&= - 63x + 45x - 35x^{2} + 81\\&= (- 63 + 45) \times x - 35x^{2} + 81\\&= - 35x^{2} - 18x + 81 + \end{align*} + \item + \begin{align*} + C &= (- 7x - 10)^{2}\\&= (- 7x - 10)(- 7x - 10)\\&= - 7x \times - 7x - 7x \times - 10 - 10 \times - 7x - 10 \times - 10\\&= - 7 \times - 7 \times x^{1 + 1} - 10 \times - 7 \times x - 10 \times - 7 \times x + 100\\&= 70x + 70x + 49x^{2} + 100\\&= (70 + 70) \times x + 49x^{2} + 100\\&= 49x^{2} + 140x + 100 + \end{align*} + \item + \begin{align*} + D &= - 9 + x(- 1x - 2)\\&= - 9 + x \times - x + x \times - 2\\&= - x^{2} - 2x - 9 + \end{align*} + \item + \begin{align*} + E &= - 4x^{2} + x(- 5x + 5)\\&= - 4x^{2} + x \times - 5x + x \times 5\\&= - 4x^{2} - 5x^{2} + 5x\\&= - 4x^{2} - 5x^{2} + 5x\\&= (- 4 - 5) \times x^{2} + 5x\\&= - 9x^{2} + 5x + \end{align*} + \item + \begin{align*} + F &= 5(x + 10)(x + 9)\\&= (5x + 5 \times 10)(x + 9)\\&= (5x + 50)(x + 9)\\&= 5x \times x + 5x \times 9 + 50x + 50 \times 9\\&= 9 \times 5 \times x + 450 + 5x^{2} + 50x\\&= 45x + 450 + 5x^{2} + 50x\\&= 5x^{2} + 45x + 50x + 450\\&= 5x^{2} + (45 + 50) \times x + 450\\&= 5x^{2} + 95x + 450 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = 7x^{2} - 56x - 63$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = 7 \times 1^{2} - 56 \times 1 - 63=7 \times 1 - 56 - 63=7 - 119=- 112 + \] + \[ + f(-1) = 7 \times - 1^{2} - 56 \times - 1 - 63=7 \times 1 + 56 - 63=7 - 7=0 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_14_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_14_2010_DM1.tex new file mode 100644 index 0000000..9dde514 --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_14_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill HIPOLITO DA SILVA Andréa} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{6}{10} - \dfrac{9}{10}$ + \item $B = \dfrac{9}{3} - \dfrac{- 7}{6}$ + + \item $C = \dfrac{7}{2} + \dfrac{- 1}{1}$ + \item $D = \dfrac{- 8}{4} - 1$ + + \item $E = \dfrac{4}{5} \times \dfrac{6}{4}$ + \item $F = \dfrac{- 9}{5} \times - 7$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{6}{10} - \dfrac{9}{10}=\dfrac{6}{10} - \dfrac{9}{10}=\dfrac{6 - 9}{10}=\dfrac{6 - 9}{10}=\dfrac{- 3}{10} + \] + \item + \[ + \dfrac{9}{3} - \dfrac{- 7}{6}=\dfrac{9}{3} + \dfrac{7}{6}=\dfrac{9 \times 2}{3 \times 2} + \dfrac{7}{6}=\dfrac{18}{6} + \dfrac{7}{6}=\dfrac{18 + 7}{6}=\dfrac{25}{6} + \] + \item + \[ + \dfrac{7}{2} + \dfrac{- 1}{1}=\dfrac{7}{2} + \dfrac{- 1 \times 2}{1 \times 2}=\dfrac{7}{2} + \dfrac{- 2}{2}=\dfrac{7 - 2}{2}=\dfrac{5}{2} + \] + \item + \[ + \dfrac{- 8}{4} - 1=\dfrac{- 8}{4} + \dfrac{- 1}{1}=\dfrac{- 8}{4} + \dfrac{- 1 \times 4}{1 \times 4}=\dfrac{- 8}{4} + \dfrac{- 4}{4}=\dfrac{- 8 - 4}{4}=\dfrac{- 12}{4} + \] + \item + \[ + \dfrac{4}{5} \times \dfrac{6}{4}=\dfrac{4 \times 6}{5 \times 4}=\dfrac{24}{20} + \] + \item + \[ + \dfrac{- 9}{5} \times - 7=\dfrac{- 9 \times - 7}{5}=\dfrac{63}{5} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (10x + 4)(- 3x + 4)$ + \item $B = (- 8x - 9)(3x - 9)$ + + \item $C = (- 10x + 1)^{2}$ + \item $D = - 2 + x(2x + 6)$ + + \item $E = 10x^{2} + x(- 1x + 7)$ + \item $F = - 7(x + 9)(x - 5)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (10x + 4)(- 3x + 4)\\&= 10x \times - 3x + 10x \times 4 + 4 \times - 3x + 4 \times 4\\&= 10 \times - 3 \times x^{1 + 1} + 4 \times 10 \times x + 4 \times - 3 \times x + 16\\&= 40x - 12x - 30x^{2} + 16\\&= (40 - 12) \times x - 30x^{2} + 16\\&= - 30x^{2} + 28x + 16 + \end{align*} + \item + \begin{align*} + B &= (- 8x - 9)(3x - 9)\\&= - 8x \times 3x - 8x \times - 9 - 9 \times 3x - 9 \times - 9\\&= - 8 \times 3 \times x^{1 + 1} - 9 \times - 8 \times x - 9 \times 3 \times x + 81\\&= 72x - 27x - 24x^{2} + 81\\&= (72 - 27) \times x - 24x^{2} + 81\\&= - 24x^{2} + 45x + 81 + \end{align*} + \item + \begin{align*} + C &= (- 10x + 1)^{2}\\&= (- 10x + 1)(- 10x + 1)\\&= - 10x \times - 10x - 10x \times 1 + 1 \times - 10x + 1 \times 1\\&= - 10 \times - 10 \times x^{1 + 1} - 10x - 10x + 1\\&= 100x^{2} - 10x - 10x + 1\\&= 100x^{2} + (- 