Devoir Libre n 2: Calcul et ordre dans IR et droite dans le plan

📅 December 01, 2024 | 👁️ Views: 890

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\begin{tabularx}{\textwidth}{@{} lCr @{}}
    Lycee Taghzirt\textbf{/}Prof MOSAID &
    2024-2025\textbf{/Devoir Libre 2 S01}&
    TCSF  \textbf{/}...h\\
    \bottomrule
\end{tabularx}
\mylink \hfill \mylink\\
\textbf{\underline{Exercice 1:}}\\
Le plan est rapporté à un repère orthonormé $(O; \vec i, \vec j)$\\
Soient les points ~$A(5,-3$~, ~$B(3,-1)$~ , ~$C(2,-3)$~ et ~$D(-1,2)$\\
\textbf{1}~$-$ Est ce que les points ~$A$~, ~$B$~ et ~$C$~ sont alignés?\\
\textbf{2}~$-$ Donner une équation cartésienne de la droite ~$(AB)$\\
\textbf{3}~$-$ Donner une représentation paramétrique de la droite ~$(AC)$\\
\textbf{4}~$-$ Donner une équation cartésienne de la droite ~$(\Delta)$~ passant par ~$C$~
et parallèle à ~$(AB)$\\
\textbf{5}~$-$ Construir la figure\\
\textbf{\underline{Exercice 2:}}\\
\textbf{1}~$-$ Soient les intervalles: ~$I=\left[\frac{13}{6},+\infty\right[$~ , ~$J=\left]2,\frac{7}{3}\right[$~
et ~$K=\left]-\infty,\frac{25}{12}\right]$\\
\hspace*{2cm} Déterminer ~$I\cap J$~, ~$I\cup J$~, ~$I\cap K$~, ~$I\cup K$~, ~$J\cap K$~, ~$J\cup K$\\
\textbf{2}~$-$ Résoudre dans ~$\mathbb{R}$:~ ~$|3x-3|=|2x-4|$~ et ~$|2x-7|\le 4$\\
\textbf{3}~$-$ Soient ~$-2<x<1$~ et ~$-3<y<-1$~ et ~$A=x^2+x-2$\\
\hspace*{0.5cm}\textbf{a}~$-$ Encadrer ~$xy$~, ~$y^2$~ et ~$A$\\
\hspace*{0.5cm}\textbf{b}~$-$ Montrer que ~$A=(x-1)(x+2)$~ puis donner un meilleur encadrement pour ~$A$\\
\textbf{\underline{Exercice 3:}}\\
\textbf{1}~$-$ Développez l'expression suivante : \( A = \left( 2\sqrt{3} - 3\sqrt{2} \right)^3 \)\\
\textbf{2}~$-$ Considérons \( a \) et \( b \) tels que : $ab = -\frac{1}{3} \quad \text{et} \quad a + b = \sqrt{2}$~~
Calculez : $a^3 + b^3 \quad \text{et} \quad a^2 + b^2$\\
\textbf{3}~$-$ Donnez l'écriture décimale et scientifique du nombre \( B \), où :
$B = \frac{3 \cdot 10^{-3} \cdot \left( 2 \cdot 10^{-6} \right)^2}{15 \cdot \left( 10^2 \right)^{-3}}$\\
\textbf{4}~$-$ Soient \( a \) et \( b \) tels que : ~$16a + 12b = 5$~
Simplifiez l'expression : ~$C = (a + 8)^2 - (a - 8)^2 + (b + 6)^2 - (b - 6)^2$\\
\textbf{5}~$-$ Considérons : ~$D = -\sqrt{2 + \sqrt{3}} + \sqrt{2 - \sqrt{3}}$\\
\hspace*{1cm} \textbf{a}~$-$ Montrez que le signe de \( D \) est négatif.\\
\hspace*{1cm} \textbf{b}~$-$ Calculez \( D^2 \), puis déduisez la valeur simplifiée de \( D \).\\
\textbf{6}~$-$ Factoriser ~$E=3\sqrt3x^3+2\sqrt2-4x(3x^2-2)+3x^2+2\sqrt{6}+4$~\\
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\usepackage{xcolor}
\usepackage[ddmmyyyy]{datetime}

