For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
Let $( u _ { n } )$ be a sequence defined for all natural integer $n$ and satisfying the following relation: $$\text{for all natural integer } n , \frac { 1 } { 2 } < u _ { n } \leqslant \frac { 3 n ^ { 2 } + 4 n + 7 } { 6 n ^ { 2 } + 1 } .$$ Statement 1: $\lim _ { n \rightarrow + \infty } u _ { n } = \frac { 1 } { 2 }$.
Let $h$ be a function defined and differentiable on the interval $[-4;4]$. The graphical representation $\mathscr { C } _ { h ^ { \prime } }$ of its derivative function $h ^ { \prime }$ is given below. Statement 2: The function $h$ is convex on $[ - 1 ; 3]$.
The code of a building is composed of 4 digits (which may be identical) followed by two distinct letters among A, B and C (example: 1232BA). Statement 3: There exist 20634 codes that contain at least one 0.
We consider the function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = x \ln x$. Statement 4: The function $f$ is a solution on $] 0 ; + \infty [$ of the differential equation $$x y ^ { \prime } - y = x .$$
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
\begin{enumerate}
\item Let $( u _ { n } )$ be a sequence defined for all natural integer $n$ and satisfying the following relation:
$$\text{for all natural integer } n , \frac { 1 } { 2 } < u _ { n } \leqslant \frac { 3 n ^ { 2 } + 4 n + 7 } { 6 n ^ { 2 } + 1 } .$$
Statement 1: $\lim _ { n \rightarrow + \infty } u _ { n } = \frac { 1 } { 2 }$.
\item Let $h$ be a function defined and differentiable on the interval $[-4;4]$. The graphical representation $\mathscr { C } _ { h ^ { \prime } }$ of its derivative function $h ^ { \prime }$ is given below.\\
Statement 2: The function $h$ is convex on $[ - 1 ; 3]$.
\item The code of a building is composed of 4 digits (which may be identical) followed by two distinct letters among A, B and C (example: 1232BA).\\
Statement 3: There exist 20634 codes that contain at least one 0.
\item We consider the function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = x \ln x$.\\
Statement 4: The function $f$ is a solution on $] 0 ; + \infty [$ of the differential equation
$$x y ^ { \prime } - y = x .$$
\end{enumerate}