\section*{Exercise 4 (5 points) -- Candidate who has NOT followed the specialization course}
Let a cube ABCDEFGH with edge length 1. In the coordinate system $(A;\,\overrightarrow{AB},\,\overrightarrow{AD},\,\overrightarrow{AE})$, we consider the points $M$, $N$ and $P$ with respective coordinates $\mathrm{M}\!\left(1\,;\,1\,;\,\tfrac{3}{4}\right)$, $\mathrm{N}\!\left(0\,;\,\tfrac{1}{2}\,;\,1\right)$, $\mathrm{P}\!\left(1\,;\,0\,;\,-\tfrac{5}{4}\right)$.
\begin{enumerate}
  \item Plot $\mathrm{M}$, $\mathrm{N}$ and $\mathrm{P}$ on the figure provided in the appendix.
  \item Determine the coordinates of the vectors $\overrightarrow{\mathrm{MN}}$ and $\overrightarrow{\mathrm{MP}}$.

Deduce that the points $\mathrm{M}$, $\mathrm{N}$ and $\mathrm{P}$ are not collinear.
  \item We consider algorithm 1 given in the appendix.\\
  a. Execute this algorithm by hand with the coordinates of the points $\mathrm{M}$, $\mathrm{N}$ and $\mathrm{P}$ given above.\\
  b. What does the result displayed by the algorithm correspond to? What can we deduce about triangle MNP?
  \item We consider algorithm 2 given in the appendix. Complete it so that it tests and displays whether a triangle MNP is right-angled and isosceles at M.
  \item We consider the vector $\vec{n}(5\,;\,-8\,;\,4)$ normal to the plane (MNP).\\
  a. Determine a Cartesian equation of the plane (MNP).\\
  b. We consider the line $\Delta$ passing through F and with direction vector $\vec{n}$.

Determine a parametric representation of the line $\Delta$.
  \item Let K be the point of intersection of the plane (MNP) and the line $\Delta$.\\
  a. Prove that the coordinates of point K are $\left(\dfrac{4}{7}\,;\,\dfrac{24}{35}\,;\,\dfrac{23}{35}\right)$.\\
  b. We are given $FK = \sqrt{\dfrac{27}{35}}$.

Calculate the volume of the tetrahedron MNPF.
\end{enumerate}