For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. In this exercise, the questions are independent of one another.
We consider the differential equation: $$(E) \quad y' = \frac{1}{2}y + 4.$$ Statement 1: The solutions of $(E)$ are the functions $f$ defined on $\mathbb{R}$ by: $$f(x) = k\mathrm{e}^{\frac{1}{2}x} - 8, \quad \text{with } k \in \mathbb{R}.$$
In a final year class, there are 18 girls and 14 boys. A volleyball team is formed by randomly choosing 3 girls and 3 boys. Statement 2: There are 297024 possibilities for forming such a team.
Let $(v_n)$ be the sequence defined for every natural integer $n$ by: $$v_n = \frac{n}{2 + \cos(n)}.$$ Statement 3: The sequence $(v_n)$ diverges to $+\infty$.
In space with respect to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points $\mathrm{A}(1; 1; 2)$, $\mathrm{B}(5; -1; 8)$ and $\mathrm{C}(2; 1; 3)$. Statement 4: $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}} = 10$ and a measure of the angle $\widehat{\mathrm{BAC}}$ is $30^\circ$.
We consider a function $h$ defined on $]0; +\infty[$ whose second derivative is defined on $]0; +\infty[$ by: $$h''(x) = x\ln x - 3x.$$ Statement 5: The function $h$ is convex on $[\mathrm{e}^3; +\infty[$.
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. In this exercise, the questions are independent of one another.
\begin{enumerate}
\item We consider the differential equation:
$$(E) \quad y' = \frac{1}{2}y + 4.$$
Statement 1: The solutions of $(E)$ are the functions $f$ defined on $\mathbb{R}$ by:
$$f(x) = k\mathrm{e}^{\frac{1}{2}x} - 8, \quad \text{with } k \in \mathbb{R}.$$
\item In a final year class, there are 18 girls and 14 boys. A volleyball team is formed by randomly choosing 3 girls and 3 boys.\\
Statement 2: There are 297024 possibilities for forming such a team.
\item Let $(v_n)$ be the sequence defined for every natural integer $n$ by:
$$v_n = \frac{n}{2 + \cos(n)}.$$
Statement 3: The sequence $(v_n)$ diverges to $+\infty$.
\item In space with respect to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points $\mathrm{A}(1; 1; 2)$, $\mathrm{B}(5; -1; 8)$ and $\mathrm{C}(2; 1; 3)$.\\
Statement 4: $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}} = 10$ and a measure of the angle $\widehat{\mathrm{BAC}}$ is $30^\circ$.
\item We consider a function $h$ defined on $]0; +\infty[$ whose second derivative is defined on $]0; +\infty[$ by:
$$h''(x) = x\ln x - 3x.$$
Statement 5: The function $h$ is convex on $[\mathrm{e}^3; +\infty[$.
\end{enumerate}