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a/amc-templates/amc_template_en/extra_section.tex +++ b/amc-templates/amc_template_en/extra_section.tex @@ -1,130 +1,128 @@ \setlength{\parindent}{0pt}% \newpage %*******************************************************************% % Suppress begin-question texts \renewcommand{\AMCbeginQuestion}[2]{\QuestionText{#1}} \def\QuestionText{\TEXT} \def\TEXT#1{} \def\NOTEXT#1{} \setlength{\parindent}{0pt}% -\addtocounter{AMCquestionaff}{1} %*******************************************************************% -\subsection*{Trois\`eme partie, questions de type ouvert} +\subsection*{Third part, open questions} \noindent -R\'epondre dans l’espace d\'edi\'e. Votre r\'eponse doit \^etre soigneusement justifi\'ee, -toutes les \'etapes de votre raisonnement doivent figurer dans votre r\'eponse. Laisser libres les cases \`a cocher\,: elles sont r\'eserv\'ees au correcteur. +Answer in the empty space below. Your answer should be carefully justified, and all the steps of your argument should be discussed in details. Leave the check-boxes empty, they are used for the grading. \bigskip %% Leave at least 2 empty lines after the \bigskip %*******************************************************************% % Question A - 5 points %*******************************************************************% \begin{description} \addtocounter{AMCquestionaff}{1} -\item[Question~\theAMCquestionaff :] \textit{Cette question est not\'ee sur 5 points.} +\item[Question~\theAMCquestionaff :] \textit{This question is worth 5 points.} \end{description} \correctorPointsPerGroup{5/3} \correctorThreeFive{q-open-A}{~} \correctorStop Soit $\Psi : \R_3[x]\to \R_3[x]$ l'application d\'efinie par $$\Psi(p)(x) = (x-1)p'(x)\text{.}$$ \begin{itemize} \item [1.] Montrer que $\Psi$ est lin\'eaire. (1 pt) \item [2.] Calculer la matrice $[\Psi]_{E,E}$ de $\Psi$ par rapport \`a la base canonique $E = (1,x, x^2, x^3)$. (2 pts) \item [3.] Calculer le rang de $\Psi$. (2 pt) \end{itemize} \OpenGrid{16cm} \FullPageOpenGrid %*******************************************************************% %*******************************************************************% % Question B - 6 points %*******************************************************************% \begin{description} -\item[Question~\theAMCquestionaff :] \textit{Cette question est not\'ee sur 6 points.} +\item[Question~\theAMCquestionaff :] \textit{This question is worth 6 points.} \end{description} \correctorPointsPerGroup{6/2} \correctorTwoFive{q-open-B}{~} \correctorStop \noindent Soient $V$ un $K$-espace vectoriel de dimension finie, et $X,Y$ deux sous-espaces vectoriels de $V$ tels que $\operatorname{dim}(X) \geq \operatorname{dim}(Y)$. Montrer qu'il existe une application lin\'eaire $T : V\to V$ telle que $T(X) = Y$. \vskip 5pt \OpenBox{22cm} %*************************** % Question C - 6points %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{description} -\item[Question~\theAMCquestionaff :] \textit{Cette question est not\'ee sur 6 points.} +\item[Question~\theAMCquestionaff :] \textit{This question is worth 6 points.} \end{description} \correctorSix{q-open-C}{~} \correctorStop Soit $\alpha\in \C$. \begin{itemize} \item [1.] Trouver la formule explicite pour les éléments de la matrice $A_n = \begin{pmatrix}\alpha &1\\ 0 &\alpha\end{pmatrix}^{n}$, o\`u $n \geq 1$ est un entier. Montrer la formule par r\'eccurence. % \item [1.] Trouver la formule pour $\begin{pmatrix}\alpha &1\\ 0 &\alpha\end{pmatrix}^{n}$, o\`u $n \geq 1$ est un entier. Montrer la formule par r\'eccurance. \item [2.] On pose $\alpha = 1+\textrm{i}$. Calculer $\alpha^{99}$ et $\alpha^{100}$. \item [3.] Calculer $\begin{pmatrix} 1+\textrm{i} &1\\ 0 &1+\textrm{i}\end{pmatrix}^{100}$. \end{itemize} \OpenGrid{17cm} % Question D - 10points %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{description} -\item[Question~\theAMCquestionaff :] \textit{Cette question est not\'ee sur 10 points.} +\item[Question~\theAMCquestionaff :] \textit{This question is worth 10 points.} \end{description} \correctorTenHalf{q-open-D}{~} \correctorStop Soit $\alpha\in \C$. \begin{itemize} \item [1.] Trouver la formule explicite pour les éléments de la matrice $A_n = \begin{pmatrix}\alpha &1\\ 0 &\alpha\end{pmatrix}^{n}$, o\`u $n \geq 1$ est un entier. Montrer la formule par r\'eccurence. % \item [1.] Trouver la formule pour $\begin{pmatrix}\alpha &1\\ 0 &\alpha\end{pmatrix}^{n}$, o\`u $n \geq 1$ est un entier. Montrer la formule par r\'eccurance. \item [2.] On pose $\alpha = 1+\textrm{i}$. Calculer $\alpha^{99}$ et $\alpha^{100}$. \item [3.] Calculer $\begin{pmatrix} 1+\textrm{i} &1\\ 0 &1+\textrm{i}\end{pmatrix}^{100}$. \end{itemize} \OpenGrid{17cm} \FullPageOpenGrid % Question E - 15points %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{description} -\item[Question~\theAMCquestionaff :] \textit{Cette question est not\'ee sur 20 points.} +\item[Question~\theAMCquestionaff :] \textit{This question is worth 20 points.} \end{description} \correctorTwenty{q-open-C}{~} \correctorStop Soit $\alpha\in \C$. \begin{itemize} \item [1.] Trouver la formule explicite pour les éléments de la matrice $A_n = \begin{pmatrix}\alpha &1\\ 0 &\alpha\end{pmatrix}^{n}$, o\`u $n \geq 1$ est un entier. Montrer la formule par r\'eccurence. % \item [1.] Trouver la formule pour $\begin{pmatrix}\alpha &1\\ 0 &\alpha\end{pmatrix}^{n}$, o\`u $n \geq 1$ est un entier. Montrer la formule par r\'eccurance. \item [2.] On pose $\alpha = 1+\textrm{i}$. Calculer $\alpha^{99}$ et $\alpha^{100}$. \item [3.] Calculer $\begin{pmatrix} 1+\textrm{i} &1\\ 0 &1+\textrm{i}\end{pmatrix}^{100}$. \end{itemize} \OpenGrid{17cm} \FullPageOpenGrid \FullPageOpenGrid \ No newline at end of file