\section*{Exercise 2 -- 3 points -- Common to all candidates}
Space is equipped with a coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$.\\
Let $\mathcal{P}$ be the plane with Cartesian equation: $2x - z - 3 = 0$.\\
We denote $A$ the point with coordinates $(1 ; a ; a^{2})$ where $a$ is a real number.

\begin{enumerate}
  \item Justify that, regardless of the value of $a$, the point $A$ does not belong to the plane $\mathcal{P}$.
  \item a. Determine a parametric representation of the line $\mathcal{D}$ (with parameter $t$) passing through the point $A$ and orthogonal to the plane $\mathcal{P}$.\\
b. Let $M$ be a point belonging to the line $\mathcal{D}$, associated with the value $t$ of the parameter in the previous parametric representation.\\
Express the distance $AM$ as a function of the real number $t$.
  \item We denote $H$ the point of intersection of the plane $\mathcal{P}$ and the line $\mathcal{D}$ orthogonal to $\mathcal{P}$ and passing through the point $A$. The point $H$ is called the orthogonal projection of the point $A$ onto the plane $\mathcal{P}$ and the distance $AH$ is called the distance from the point $A$ to the plane $\mathcal{P}$.\\
Is there a value of $a$ for which the distance $AH$ from the point $A$ with coordinates $(1 ; a ; a^{2})$ to the plane $\mathcal{P}$ is minimal? Justify the answer.
\end{enumerate}