ABCDEFGH is a cube.

I is the midpoint of segment $[\mathrm{AB}]$, J is the midpoint of segment $[\mathrm{EH}]$, K is the midpoint of segment [BC] and L is the midpoint of segment [CG].\\
We equip space with the orthonormal coordinate system (A ; $\overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}}$).

\begin{enumerate}
  \item a) Prove that the line (FD) is orthogonal to the plane (IJK).\\
b) Deduce a Cartesian equation of the plane (IJK).
  \item Determine a parametric representation of the line (FD).
  \item Let $M$ be the point of intersection of the line (FD) and the plane (IJK). Determine the coordinates of point $M$.
  \item Determine the nature of triangle IJK and calculate its area.
  \item Calculate the volume of the tetrahedron FIJK.
  \item Are the lines (IJ) and (KL) intersecting?
\end{enumerate}