10 - 10) \times x + 1\\&= 100x^{2} - 20x + 1 + \end{align*} + \item + \begin{align*} + D &= - 2 + x(2x + 6)\\&= - 2 + x \times 2x + x \times 6\\&= 2x^{2} + 6x - 2 + \end{align*} + \item + \begin{align*} + E &= 10x^{2} + x(- 1x + 7)\\&= 10x^{2} + x \times - x + x \times 7\\&= 10x^{2} - x^{2} + 7x\\&= 10x^{2} - x^{2} + 7x\\&= (10 - 1) \times x^{2} + 7x\\&= 9x^{2} + 7x + \end{align*} + \item + \begin{align*} + F &= - 7(x + 9)(x - 5)\\&= (- 7x - 7 \times 9)(x - 5)\\&= (- 7x - 63)(x - 5)\\&= - 7x \times x - 7x \times - 5 - 63x - 63 \times - 5\\&= - 5 \times - 7 \times x + 315 - 7x^{2} - 63x\\&= 35x + 315 - 7x^{2} - 63x\\&= - 7x^{2} + 35x - 63x + 315\\&= - 7x^{2} + (35 - 63) \times x + 315\\&= - 7x^{2} - 28x + 315 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = x^{2} + 5x - 6$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = 1^{2} + 5 \times 1 - 6=1 + 5 - 6=1 - 1=0 + \] + \[ + f(-1) = - 1^{2} + 5 \times - 1 - 6=1 - 5 - 6=1 - 11=- 10 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_15_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_15_2010_DM1.tex new file mode 100644 index 0000000..020d045 --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_15_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill HUMBERT Rayan} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{- 9}{5} - \dfrac{7}{5}$ + \item $B = \dfrac{- 1}{6} - \dfrac{- 6}{54}$ + + \item $C = \dfrac{4}{10} + \dfrac{10}{9}$ + \item $D = \dfrac{9}{5} + 8$ + + \item $E = \dfrac{- 2}{4} \times \dfrac{8}{3}$ + \item $F = \dfrac{- 7}{4} \times 10$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{- 9}{5} - \dfrac{7}{5}=\dfrac{- 9}{5} - \dfrac{7}{5}=\dfrac{- 9 - 7}{5}=\dfrac{- 9 - 7}{5}=\dfrac{- 16}{5} + \] + \item + \[ + \dfrac{- 1}{6} - \dfrac{- 6}{54}=\dfrac{- 1}{6} + \dfrac{6}{54}=\dfrac{- 1 \times 9}{6 \times 9} + \dfrac{6}{54}=\dfrac{- 9}{54} + \dfrac{6}{54}=\dfrac{- 9 + 6}{54}=\dfrac{- 3}{54} + \] + \item + \[ + \dfrac{4}{10} + \dfrac{10}{9}=\dfrac{4 \times 9}{10 \times 9} + \dfrac{10 \times 10}{9 \times 10}=\dfrac{36}{90} + \dfrac{100}{90}=\dfrac{36 + 100}{90}=\dfrac{136}{90} + \] + \item + \[ + \dfrac{9}{5} + 8=\dfrac{9}{5} + \dfrac{8}{1}=\dfrac{9}{5} + \dfrac{8 \times 5}{1 \times 5}=\dfrac{9}{5} + \dfrac{40}{5}=\dfrac{9 + 40}{5}=\dfrac{49}{5} + \] + \item + \[ + \dfrac{- 2}{4} \times \dfrac{8}{3}=\dfrac{- 2 \times 8}{4 \times 3}=\dfrac{- 16}{12} + \] + \item + \[ + \dfrac{- 7}{4} \times 10=\dfrac{- 7 \times 10}{4}=\dfrac{- 70}{4} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (3x - 5)(- 3x - 5)$ + \item $B = (3x + 4)(- 10x + 4)$ + + \item $C = (3x - 3)^{2}$ + \item $D = 8 + x(- 6x + 5)$ + + \item $E = - 3x^{2} + x(2x - 3)$ + \item $F = 5(x - 9)(x + 4)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (3x - 5)(- 3x - 5)\\&= 3x \times - 3x + 3x \times - 5 - 5 \times - 3x - 5 \times - 5\\&= 3 \times - 3 \times x^{1 + 1} - 5 \times 3 \times x - 5 \times - 3 \times x + 25\\&= - 15x + 15x - 9x^{2} + 25\\&= (- 15 + 15) \times x - 9x^{2} + 25\\&= 0x - 9x^{2} + 25\\&= - 9x^{2} + 25 + \end{align*} + \item + \begin{align*} + B &= (3x + 4)(- 10x + 4)\\&= 3x \times - 10x + 3x \times 4 + 4 \times - 10x + 4 \times 4\\&= 3 \times - 10 \times x^{1 + 1} + 4 \times 3 \times x + 4 \times - 10 \times x + 16\\&= 12x - 40x - 30x^{2} + 16\\&= (12 - 40) \times x - 30x^{2} + 16\\&= - 30x^{2} - 28x + 16 + \end{align*} + \item + \begin{align*} + C &= (3x - 3)^{2}\\&= (3x - 3)(3x - 3)\\&= 3x \times 3x + 3x \times - 3 - 3 \times 3x - 3 \times - 3\\&= 3 \times 3 \times x^{1 + 1} - 3 \times 3 \times x - 3 \times 3 \times x + 9\\&= - 9x - 9x + 9x^{2} + 9\\&= (- 9 - 9) \times x + 9x^{2} + 9\\&= 9x^{2} - 18x + 9 + \end{align*} + \item + \begin{align*} + D &= 8 + x(- 6x + 5)\\&= 8 + x \times - 6x + x \times 5\\&= - 6x^{2} + 5x + 8 + \end{align*} + \item + \begin{align*} + E &= - 3x^{2} + x(2x - 3)\\&= - 3x^{2} + x \times 2x + x \times - 3\\&= - 3x^{2} + 2x^{2} - 3x\\&= - 3x^{2} + 2x^{2} - 3x\\&= (- 3 + 2) \times x^{2} - 3x\\&= - x^{2} - 3x + \end{align*} + \item + \begin{align*} + F &= 5(x - 9)(x + 4)\\&= (5x + 5 \times - 9)(x + 4)\\&= (5x - 45)(x + 4)\\&= 5x \times x + 5x \times 4 - 45x - 45 \times 4\\&= 4 \times 5 \times x - 180 + 5x^{2} - 45x\\&= 20x - 180 + 5x^{2} - 45x\\&= 5x^{2} + 20x - 45x - 180\\&= 5x^{2} + (20 - 45) \times x - 180\\&= 5x^{2} - 25x - 180 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = 9x^{2} + 153x + 648$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = 9 \times 1^{2} + 153 \times 1 + 648=9 \times 1 + 153 + 648=9 + 801=810 + \] + \[ + f(-1) = 9 \times - 1^{2} + 153 \times - 1 + 648=9 \times 1 - 153 + 648=9 + 495=504 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_16_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_16_2010_DM1.tex new file mode 100644 index 0000000..9ed597b --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_16_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill MASSON Grace} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{- 7}{8} - \dfrac{- 7}{8}$ + \item $B = \dfrac{- 6}{7} - \dfrac{- 5}{70}$ + + \item $C = \dfrac{- 2}{2} + \dfrac{4}{1}$ + \item $D = \dfrac{3}{10} + 5$ + + \item $E = \dfrac{8}{3} \times \dfrac{- 10}{2}$ + \item $F = \dfrac{4}{9} \times - 4$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{- 7}{8} - \dfrac{- 7}{8}=\dfrac{- 7}{8} + \dfrac{7}{8}=\dfrac{- 7 + 7}{8}=\dfrac{0}{8} + \] + \item + \[ + \dfrac{- 6}{7} - \dfrac{- 5}{70}=\dfrac{- 6}{7} + \dfrac{5}{70}=\dfrac{- 6 \times 10}{7 \times 10} + \dfrac{5}{70}=\dfrac{- 60}{70} + \dfrac{5}{70}=\dfrac{- 60 + 5}{70}=\dfrac{- 55}{70} + \] + \item + \[ + \dfrac{- 2}{2} + \dfrac{4}{1}=\dfrac{- 2}{2} + \dfrac{4 \times 2}{1 \times 2}=\dfrac{- 2}{2} + \dfrac{8}{2}=\dfrac{- 2 + 8}{2}=\dfrac{6}{2} + \] + \item + \[ + \dfrac{3}{10} + 5=\dfrac{3}{10} + \dfrac{5}{1}=\dfrac{3}{10} + \dfrac{5 \times 10}{1 \times 10}=\dfrac{3}{10} + \dfrac{50}{10}=\dfrac{3 + 50}{10}=\dfrac{53}{10} + \] + \item + \[ + \dfrac{8}{3} \times \dfrac{- 10}{2}=\dfrac{8 \times - 10}{3 \times 2}=\dfrac{- 80}{6} + \] + \item + \[ + \dfrac{4}{9} \times - 4=\dfrac{4 \times - 4}{9}=\dfrac{- 16}{9} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (- 10x + 7)(- 1x + 7)$ + \item $B = (2x - 10)(- 4x - 10)$ + + \item $C = (- 2x + 1)^{2}$ + \item $D = - 6 + x(- 8x - 9)$ + + \item $E = - 1x^{2} + x(- 6x + 6)$ + \item $F = - 2(x + 5)(x - 5)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (- 10x + 7)(- 1x + 7)\\&= - 10x \times - x - 10x \times 7 + 7 \times - x + 7 \times 7\\&= - 10 \times - 1 \times x^{1 + 1} + 7 \times - 10 \times x + 7 \times - 1 \times x + 49\\&= - 70x - 7x + 10x^{2} + 49\\&= (- 70 - 7) \times x + 10x^{2} + 49\\&= 10x^{2} - 77x + 49 + \end{align*} + \item + \begin{align*} + B &= (2x - 10)(- 4x - 10)\\&= 2x \times - 4x + 2x \times - 10 - 10 \times - 4x - 10 \times - 10\\&= 2 \times - 4 \times x^{1 + 1} - 10 \times 2 \times x - 10 \times - 4 \times x + 100\\&= - 20x + 40x - 8x^{2} + 100\\&= (- 20 + 40) \times x - 8x^{2} + 100\\&= - 8x^{2} + 20x + 100 + \end{align*} + \item + \begin{align*} + C &= (- 2x + 1)^{2}\\&= (- 2x + 1)(- 2x + 1)\\&= - 2x \times - 2x - 2x \times 1 + 1 \times - 2x + 1 \times 1\\&= - 2 \times - 2 \times x^{1 + 1} - 2x - 2x + 1\\&= 4x^{2} - 2x - 2x + 1\\&= 4x^{2} + (- 2 - 2) \times x + 1\\&= 4x^{2} - 4x + 1 + \end{align*} + \item + \begin{align*} + D &= - 6 + x(- 8x - 9)\\&= - 6 + x \times - 8x + x \times - 9\\&= - 8x^{2} - 9x - 6 + \end{align*} + \item + \begin{align*} + E &= - 1x^{2} + x(- 6x + 6)\\&= - x^{2} + x \times - 6x + x \times 6\\&= - x^{2} - 6x^{2} + 6x\\&= - x^{2} - 6x^{2} + 6x\\&= (- 1 - 6) \times x^{2} + 6x\\&= - 7x^{2} + 6x + \end{align*} + \item + \begin{align*} + F &= - 2(x + 5)(x - 5)\\&= (- 2x - 2 \times 5)(x - 5)\\&= (- 2x - 10)(x - 5)\\&= - 2x \times x - 2x \times - 5 - 10x - 10 \times - 5\\&= - 5 \times - 2 \times x + 50 - 2x^{2} - 10x\\&= 10x + 50 - 2x^{2} - 10x\\&= - 2x^{2} + 10x - 10x + 50\\&= - 2x^{2} + (10 - 10) \times x + 50\\&= - 2x^{2} + 50 + 0x\\&= - 2x^{2} + 50 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = - 6x^{2} - 66x - 144$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = - 6 \times 1^{2} - 66 \times 1 - 144=- 6 \times 1 - 66 - 144=- 6 - 210=- 216 + \] + \[ + f(-1) = - 6 \times - 1^{2} - 66 \times - 1 - 144=- 6 \times 1 + 66 - 144=- 6 - 78=- 84 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_17_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_17_2010_DM1.tex new file mode 100644 index 0000000..f393d10 --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_17_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill MOKHTARI Nissrine} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{7}{9} - \dfrac{- 6}{9}$ + \item $B = \dfrac{6}{3} - \dfrac{3}{27}$ + + \item $C = \dfrac{- 9}{7} + \dfrac{4}{6}$ + \item $D = \dfrac{- 4}{5} + 7$ + + \item $E = \dfrac{2}{4} \times \dfrac{6}{3}$ + \item $F = \dfrac{1}{2} \times 5$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{7}{9} - \dfrac{- 6}{9}=\dfrac{7}{9} + \dfrac{6}{9}=\dfrac{7 + 6}{9}=\dfrac{13}{9} + \] + \item + \[ + \dfrac{6}{3} - \dfrac{3}{27}=\dfrac{6}{3} - \dfrac{3}{27}=\dfrac{6 \times 9}{3 \times 9} - \dfrac{3}{27}=\dfrac{54}{27} - \dfrac{3}{27}=\dfrac{54 - 3}{27}=\dfrac{54 - 3}{27}=\dfrac{51}{27} + \] + \item + \[ + \dfrac{- 9}{7} + \dfrac{4}{6}=\dfrac{- 9 \times 6}{7 \times 6} + \dfrac{4 \times 7}{6 \times 7}=\dfrac{- 54}{42} + \dfrac{28}{42}=\dfrac{- 54 + 28}{42}=\dfrac{- 26}{42} + \] + \item + \[ + \dfrac{- 4}{5} + 7=\dfrac{- 4}{5} + \dfrac{7}{1}=\dfrac{- 4}{5} + \dfrac{7 \times 5}{1 \times 5}=\dfrac{- 4}{5} + \dfrac{35}{5}=\dfrac{- 4 + 35}{5}=\dfrac{31}{5} + \] + \item + \[ + \dfrac{2}{4} \times \dfrac{6}{3}=\dfrac{2 \times 6}{4 \times 3}=\dfrac{12}{12} + \] + \item + \[ + \dfrac{1}{2} \times 5=\dfrac{1 \times 5}{2}=\dfrac{5}{2} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (8x + 8)(9x + 8)$ + \item $B = (- 9x - 7)(3x - 7)$ + + \item $C = (8x - 10)^{2}$ + \item $D = - 1 + x(- 3x - 7)$ + + \item $E = - 7x^{2} + x(3x + 1)$ + \item $F = - 6(x + 9)(x - 4)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (8x + 8)(9x + 8)\\&= 8x \times 9x + 8x \times 8 + 8 \times 9x + 8 \times 8\\&= 8 \times 9 \times x^{1 + 1} + 8 \times 8 \times x + 8 \times 9 \times x + 64\\&= 64x + 72x + 72x^{2} + 64\\&= (64 + 72) \times x + 72x^{2} + 64\\&= 72x^{2} + 136x + 64 + \end{align*} + \item + \begin{align*} + B &= (- 9x - 7)(3x - 7)\\&= - 9x \times 3x - 9x \times - 7 - 7 \times 3x - 7 \times - 7\\&= - 9 \times 3 \times x^{1 + 1} - 7 \times - 9 \times x - 7 \times 3 \times x + 49\\&= 63x - 21x - 27x^{2} + 49\\&= (63 - 21) \times x - 27x^{2} + 49\\&= - 27x^{2} + 42x + 49 + \end{align*} + \item + \begin{align*} + C &= (8x - 10)^{2}\\&= (8x - 10)(8x - 10)\\&= 8x \times 8x + 8x \times - 10 - 10 \times 8x - 10 \times - 10\\&= 8 \times 8 \times x^{1 + 1} - 10 \times 8 \times x - 10 \times 8 \times x + 100\\&= - 80x - 80x + 64x^{2} + 100\\&= (- 80 - 80) \times x + 64x^{2} + 100\\&= 64x^{2} - 160x + 100 + \end{align*} + \item + \begin{align*} + D &= - 1 + x(- 3x - 7)\\&= - 1 + x \times - 3x + x \times - 7\\&= - 3x^{2} - 7x - 1 + \end{align*} + \item + \begin{align*} + E &= - 7x^{2} + x(3x + 1)\\&= - 7x^{2} + x \times 3x + x \times 1\\&= - 7x^{2} + 3x^{2} + x\\&= - 7x^{2} + 3x^{2} + x\\&= (- 7 + 3) \times x^{2} + x\\&= - 4x^{2} + x + \end{align*} + \item + \begin{align*} + F &= - 6(x + 9)(x - 4)\\&= (- 6x - 6 \times 9)(x - 4)\\&= (- 6x - 54)(x - 4)\\&= - 6x \times x - 6x \times - 4 - 54x - 54 \times - 4\\&= - 4 \times - 6 \times x + 216 - 6x^{2} - 54x\\&= 24x + 216 - 6x^{2} - 54x\\&= - 6x^{2} + 24x - 54x + 216\\&= - 6x^{2} + (24 - 54) \times x + 216\\&= - 6x^{2} - 30x + 216 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = x^{2} + 6x - 7$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = 1^{2} + 6 \times 1 - 7=1 + 6 - 7=1 - 1=0 + \] + \[ + f(-1) = - 1^{2} + 6 \times - 1 - 7=1 - 6 - 7=1 - 13=- 12 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_18_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_18_2010_DM1.tex new file mode 100644 index 