For each of the following statements, indicate whether it is true or false. Each answer must be justified.
An unjustified answer earns no points.

\section*{PART A}
ABCDEFGH is a cube with edge length 1. The points I, J, K, L and M are the midpoints respectively of the edges [AB], [BF], [AE], [CD] and [DH].

Statement 1: $\ll \overrightarrow { \mathrm { JH } } = 2 \overrightarrow { \mathrm { BI } } + \overrightarrow { \mathrm { DM } } - \overrightarrow { \mathrm { CB } } \gg$\\
Statement 2: ``The triplet of vectors ( $\overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AH } } , \overrightarrow { \mathrm { AG } }$ ) is a basis of space.''\\
Statement 3: ``$\overrightarrow { \mathrm { IB } } \cdot \overrightarrow { \mathrm { LM } } = - \frac { 1 } { 4 }$.''

\section*{PART B}
In space equipped with an orthonormal coordinate system, we consider:
\begin{itemize}
  \item the plane $\mathscr { P }$ with Cartesian equation $2 x - y + 3 z + 6 = 0$
  \item the points $\mathrm { A } ( 2 ; 0 ; - 1 )$ and $\mathrm { B } ( 5 ; - 3 ; 7 )$
\end{itemize}

Statement 4: ``The plane $\mathscr { P }$ and the line ( AB ) are parallel.''\\
Statement 5: ``The plane $\mathscr { P } ^ { \prime }$ parallel to $\mathscr { P }$ passing through B has Cartesian equation $- 2 x + y - 3 z + 34 = 0$''\\
Statement 6: ``The distance from point A to plane $\mathscr { P }$ is equal to $\frac { \sqrt { 14 } } { 2 }$.''\\
We denote by (d) the line with parametric representation

$$\left\{ \begin{array} { r l } 
x & = - 12 + 2 k \\
y & = 6 \\
z & = 3 - 5 k
\end{array} , \text { where } k \in \mathbb { R } \right.$$

Statement 7: ``The lines (AB) and (d) are not coplanar.''