For each of the five following statements, indicate whether it is true or false, by justifying the answer.\\
An unjustified answer is not taken into account.\\
An absence of answer is not penalized.

\begin{enumerate}
  \item We consider the function $f$ defined on the interval $]0; +\infty[$ by:
$$f(x) = \ln(x) - x^2$$
Statement 1: $\lim_{x \to +\infty} f(x) = -\infty$.

  \item We consider the differential equation
$$(E): \quad -2y' + 3y = \sin x + 8\cos x$$
We consider the function $f$ defined on $\mathbb{R}$ by:
$$f(x) = 2\cos x - \sin x$$
Statement 2: The function $f$ is a solution of the differential equation $(E)$.

  \item We consider the function $g$ defined on the interval $]0; +\infty[$ by:
$$g(x) = \ln(3x + 1) + 8$$
We consider the sequence $(u_n)$ defined by $u_0 = 25$ and for all natural integers $n$:
$$u_{n+1} = g(u_n).$$
We admit that the sequence $(u_n)$ is strictly positive.\\
Statement 3: The sequence $(u_n)$ is decreasing.

  \item We consider an affine function $h$ defined on $\mathbb{R}$. We denote $k$ the function defined on $\mathbb{R}$ by $k(x) = x^4 + x^2 + h(x)$.\\
Statement 4: The function $k$ is convex on $\mathbb{R}$.

  \item An anagram of a word is the result of a permutation of the letters of that word. Example: the word BAC has 6 anagrams: $BAC, BCA, ABC, ACB, CAB, CBA$.\\
Statement 5: The word EULER has 120 anagrams.
\end{enumerate}