Consider a cube ABCDEFGH whose graphical representation in cavalier perspective is given below.
The edges have length 1.
Space is referred to the orthonormal coordinate system $( \mathrm { D } ; \overrightarrow { \mathrm { DA } } , \overrightarrow { \mathrm { DC } } , \overrightarrow { \mathrm { DH } } )$.

\section*{Part A}
\begin{enumerate}
  \item Show that the vector $\overrightarrow { \mathrm { DF } }$ is normal to the plane (EBG).
  \item Determine a Cartesian equation of the plane (EBG).
  \item Deduce the coordinates of point I, the intersection of line (DF) and plane (EBG).
\end{enumerate}

One would show in the same way that point J, the intersection of line (DF) and plane (AHC), has coordinates $\left( \frac { 1 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.

\section*{Part B}
For any real number $x$ in the interval $[ 0 ; 1 ]$, we associate the point $M$ of segment $[ \mathrm{DF} ]$ such that $\overrightarrow { \mathrm { DM } } = x \overrightarrow { \mathrm { DF } }$.
We are interested in the evolution of the measure $\theta$ in radians of angle $\widehat { \mathrm { EMB } }$ as point $M$ moves along segment [DF]. We have $0 \leqslant \theta \leqslant \pi$.

\begin{enumerate}
  \item What is the value of $\theta$ if point $M$ coincides with point D? with point F?
  \item a. Justify that the coordinates of point $M$ are $( x ; x ; x )$.\\
b. Show that $\cos ( \theta ) = \frac { 3 x ^ { 2 } - 4 x + 1 } { 3 x ^ { 2 } - 4 x + 2 }$. For this, one may consider the dot product of vectors $\overrightarrow { M \mathrm { E } }$ and $\overrightarrow { M \mathrm {~B} }$.
  \item The table of variations of the function below has been constructed
$$f : x \longmapsto \frac { 3 x ^ { 2 } - 4 x + 1 } { 3 x ^ { 2 } - 4 x + 2 }$$
\begin{center}
\begin{tabular}{ | c | l c c c | }
\hline
$x$ & 0 & $\frac { 1 } { 3 }$ & $\frac { 2 } { 3 }$ & 1 \\
\hline
\begin{tabular}{ c }
Variations \\
of $f$ \\
\end{tabular} & $\frac { 1 } { 2 }$ &  &  &  \\
 &  &  & 0 &  \\
\hline
\end{tabular}
\end{center}
For which positions of point $M$ on segment [DF]:
a. is triangle $MEB$ right-angled at $M$?\\
b. is angle $\theta$ maximal?
\end{enumerate}