This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. The candidate will indicate on their answer sheet the number of the question and the chosen answer. No justification is required. A wrong answer, multiple answers, or the absence of an answer to a question neither awards nor deducts points. The five questions are independent. Throughout the exercise, $\mathbb { R }$ denotes the set of real numbers.
A primitive of the function $f$, defined on $\mathbb { R }$ by $f ( x ) = x \mathrm { e } ^ { x }$, is the function $F$, defined on $\mathbb { R }$, by: a. $F ( x ) = \frac { x ^ { 2 } } { 2 } \mathrm { e } ^ { x }$ b. $F ( x ) = ( x - 1 ) \mathrm { e } ^ { x }$ c. $F ( x ) = ( x + 1 ) \mathrm { e } ^ { x }$ d. $F ( x ) = x ^ { 2 } \mathrm { e } ^ { x ^ { 2 } }$
We consider the function $g$ defined by $g ( x ) = \ln \left( \frac { x - 1 } { 2 x + 4 } \right)$. The function $g$ is defined on: a. $\mathbb { R }$ b. $] - \infty ; - 2 [ \cup ] 1 ; + \infty [$ c. $] - 2 ; + \infty [$ d. $] - 2 ; 1 [$
The function $h$ defined on $\mathbb { R }$ by $h ( x ) = ( x + 1 ) \mathrm { e } ^ { x }$ is: a. concave on $\mathbb { R }$ b. convex on $\mathbb { R }$ c. convex on $] - \infty ; - 3 ]$ and concave on $[-3; + \infty [$ d. concave on $] - \infty ; - 3 ]$ and convex on $[-3; + \infty [$
A sequence ( $u _ { n }$ ) is bounded below by 3 and converges to a real number $\ell$. We can affirm that: a. $\ell = 3$ b. $\ell \geqslant 3$ c. The sequence ( $u _ { n }$ ) is decreasing. d. The sequence ( $u _ { n }$ ) is constant from a certain rank onwards.
The sequence ( $w _ { n }$ ) is defined by $w _ { 1 } = 2$ and for every strictly positive natural number $n$, $w _ { n + 1 } = \frac { 1 } { n } w _ { n }$. a. The sequence ( $w _ { n }$ ) is geometric b. The sequence ( $w _ { n }$ ) does not have a limit c. $w _ { 5 } = \frac { 1 } { 15 }$ d. The sequence ( $w _ { n }$ ) converges to 0.
\section*{Exercise 3}
This exercise is a multiple choice questionnaire.\\
For each question, only one of the four proposed answers is correct.\\
The candidate will indicate on their answer sheet the number of the question and the chosen answer.\\
No justification is required.\\
A wrong answer, multiple answers, or the absence of an answer to a question neither awards nor deducts points.\\
The five questions are independent.\\
Throughout the exercise, $\mathbb { R }$ denotes the set of real numbers.
\begin{enumerate}
\item A primitive of the function $f$, defined on $\mathbb { R }$ by $f ( x ) = x \mathrm { e } ^ { x }$, is the function $F$, defined on $\mathbb { R }$, by:\\
a. $F ( x ) = \frac { x ^ { 2 } } { 2 } \mathrm { e } ^ { x }$\\
b. $F ( x ) = ( x - 1 ) \mathrm { e } ^ { x }$\\
c. $F ( x ) = ( x + 1 ) \mathrm { e } ^ { x }$\\
d. $F ( x ) = x ^ { 2 } \mathrm { e } ^ { x ^ { 2 } }$
\item We consider the function $g$ defined by $g ( x ) = \ln \left( \frac { x - 1 } { 2 x + 4 } \right)$.\\
The function $g$ is defined on:\\
a. $\mathbb { R }$\\
b. $] - \infty ; - 2 [ \cup ] 1 ; + \infty [$\\
c. $] - 2 ; + \infty [$\\
d. $] - 2 ; 1 [$
\item The function $h$ defined on $\mathbb { R }$ by $h ( x ) = ( x + 1 ) \mathrm { e } ^ { x }$ is:\\
a. concave on $\mathbb { R }$\\
b. convex on $\mathbb { R }$\\
c. convex on $] - \infty ; - 3 ]$ and concave on $[-3; + \infty [$\\
d. concave on $] - \infty ; - 3 ]$ and convex on $[-3; + \infty [$
\item A sequence ( $u _ { n }$ ) is bounded below by 3 and converges to a real number $\ell$.\\
We can affirm that:\\
a. $\ell = 3$\\
b. $\ell \geqslant 3$\\
c. The sequence ( $u _ { n }$ ) is decreasing.\\
d. The sequence ( $u _ { n }$ ) is constant from a certain rank onwards.
\item The sequence ( $w _ { n }$ ) is defined by $w _ { 1 } = 2$ and for every strictly positive natural number $n$, $w _ { n + 1 } = \frac { 1 } { n } w _ { n }$.\\
a. The sequence ( $w _ { n }$ ) is geometric\\
b. The sequence ( $w _ { n }$ ) does not have a limit\\
c. $w _ { 5 } = \frac { 1 } { 15 }$\\
d. The sequence ( $w _ { n }$ ) converges to 0.
\end{enumerate}