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 $E$ and $F$ be the sets $E = \{1; 2; 3; 4; 5; 6; 7\}$ and $F = \{0; 1; 2; 3; 4; 5; 6; 7; 8; 9\}$. Statement $\mathbf{n^\circ 1}$: There are more 3-tuples of distinct elements of $E$ than 4-element combinations of $F$.
In the orthonormal coordinate system, we have represented the square function, denoted $f$, as well as the square ABCD with side 3. Statement $\mathbf{n^\circ 2}$: The shaded region and the square ABCD have the same area.
We consider the integral $J$ below: $$J = \int_1^2 x\ln(x)\,\mathrm{d}x$$ Statement $\mathbf{n^\circ 3}$: Integration by parts makes it possible to obtain: $J = \dfrac{7}{11}$.
On $\mathbb{R}$, we consider the differential equation $$(E): \quad y' = 2y - \mathrm{e}^x.$$ Statement $\mathbf{n^\circ 4}$: The function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^x + \mathrm{e}^{2x}$ is a solution of the differential equation $(E)$.
Let $x$ be given in $[0; 1[$. We consider the sequence $(u_n)$ defined for any natural integer $n$ by: $$u_n = (x-1)\mathrm{e}^n + \cos(n).$$ Statement $\mathbf{n^\circ 5}$: The sequence $(u_n)$ diverges to $-\infty$.
For each of the following statements, indicate whether it is true or false.\\
Each answer must be justified. An unjustified answer earns no points.
\begin{enumerate}
\item Let $E$ and $F$ be the sets $E = \{1; 2; 3; 4; 5; 6; 7\}$ and $F = \{0; 1; 2; 3; 4; 5; 6; 7; 8; 9\}$.\\
\textbf{Statement $\mathbf{n^\circ 1}$}: There are more 3-tuples of distinct elements of $E$ than 4-element combinations of $F$.
\item In the orthonormal coordinate system, we have represented the square function, denoted $f$, as well as the square ABCD with side 3.\\
\textbf{Statement $\mathbf{n^\circ 2}$}: The shaded region and the square ABCD have the same area.
\item We consider the integral $J$ below:
$$J = \int_1^2 x\ln(x)\,\mathrm{d}x$$
\textbf{Statement $\mathbf{n^\circ 3}$}: Integration by parts makes it possible to obtain: $J = \dfrac{7}{11}$.
\item On $\mathbb{R}$, we consider the differential equation
$$(E): \quad y' = 2y - \mathrm{e}^x.$$
\textbf{Statement $\mathbf{n^\circ 4}$}: The function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^x + \mathrm{e}^{2x}$ is a solution of the differential equation $(E)$.
\item Let $x$ be given in $[0; 1[$. We consider the sequence $(u_n)$ defined for any natural integer $n$ by:
$$u_n = (x-1)\mathrm{e}^n + \cos(n).$$
\textbf{Statement $\mathbf{n^\circ 5}$}: The sequence $(u_n)$ diverges to $-\infty$.
\end{enumerate}