\textbf{(For candidates who have not followed the specialization course)}

We connect the centres of each face of a cube ABCDEFGH to form a solid IJKLMN. More precisely, the points I, J, K, L, M and N are the centres respectively of the square faces ABCD, BCGF, CDHG, ADHE, ABFE and EFGH (thus the midpoints of the diagonals of these squares).

\begin{enumerate}
  \item Without using a coordinate system (and thus coordinates) in the reasoning, justify that the lines (IN) and (ML) are orthogonal.
\end{enumerate}

In what follows, we consider the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}; \overrightarrow{\mathrm{AD}}; \overrightarrow{\mathrm{AE}})$ in which, for example, the point N has coordinates $\left(\frac{1}{2}; \frac{1}{2}; 1\right)$.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item a. Give the coordinates of the vectors $\overrightarrow{\mathrm{NC}}$ and $\overrightarrow{\mathrm{ML}}$.\\
b. Deduce that the lines (NC) and (ML) are orthogonal.\\
c. From the previous questions, deduce a Cartesian equation of the plane (NCI).
  \item a. Show that a Cartesian equation of the plane (NJM) is: $x - y + z = 1$.\\
b. Is the line (DF) perpendicular to the plane (NJM)? Justify.\\
c. Show that the intersection of the planes (NJM) and (NCI) is a line for which you will give a point and a direction vector. Name the line thus obtained using two points from the figure.
\end{enumerate}