0000000..e33c1b8 --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_18_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill MOUFAQ Amine} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{10}{2} - \dfrac{4}{2}$ + \item $B = \dfrac{- 7}{3} - \dfrac{5}{15}$ + + \item $C = \dfrac{- 10}{6} + \dfrac{5}{5}$ + \item $D = \dfrac{- 10}{9} + 10$ + + \item $E = \dfrac{4}{2} \times \dfrac{- 3}{1}$ + \item $F = \dfrac{- 10}{3} \times 10$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{10}{2} - \dfrac{4}{2}=\dfrac{10}{2} - \dfrac{4}{2}=\dfrac{10 - 4}{2}=\dfrac{10 - 4}{2}=\dfrac{6}{2} + \] + \item + \[ + \dfrac{- 7}{3} - \dfrac{5}{15}=\dfrac{- 7}{3} - \dfrac{5}{15}=\dfrac{- 7 \times 5}{3 \times 5} - \dfrac{5}{15}=\dfrac{- 35}{15} - \dfrac{5}{15}=\dfrac{- 35 - 5}{15}=\dfrac{- 35 - 5}{15}=\dfrac{- 40}{15} + \] + \item + \[ + \dfrac{- 10}{6} + \dfrac{5}{5}=\dfrac{- 10 \times 5}{6 \times 5} + \dfrac{5 \times 6}{5 \times 6}=\dfrac{- 50}{30} + \dfrac{30}{30}=\dfrac{- 50 + 30}{30}=\dfrac{- 20}{30} + \] + \item + \[ + \dfrac{- 10}{9} + 10=\dfrac{- 10}{9} + \dfrac{10}{1}=\dfrac{- 10}{9} + \dfrac{10 \times 9}{1 \times 9}=\dfrac{- 10}{9} + \dfrac{90}{9}=\dfrac{- 10 + 90}{9}=\dfrac{80}{9} + \] + \item + \[ + \dfrac{4}{2} \times \dfrac{- 3}{1}=\dfrac{4 \times - 3}{2 \times 1}=\dfrac{- 12}{2} + \] + \item + \[ + \dfrac{- 10}{3} \times 10=\dfrac{- 10 \times 10}{3}=\dfrac{- 100}{3} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (- 9x + 2)(- 8x + 2)$ + \item $B = (- 7x - 7)(- 8x - 7)$ + + \item $C = (2x - 8)^{2}$ + \item $D = - 6 + x(5x - 3)$ + + \item $E = 8x^{2} + x(- 4x - 7)$ + \item $F = 5(x - 4)(x + 2)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (- 9x + 2)(- 8x + 2)\\&= - 9x \times - 8x - 9x \times 2 + 2 \times - 8x + 2 \times 2\\&= - 9 \times - 8 \times x^{1 + 1} + 2 \times - 9 \times x + 2 \times - 8 \times x + 4\\&= - 18x - 16x + 72x^{2} + 4\\&= (- 18 - 16) \times x + 72x^{2} + 4\\&= 72x^{2} - 34x + 4 + \end{align*} + \item + \begin{align*} + B &= (- 7x - 7)(- 8x - 7)\\&= - 7x \times - 8x - 7x \times - 7 - 7 \times - 8x - 7 \times - 7\\&= - 7 \times - 8 \times x^{1 + 1} - 7 \times - 7 \times x - 7 \times - 8 \times x + 49\\&= 49x + 56x + 56x^{2} + 49\\&= (49 + 56) \times x + 56x^{2} + 49\\&= 56x^{2} + 105x + 49 + \end{align*} + \item + \begin{align*} + C &= (2x - 8)^{2}\\&= (2x - 8)(2x - 8)\\&= 2x \times 2x + 2x \times - 8 - 8 \times 2x - 8 \times - 8\\&= 2 \times 2 \times x^{1 + 1} - 8 \times 2 \times x - 8 \times 2 \times x + 64\\&= - 16x - 16x + 4x^{2} + 64\\&= (- 16 - 16) \times x + 4x^{2} + 64\\&= 4x^{2} - 32x + 64 + \end{align*} + \item + \begin{align*} + D &= - 6 + x(5x - 3)\\&= - 6 + x \times 5x + x \times - 3\\&= 5x^{2} - 3x - 6 + \end{align*} + \item + \begin{align*} + E &= 8x^{2} + x(- 4x - 7)\\&= 8x^{2} + x \times - 4x + x \times - 7\\&= 8x^{2} - 4x^{2} - 7x\\&= 8x^{2} - 4x^{2} - 7x\\&= (8 - 4) \times x^{2} - 7x\\&= 4x^{2} - 7x + \end{align*} + \item + \begin{align*} + F &= 5(x - 4)(x + 2)\\&= (5x + 5 \times - 4)(x + 2)\\&= (5x - 20)(x + 2)\\&= 5x \times x + 5x \times 2 - 20x - 20 \times 2\\&= 2 \times 5 \times x - 40 + 5x^{2} - 20x\\&= 10x - 40 + 5x^{2} - 20x\\&= 5x^{2} + 10x - 20x - 40\\&= 5x^{2} + (10 - 20) \times x - 40\\&= 5x^{2} - 10x - 40 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = - 9x^{2} + 90x - 216$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = - 9 \times 1^{2} + 90 \times 1 - 216=- 9 \times 1 + 90 - 216=- 9 - 126=- 135 + \] + \[ + f(-1) = - 9 \times - 1^{2} + 90 \times - 1 - 216=- 9 \times 1 - 90 - 216=- 9 - 306=- 315 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_19_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_19_2010_DM1.tex new file mode 100644 index 0000000..8c4d0c1 --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_19_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill ONAL Yakub} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{9}{6} - \dfrac{9}{6}$ + \item $B = \dfrac{3}{4} - \dfrac{1}{32}$ + + \item $C = \dfrac{- 3}{7} + \dfrac{- 10}{6}$ + \item $D = \dfrac{- 7}{2} - 1$ + + \item $E = \dfrac{5}{10} \times \dfrac{- 7}{9}$ + \item $F = \dfrac{10}{7} \times 9$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{9}{6} - \dfrac{9}{6}=\dfrac{9}{6} - \dfrac{9}{6}=\dfrac{9 - 9}{6}=\dfrac{9 - 9}{6}=\dfrac{0}{6} + \] + \item + \[ + \dfrac{3}{4} - \dfrac{1}{32}=\dfrac{3}{4} - \dfrac{1}{32}=\dfrac{3 \times 8}{4 \times 8} - \dfrac{1}{32}=\dfrac{24}{32} - \dfrac{1}{32}=\dfrac{24 - 1}{32}=\dfrac{24 - 1}{32}=\dfrac{23}{32} + \] + \item + \[ + \dfrac{- 3}{7} + \dfrac{- 10}{6}=\dfrac{- 3 \times 6}{7 \times 6} + \dfrac{- 10 \times 7}{6 \times 7}=\dfrac{- 18}{42} + \dfrac{- 70}{42}=\dfrac{- 18 - 70}{42}=\dfrac{- 88}{42} + \] + \item + \[ + \dfrac{- 7}{2} - 1=\dfrac{- 7}{2} + \dfrac{- 1}{1}=\dfrac{- 7}{2} + \dfrac{- 1 \times 2}{1 \times 2}=\dfrac{- 7}{2} + \dfrac{- 2}{2}=\dfrac{- 7 - 2}{2}=\dfrac{- 9}{2} + \] + \item + \[ + \dfrac{5}{10} \times \dfrac{- 7}{9}=\dfrac{5 \times - 7}{10 \times 9}=\dfrac{- 35}{90} + \] + \item + \[ + \dfrac{10}{7} \times 9=\dfrac{10 \times 9}{7}=\dfrac{90}{7} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (1x + 7)(3x + 7)$ + \item $B = (- 7x - 3)(- 9x - 3)$ + + \item $C = (- 2x + 9)^{2}$ + \item $D = - 3 + x(6x + 1)$ + + \item $E = - 7x^{2} + x(- 10x - 10)$ + \item $F = - 7(x - 6)(x - 6)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (1x + 7)(3x + 7)\\&= x \times 3x + x \times 7 + 7 \times 3x + 7 \times 7\\&= 7 \times 3 \times x + 49 + 3x^{2} + 7x\\&= 21x + 49 + 3x^{2} + 7x\\&= 3x^{2} + 21x + 7x + 49\\&= 3x^{2} + (21 + 7) \times x + 49\\&= 3x^{2} + 28x + 49 + \end{align*} + \item + \begin{align*} + B &= (- 7x - 3)(- 9x - 3)\\&= - 7x \times - 9x - 7x \times - 3 - 3 \times - 9x - 3 \times - 3\\&= - 7 \times - 9 \times x^{1 + 1} - 3 \times - 7 \times x - 3 \times - 9 \times x + 9\\&= 21x + 27x + 63x^{2} + 9\\&= (21 + 27) \times x + 63x^{2} + 9\\&= 63x^{2} + 48x + 9 + \end{align*} + \item + \begin{align*} + C &= (- 2x + 9)^{2}\\&= (- 2x + 9)(- 2x + 9)\\&= - 2x \times - 2x - 2x \times 9 + 9 \times - 2x + 9 \times 9\\&= - 2 \times - 2 \times x^{1 + 1} + 9 \times - 2 \times x + 9 \times - 2 \times x + 81\\&= - 18x - 18x + 4x^{2} + 81\\&= (- 18 - 18) \times x + 4x^{2} + 81\\&= 4x^{2} - 36x + 81 + \end{align*} + \item + \begin{align*} + D &= - 3 + x(6x + 1)\\&= - 3 + x \times 6x + x \times 1\\&= 6x^{2} + x - 3 + \end{align*} + \item + \begin{align*} + E &= - 7x^{2} + x(- 10x - 10)\\&= - 7x^{2} + x \times - 10x + x \times - 10\\&= - 7x^{2} - 10x^{2} - 10x\\&= - 7x^{2} - 10x^{2} - 10x\\&= (- 7 - 10) \times x^{2} - 10x\\&= - 17x^{2} - 10x + \end{align*} + \item + \begin{align*} + F &= - 7(x - 6)(x - 6)\\&= (- 7x - 7 \times - 6)(x - 6)\\&= (- 7x + 42)(x - 6)\\&= - 7x \times x - 7x \times - 6 + 42x + 42 \times - 6\\&= - 6 \times - 7 \times x - 252 - 7x^{2} + 42x\\&= 42x - 252 - 7x^{2} + 42x\\&= - 7x^{2} + 42x + 42x - 252\\&= - 7x^{2} + (42 + 42) \times x - 252\\&= - 7x^{2} + 84x - 252 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = - x^{2} + 4x + 60$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = - 1 \times 1^{2} + 4 \times 1 + 60=- 1 \times 1 + 4 + 60=- 1 + 64=63 + \] + \[ + f(-1) = - 1 \times - 1^{2} + 4 \times - 1 + 60=- 1 \times 1 - 4 + 60=- 1 + 56=55 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_20_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_20_2010_DM1.tex new file mode 100644 index 0000000..f0f100b --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_20_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill SORIANO Laura} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{- 7}{6} - \dfrac{9}{6}$ + \item $B = \dfrac{2}{4} - \dfrac{4}{20}$ + + \item $C = \dfrac{7}{10} + \dfrac{- 7}{9}$ + \item $D = \dfrac{6}{4} - 3$ + + \item $E = \dfrac{- 6}{8} \times \dfrac{7}{7}$ + \item $F = \dfrac{- 5}{7} \times - 2$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{- 7}{6} - \dfrac{9}{6}=\dfrac{- 7}{6} - \dfrac{9}{6}=\dfrac{- 7 - 9}{6}=\dfrac{- 7 - 9}{6}=\dfrac{- 16}{6} + \] + \item + \[ + \dfrac{2}{4} - \dfrac{4}{20}=\dfrac{2}{4} - \dfrac{4}{20}=\dfrac{2 \times 5}{4 \times 5} - \dfrac{4}{20}=\dfrac{10}{20} - \dfrac{4}{20}=\dfrac{10 - 4}{20}=\dfrac{10 - 4}{20}=\dfrac{6}{20} + \] + \item + \[ + \dfrac{7}{10} + \dfrac{- 7}{9}=\dfrac{7 \times 9}{10 \times 9} + \dfrac{- 7 \times 10}{9 \times 10}=\dfrac{63}{90} + \dfrac{- 70}{90}=\dfrac{63 - 70}{90}=\dfrac{- 7}{90} + \] + \item + \[ + \dfrac{6}{4} - 3=\dfrac{6}{4} + \dfrac{- 3}{1}=\dfrac{6}{4} + \dfrac{- 3 \times 4}{1 \times 4}=\dfrac{6}{4} + \dfrac{- 12}{4}=\dfrac{6 - 12}{4}=\dfrac{- 6}{4} + \] + \item + \[ + \dfrac{- 6}{8} \times \dfrac{7}{7}=\dfrac{- 6 \times 7}{8 \times 7}=\dfrac{- 42}{56} + \] + \item + \[ + \dfrac{- 5}{7} \times - 2=\dfrac{- 5 \times - 2}{7}=\dfrac{10}{7} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (2x + 4)(- 2x + 4)$ + \item $B = (- 1x + 6)(- 7x + 6)$ + + \item $C = (7x - 7)^{2}$ + \item $D = - 5 + x(- 9x - 9)$ + + \item $E = 7x^{2} + x(- 5x - 4)$ + \item $F = - 10(x - 8)(x + 9)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (2x + 4)(- 2x + 4)\\&= 2x \times - 2x + 2x \times 4 + 4 \times - 2x + 4 \times 4\\&= 2 \times - 2 \times x^{1 + 1} + 4 \times 2 \times x + 4 \times - 2 \times x + 16\\&= 8x - 8x - 4x^{2} + 16\\&= (8 - 8) \times x - 4x^{2} + 16\\&= 0x - 4x^{2} + 16\\&= - 4x^{2} + 16 + \end{align*} + \item + \begin{align*} + B &= (- 1x + 6)(- 7x + 6)\\&= - x \times - 7x - x \times 6 + 6 \times - 7x + 6 \times 6\\&= - 1 \times - 7 \times x^{1 + 1} + 6 \times - 1 \times x + 6 \times - 7 \times x + 36\\&= - 6x - 42x + 7x^{2} + 36\\&= (- 6 - 42) \times x + 7x^{2} + 36\\&= 7x^{2} - 48x + 36 + \end{align*} + \item + \begin{align*} + C &= (7x - 7)^{2}\\&= (7x - 7)(7x - 7)\\&= 7x \times 7x + 7x \times - 7 - 7 \times 7x - 7 \times - 7\\&= 7 \times 7 \times x^{1 + 1} - 7 \times 7 \times x - 7 \times 7 \times x + 49\\&= - 49x - 49x + 49x^{2} + 49\\&= (- 49 - 49) \times x + 49x^{2} + 49\\&= 49x^{2} - 98x + 49 + \end{align*} + \item + \begin{align*} + D &= - 5 + x(- 9x - 9)\\&= - 5 + x \times - 9x + x \times - 9\\&= - 9x^{2} - 9x - 5 + \end{align*} + \item + \begin{align*} + E &= 7x^{2} + x(- 5x - 4)\\&= 7x^{2} + x \times - 5x + x \times - 4\\&= 7x^{2} - 5x^{2} - 4x\\&= 7x^{2} - 5x^{2} - 4x\\&= (7 - 5) \times x^{2} - 4x\\&= 2x^{2} - 4x + \end{align*} + \item + \begin{align*} + F &= - 10(x - 8)(x + 9)\\&= (- 10x - 10 \times - 8)(x + 9)\\&= (- 10x + 80)(x + 9)\\&= - 10x \times x - 10x \times 9 + 80x + 80 \times 9\\&= 9 \times - 10 \times x + 720 - 10x^{2} + 80x\\&= - 90x + 720 - 10x^{2} + 80x\\&= - 10x^{2} - 90x + 80x + 720\\&= - 10x^{2} + (- 90 + 80) \times x + 720\\&= - 10x^{2} - 10x + 720 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = 6x^{2} - 48x - 54$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = 6 \times 1^{2} - 48 \times 1 - 54=6 \times 1 - 48 - 54=6 - 102=- 96 + \] + \[ + f(-1) = 6 \times - 1^{2} - 48 \times - 1 - 54=6 \times 1 + 48 - 54=6 - 6=0 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_21_2010_DM1.tex b/TST/DM/2010_DM1/TST1/corr_21_2010_DM1.tex new file mode 100644 index 0000000..585a62e --- /dev/null +++ b/TST/DM/2010_DM1/TST1/corr_21_2010_DM1.tex @@ -0,0 +1,141 @@ +\documentclass[a5paper,10pt]{article} +\usepackage{myXsim} +\usepackage{tasks} + +% Title Page +\title{DM1 \hfill VECCHIO Léa} +\tribe{TST} +\date{Toussain 2020} + +\begin{document} +\maketitle + +\begin{exercise}[subtitle={Fractions}] + Faire les calculs avec les fraction suivants + \begin{multicols}{3} + \begin{enumerate} + \item $A = \dfrac{5}{9} - \dfrac{9}{9}$ + \item $B = \dfrac{7}{4} - \dfrac{8}{36}$ + + \item $C = \dfrac{- 10}{10} + \dfrac{2}{9}$ + \item $D = \dfrac{5}{4} + 6$ + + \item $E = \dfrac{- 7}{7} \times \dfrac{1}{6}$ + \item $F = \dfrac{7}{10} \times - 8$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \[ + \dfrac{5}{9} - \dfrac{9}{9}=\dfrac{5}{9} - \dfrac{9}{9}=\dfrac{5 - 9}{9}=\dfrac{5 - 9}{9}=\dfrac{- 4}{9} + \] + \item + \[ + \dfrac{7}{4} - \dfrac{8}{36}=\dfrac{7}{4} - \dfrac{8}{36}=\dfrac{7 \times 9}{4 \times 