For each of the following propositions, indicate whether it is true or false and justify your chosen answer. One point is awarded for each correct answer that is properly justified. An answer without justification is not taken into account. An absence of an answer is not penalized. Proposition 1 Every positive increasing sequence tends to $+ \infty$. Proposition 2 $g$ is the function defined on $] - \frac { 1 } { 2 } ; + \infty [$ by $$g ( x ) = 2 x \ln ( 2 x + 1 ) .$$ On $] - \frac { 1 } { 2 } ; + \infty$ [, the equation $g ( x ) = 2 x$ has a unique solution: $\frac { \mathrm { e } - 1 } { 2 }$. Proposition 3 The slope of the tangent line to the curve representing the function $g$ at the point with abscissa $\frac { 1 } { 2 }$ is: $1 + \ln 4$. Proposition 4 Space is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k }$ ). $\mathscr { P }$ and $\mathscr { R }$ are the planes with equations respectively: $2 x + 3 y - z - 11 = 0$ and $x + y + 5 z - 11 = 0$. The planes $\mathscr { P }$ and $\mathscr { R }$ intersect perpendicularly.
For each of the following propositions, indicate whether it is true or false and justify your chosen answer.\\
One point is awarded for each correct answer that is properly justified.\\
An answer without justification is not taken into account.\\
An absence of an answer is not penalized.
\textbf{Proposition 1}\\
Every positive increasing sequence tends to $+ \infty$.
\textbf{Proposition 2}\\
$g$ is the function defined on $] - \frac { 1 } { 2 } ; + \infty [$ by
$$g ( x ) = 2 x \ln ( 2 x + 1 ) .$$
On $] - \frac { 1 } { 2 } ; + \infty$ [, the equation $g ( x ) = 2 x$ has a unique solution: $\frac { \mathrm { e } - 1 } { 2 }$.
\textbf{Proposition 3}\\
The slope of the tangent line to the curve representing the function $g$ at the point with abscissa $\frac { 1 } { 2 }$ is: $1 + \ln 4$.
\textbf{Proposition 4}\\
Space is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k }$ ).\\
$\mathscr { P }$ and $\mathscr { R }$ are the planes with equations respectively: $2 x + 3 y - z - 11 = 0$ and $x + y + 5 z - 11 = 0$.\\
The planes $\mathscr { P }$ and $\mathscr { R }$ intersect perpendicularly.