This exercise contains 5 statements. For each statement, answer TRUE or FALSE by justifying the answer. Any lack of justification or incorrect justification will not be taken into account in the grading.
Part 1
We consider the sequence $(u_n)$ defined by: $$u_0 = 10 \text{ and for all natural integer } n,\ u_{n+1} = \frac{1}{3}u_n + 2.$$

1. Statement 1: The sequence $(u_n)$ is decreasing and bounded below by 0.
2. Statement 2: $\lim_{n \rightarrow +\infty} u_n = 0$.
3. Statement 3: The sequence $(v_n)$ defined for all natural integer $n$ by $v_n = u_n - 3$ is geometric.

Part 2
We consider the differential equation $(E): y' = \frac{3}{2}y + 2$ with unknown $y$, a function defined and differentiable on $\mathbb{R}$.

1. Statement 4: There exists a constant function that is a solution to the differential equation $(E)$.
2. In an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$ we denote $\mathscr{C}_f$ the representative curve of the function $f$ solution of $(E)$ such that $f(0) = 0$. Statement 5: The tangent line at the point with abscissa 1 of $\mathscr{C}_f$ has slope $2\mathrm{e}^{\frac{3}{2}}$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

1. Consider the function $f$ defined on $\mathbb { R }$ by: $f ( x ) = 5 x \mathrm { e } ^ { - x }$.

We denote by $C _ { f }$ the representative curve of $f$ in an orthonormal coordinate system.
Statement 1: The $x$-axis is a horizontal asymptote to the curve $C _ { f }$.
Statement 2: The function $f$ is a solution on $\mathbb { R }$ of the differential equation $( E ) : y ^ { \prime } + y = 5 \mathrm { e } ^ { - x }$.
2. Consider the sequences $\left( u _ { n } \right) , \left( v _ { n } \right)$ and $\left( w _ { n } \right)$, such that, for every natural integer $n$ :
$$u _ { n } \leqslant v _ { n } \leqslant w _ { n } .$$
Moreover, the sequence $( u _ { n } )$ converges to $- 1$ and the sequence $( w _ { n } )$ converges to $1$.
Statement 3: The sequence $\left( \nu _ { n } \right)$ converges to a real number $\ell$ belonging to the interval $[ - 1 ; 1 ]$.
We further assume that the sequence $( u _ { n } )$ is increasing and that the sequence $( w _ { n } )$ is decreasing.
Statement 4: For every natural integer $n$, we then have: $\quad u _ { 0 } \leqslant v _ { n } \leqslant w _ { 0 }$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer receives no points.

1. Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \mathrm{e}^{x} + x.$$ Statement A: The function $f$ has the following variation table:
   

   $x$	$-\infty$
   variations of $f$	$+\infty$

   
   Statement B: The equation $f(x) = -2$ has two solutions in $\mathbb{R}$.
2. Statement C: $$\lim_{x \rightarrow +\infty} \frac{\ln(x) - x^{2} + 2}{3x^{2}} = -\frac{1}{3}.$$
3. Consider the function $k$ defined and continuous on $\mathbb{R}$ by $$k(x) = 1 + 2\mathrm{e}^{-x^{2}+1}$$ Statement D: There exists a primitive of the function $k$ that is decreasing on $\mathbb{R}$.
4. Consider the differential equation $$(E): \quad 3y' + y = 1.$$ Statement E: The function $g$ defined on $\mathbb{R}$ by $$g(x) = 4\mathrm{e}^{-\frac{1}{3}x} + 1$$ is a solution of the differential equation $(E)$ with $g(0) = 5$.
5. Statement F: Integration by parts allows us to obtain: $$\int_{0}^{1} x\mathrm{e}^{-x}\,\mathrm{d}x = 1 - 2\mathrm{e}^{-1}$$

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
Statement 1: Let (E) be the differential equation: $y ^ { \prime } - 2 y = - 6 x + 1$. The function $f$ defined on $\mathbb { R }$ by: $f ( x ) = \mathrm { e } ^ { 2 x } - 6 x + 1$ is a solution of the differential equation (E).
Statement 2: Consider the sequence $\left( u _ { n } \right)$ defined on $\mathbb { N }$ by $$u _ { n } = 1 + \frac { 3 } { 4 } + \left( \frac { 3 } { 4 } \right) ^ { 2 } + \cdots + \left( \frac { 3 } { 4 } \right) ^ { n }$$ The sequence $(u _ { n })$ has limit $+ \infty$.
Statement 3: Consider the sequence $(u _ { n })$ defined in Statement 2. The instruction suite(50) below, written in Python language, returns $u _ { 50 }$. \begin{verbatim} def suite(k): S=0 for i in range(k): S=S+(3/4)**k return S \end{verbatim}
Statement 4: Let $a$ be a real number and $f$ the function defined on $] 0 ; + \infty [$ by: $$f ( x ) = a \ln ( x ) - 2 x$$ Let $C$ be the representative curve of the function $f$ in a coordinate system $(O ; \vec { \imath } , \vec { \jmath })$. There exists a value of $a$ for which the tangent to $C$ at the point with abscissa 1 is parallel to the horizontal axis.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

1. In a class of 24 students, there are 14 girls and 10 boys.
   Statement 1: It is possible to form 272 different groups of four students composed of two girls and two boys.
   