9} - \dfrac{8}{36}=\dfrac{63}{36} - \dfrac{8}{36}=\dfrac{63 - 8}{36}=\dfrac{63 - 8}{36}=\dfrac{55}{36} + \] + \item + \[ + \dfrac{- 10}{10} + \dfrac{2}{9}=\dfrac{- 10 \times 9}{10 \times 9} + \dfrac{2 \times 10}{9 \times 10}=\dfrac{- 90}{90} + \dfrac{20}{90}=\dfrac{- 90 + 20}{90}=\dfrac{- 70}{90} + \] + \item + \[ + \dfrac{5}{4} + 6=\dfrac{5}{4} + \dfrac{6}{1}=\dfrac{5}{4} + \dfrac{6 \times 4}{1 \times 4}=\dfrac{5}{4} + \dfrac{24}{4}=\dfrac{5 + 24}{4}=\dfrac{29}{4} + \] + \item + \[ + \dfrac{- 7}{7} \times \dfrac{1}{6}=\dfrac{- 7 \times 1}{7 \times 6}=\dfrac{- 7}{42} + \] + \item + \[ + \dfrac{7}{10} \times - 8=\dfrac{7 \times - 8}{10}=\dfrac{- 56}{10} + \] + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Développer réduire}] + Développer puis réduire les expressions suivantes + \begin{multicols}{2} + \begin{enumerate} + \item $A = (- 5x + 5)(- 8x + 5)$ + \item $B = (1x + 7)(- 10x + 7)$ + + \item $C = (2x + 4)^{2}$ + \item $D = 1 + x(- 7x + 1)$ + + \item $E = - 3x^{2} + x(- 2x + 6)$ + \item $F = 5(x - 6)(x + 8)$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item + \begin{align*} + A &= (- 5x + 5)(- 8x + 5)\\&= - 5x \times - 8x - 5x \times 5 + 5 \times - 8x + 5 \times 5\\&= - 5 \times - 8 \times x^{1 + 1} + 5 \times - 5 \times x + 5 \times - 8 \times x + 25\\&= - 25x - 40x + 40x^{2} + 25\\&= (- 25 - 40) \times x + 40x^{2} + 25\\&= 40x^{2} - 65x + 25 + \end{align*} + \item + \begin{align*} + B &= (1x + 7)(- 10x + 7)\\&= x \times - 10x + x \times 7 + 7 \times - 10x + 7 \times 7\\&= 7 \times - 10 \times x + 49 - 10x^{2} + 7x\\&= - 70x + 49 - 10x^{2} + 7x\\&= - 10x^{2} - 70x + 7x + 49\\&= - 10x^{2} + (- 70 + 7) \times x + 49\\&= - 10x^{2} - 63x + 49 + \end{align*} + \item + \begin{align*} + C &= (2x + 4)^{2}\\&= (2x + 4)(2x + 4)\\&= 2x \times 2x + 2x \times 4 + 4 \times 2x + 4 \times 4\\&= 2 \times 2 \times x^{1 + 1} + 4 \times 2 \times x + 4 \times 2 \times x + 16\\&= 8x + 8x + 4x^{2} + 16\\&= (8 + 8) \times x + 4x^{2} + 16\\&= 4x^{2} + 16x + 16 + \end{align*} + \item + \begin{align*} + D &= 1 + x(- 7x + 1)\\&= 1 + x \times - 7x + x \times 1\\&= - 7x^{2} + x + 1 + \end{align*} + \item + \begin{align*} + E &= - 3x^{2} + x(- 2x + 6)\\&= - 3x^{2} + x \times - 2x + x \times 6\\&= - 3x^{2} - 2x^{2} + 6x\\&= - 3x^{2} - 2x^{2} + 6x\\&= (- 3 - 2) \times x^{2} + 6x\\&= - 5x^{2} + 6x + \end{align*} + \item + \begin{align*} + F &= 5(x - 6)(x + 8)\\&= (5x + 5 \times - 6)(x + 8)\\&= (5x - 30)(x + 8)\\&= 5x \times x + 5x \times 8 - 30x - 30 \times 8\\&= 8 \times 5 \times x - 240 + 5x^{2} - 30x\\&= 40x - 240 + 5x^{2} - 30x\\&= 5x^{2} + 40x - 30x - 240\\&= 5x^{2} + (40 - 30) \times x - 240\\&= 5x^{2} + 10x - 240 + \end{align*} + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Étude de fonctions}] + Soit $f(x) = - 6x^{2} - 108x - 480$ une fonction définie sur $\R$. + \begin{enumerate} + \item Calculer les valeurs suivantes + \[ + f(1) \qquad f(-2) + \] + \item Dériver la fonction $f$ + \item Étudier le signe de $f'$ puis en déduire les variations de $f$. + \item Est-ce que $f$ admet un maximum? un minimum? Calculer sa valeur. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item On remplace $x$ par les valeurs demandées + \[ + f(1) = - 6 \times 1^{2} - 108 \times 1 - 480=- 6 \times 1 - 108 - 480=- 6 - 588=- 594 + \] + \[ + f(-1) = - 6 \times - 1^{2} - 108 \times - 1 - 480=- 6 \times 1 + 108 - 480=- 6 - 372=- 378 + \] + \item Pas de solutions automatiques. + \item Pas de solutions automatiques. + \end{enumerate} +\end{solution} + + + +\printsolutionstype{exercise} + + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "master" +%%% End: diff --git a/TST/DM/2010_DM1/TST1/corr_all_2010_DM1.pdf b/TST/DM/2010_DM1/TST1/corr_all_2010_DM1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1737ae691484a32bcb354ed9c9d85fbb157fab97 GIT binary patch literal 239321 zcmdSBW0YoFmWCU)%?#T{hHcxnZQHhO+sMeUnPD3l&hU*>U8m}F_36>){;0lvzaKHy z+GDRhW~{x(H{W+YdwwKxLLxK_v@B30GYdm2P^@hD^!T<07EoMVP;^qpHl|Ky_^j;s 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