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 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.
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 $y = 6 x - 3 \pi$.
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 $f ( x ) = \frac { 2 } { x }$.
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 $g ( 0 ) = 65$.
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 $\mathbb { R }$.
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 In a class of 24 students, there are 14 girls and 10 boys.
\textbf{Statement 1:}\\
It is possible to form 272 different groups of four students composed of two girls and two boys.
\item 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.
\textbf{Statement 2:}\\
An equation of the tangent line to $C$ at the point with abscissa $\frac { \pi } { 2 }$ is $y = 6 x - 3 \pi$.
\item We consider the function $F$ defined on $] 0$; $+ \infty [$ by $F ( x ) = ( 2 x + 1 ) \ln ( x )$.
\textbf{Statement 3:}\\
The function $F$ is an antiderivative of the function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = \frac { 2 } { x }$.
\item We consider the function $g$ defined on $\mathbb { R }$ by $g ( t ) = 45 \mathrm { e } ^ { 0.06 t } + 20$.
\textbf{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 $g ( 0 ) = 65$.
\item 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 $y$.
\textbf{Statement 5:}\\
The solutions of the equation $\left( E _ { 2 } \right)$ are convex functions on $\mathbb { R }$.
\end{enumerate}