2. Let $f$ be the function defined on $\mathbb { R }$ by $f ( x ) = 3 \sin ( 2 x + \pi )$ and $C$ its representative curve in a given coordinate system.
   Statement 2: An equation of the tangent line to $C$ at the point with abscissa $\frac { \pi } { 2 }$ is $y = 6 x - 3 \pi$.
   
3. We consider the function $F$ defined on $] 0$; $+ \infty [$ by $F ( x ) = ( 2 x + 1 ) \ln ( x )$.
   Statement 3: The function $F$ is an antiderivative of the function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = \frac { 2 } { x }$.
   
4. We consider the function $g$ defined on $\mathbb { R }$ by $g ( t ) = 45 \mathrm { e } ^ { 0.06 t } + 20$.
   Statement 4: The function $g$ is the unique solution of the differential equation $\left( E _ { 1 } \right) y ^ { \prime } + 0.06 y = 1.2$ satisfying $g ( 0 ) = 65$.
   
5. We consider the differential equation: $$\left( E _ { 2 } \right) : \quad y ^ { \prime } - y = 3 \mathrm { e } ^ { 0.4 x }$$ where $y$ is a positive function of the real variable $x$, defined and differentiable on $\mathbb { R }$ and $y ^ { \prime }$ the derivative function of the function $y$.
   Statement 5: The solutions of the equation $\left( E _ { 2 } \right)$ are convex functions on $\mathbb { R }$.

Show that the function $\left\lvert\, \begin{aligned} & \mathbb{R}_{+}^{*} \times \mathbb{R} \rightarrow \mathbb{R} \\ & (t, x) \mapsto g_{\sqrt{\sigma^{2}+2t}}(x) \end{aligned}\right.$ satisfies conditions i and iii, where:

• [i.] the diffusion equation: $\forall(t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R},\ \frac{\partial f}{\partial t}(t, x) = \frac{\partial^{2} f}{\partial x^{2}}(t, x)$;
• [iii.] the boundary condition: $\forall x \in \mathbb{R},\ \lim_{t \rightarrow 0^{+}} f(t, x) = g_{\sigma}(x)$.

We assume (in this question only) that $c ( x ) = 0$ and $f ( x ) = 1$ for all $x \in [ 0,1 ]$. Let $n \in \mathbb{N}^*$, $h = \frac{1}{n+1}$, and $x_i = ih$. We denote by $u$ the exact solution of problem (1): $$\left\{ \begin{array} { l } - u ^ { \prime \prime } ( x ) + c ( x ) u ( x ) = f ( x ) , x \in [ 0,1 ] \\ u ( 0 ) = u ( 1 ) = 0 \end{array} \right.$$ and $(u_i)_{0 \leq i \leq n+1}$ the unique family satisfying $$\left\{ \begin{array} { l } - \frac { 1 } { h ^ { 2 } } \left( u _ { i + 1 } + u _ { i - 1 } - 2 u _ { i } \right) + c \left( x _ { i } \right) u _ { i } = f \left( x _ { i } \right) , \text { for } 1 \leq i \leq n \\ u _ { 0 } = u _ { n + 1 } = 0 \end{array} \right.$$ Show that for all $i \in \{ 0 , \ldots , n + 1 \}$, we have
$$u _ { i } = u \left( x _ { i } \right) = \frac { 1 } { 2 } x _ { i } \left( 1 - x _ { i } \right)$$

Let $g$ be the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$ with radius of convergence $R \geqslant \pi/2$. Show $$\forall x \in I, \quad 2g^{\prime}(x) = g(x)^2 + 1.$$

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. We consider the function $$h : \begin{array}{ccc} ]-R,R[ & \rightarrow & \mathbb{R} \\ x & \mapsto & S(x)\mathrm{e}^{S(x)} \end{array}$$ Prove that $h$ is a solution on $]-R, R[$ of the differential equation $xy' - y = 0$.

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ Deduce that $L_n$ is a solution of the differential equation $$x L_n''(x) + (1-x) L_n'(x) + n L_n(x) = 0.$$

The function $y = e ^ { k x }$ satisfies
$$\left( \frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } \right) \left( \frac { d y } { d x } - y \right) = y \frac { d y } { d x }$$
for
(A) exactly one value of $k$.
(B) two distinct values of $k$.
(C) three distinct values of $k$.
(D) infinitely many values of $k$.