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    Lycee Taghzirt\textbf{/}Prof MOSAID &
    2024-2025\textbf{/Devoir Libre 2 S01}&
    TCSF  \textbf{/}...h\\
    \bottomrule
\end{tabularx}
\mylink \hfill \mylink\\
\textbf{\underline{Exercice 1:}}\\
Le plan est rapporté à un repère orthonormé $(O; \vec i, \vec j)$\\
Soient les points ~$A(5,-3$~, ~$B(3,-1)$~ , ~$C(2,-3)$~ et ~$D(-1,2)$\\
\textbf{1}~$-$ Est ce que les points ~$A$~, ~$B$~ et ~$C$~ sont alignés?\\
\textbf{2}~$-$ Donner une équation cartésienne de la droite ~$(AB)$\\
\textbf{3}~$-$ Donner une représentation paramétrique de la droite ~$(AC)$\\
\textbf{4}~$-$ Donner une équation cartésienne de la droite ~$(\Delta)$~ passant par ~$C$~
et parallèle à ~$(AB)$\\
\textbf{5}~$-$ Construir la figure\\
\textbf{\underline{Exercice 2:}}\\
\textbf{1}~$-$ Soient les intervalles: ~$I=\left[\frac{13}{6},+\infty\right[$~ , ~$J=\left]2,\frac{7}{3}\right[$~
et ~$K=\left]-\infty,\frac{25}{12}\right]$\\
\hspace*{2cm} Déterminer ~$I\cap J$~, ~$I\cup J$~, ~$I\cap K$~, ~$I\cup K$~, ~$J\cap K$~, ~$J\cup K$\\
\textbf{2}~$-$ Résoudre dans ~$\mathbb{R}$:~ ~$|3x-3|=|2x-4|$~ et ~$|2x-7|\le 4$\\
\textbf{3}~$-$ Soient ~$-2<x<1$~ et ~$-3<y<-1$~ et ~$A=x^2+x-2$\\
\hspace*{0.5cm}\textbf{a}~$-$ Encadrer ~$xy$~, ~$y^2$~ et ~$A$\\
\hspace*{0.5cm}\textbf{b}~$-$ Montrer que ~$A=(x-1)(x+2)$~ puis donner un meilleur encadrement pour ~$A$\\
\textbf{\underline{Exercice 3:}}\\
\textbf{1}~$-$ Développez l'expression suivante : \( A = \left( 2\sqrt{3} - 3\sqrt{2} \right)^3 \)\\
\textbf{2}~$-$ Considérons \( a \) et \( b \) tels que : $ab = -\frac{1}{3} \quad \text{et} \quad a + b = \sqrt{2}$~~
Calculez : $a^3 + b^3 \quad \text{et} \quad a^2 + b^2$\\
\textbf{3}~$-$ Donnez l'écriture décimale et scientifique du nombre \( B \), où :
$B = \frac{3 \cdot 10^{-3} \cdot \left( 2 \cdot 10^{-6} \right)^2}{15 \cdot \left( 10^2 \right)^{-3}}$\\
\textbf{4}~$-$ Soient \( a \) et \( b \) tels que : ~$16a + 12b = 5$~
Simplifiez l'expression : ~$C = (a + 8)^2 - (a - 8)^2 + (b + 6)^2 - (b - 6)^2$\\
\textbf{5}~$-$ Considérons : ~$D = -\sqrt{2 + \sqrt{3}} + \sqrt{2 - \sqrt{3}}$\\
\hspace*{1cm} \textbf{a}~$-$ Montrez que le signe de \( D \) est négatif.\\
\hspace*{1cm} \textbf{b}~$-$ Calculez \( D^2 \), puis déduisez la valeur simplifiée de \( D \).\\
\textbf{6}~$-$ Factoriser ~$E=3\sqrt3x^3+2\sqrt2-4x(3x^2-2)+3x^2+2\sqrt{6}+4$~\\
\begin{center}
    \stamp{0}{-1}
\end{center}

\textcolor{white}{.}\hfill \underline{MOSAID le \today}\\
\textcolor{white}{.}\hfill \mylink
\end{document}

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Frequently Asked Questions

What chapters or courses does this exam cover?
This exam covers: مجموعات الأعداد, Ensembles des nombres, مجموعات الأعداد نسخة 2, L'ordre dans IR - partie 2, L'ordre dans IR - partie 1, الترتيب في IR, الترتيب في IR نسخة 2, المستقيم في المستوى نسخة 2, المستقيم في المستوى. It is designed to test understanding of these topics.

How many questions are in this exam?
The exam contains approximately several questions.

Is this exam aligned with the official curriculum?
Yes, it follows the tronc-commun-sciences maths guidelines.

What topics are covered in this course?
The course "Droite dans Le Plan" covers key concepts of maths for tronc-commun-sciences. Designed to help students master the curriculum.

Is this course suitable for beginners?
Yes, the material is structured to be accessible while providing depth for advanced learners.

Are there exercises or practice problems?
Exercises are included to help you practice.

Does this course include solutions?
Solutions are available separately.