Specify whether each of the following statements is true or false by justifying your answer.

\begin{enumerate}
  \item Let $m$ be a real number and let the equation $( E )$ : $2 z ^ { 2 } + ( m - 5 ) z + m = 0$.\\
a. Statement 1 :\\
``For $m = 4$, the equation ( $E$ ) admits two real solutions.''\\
b. Statement 2 :\\
``There exists only one value of $m$ such that ( $E$ ) admits two complex solutions that are pure imaginary numbers.''
  \item In the complex plane, we consider the set $S$ of points $M$ with affixe $z$ satisfying:
$$| z - 6 | = | z + 5 i |$$
Statement 3 :\\
``The set $S$ is a circle.''
  \item We equip space with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We denote $d$ the line with parametric representation:
$$d : \left\{ \begin{aligned}
x & = - 1 + t \\
y & = 2 - t \quad t \in \mathbb { R } . \\
z & = 3 + t
\end{aligned} \right.$$
We denote $d ^ { \prime }$ the line passing through the point $\mathrm { B } ( 4 ; 4 ; - 6 )$ and with direction vector $\vec { v } ( 5 ; 2 ; - 9 )$.\\
Statement 4 :\\
``The lines $d$ and $d ^ { \prime }$ are coplanar.''
  \item We consider the cube ABCDEFGH.\\
Statement 5 :\\
``The vector $\overrightarrow { \mathrm { DE } }$ is a normal vector to the plane (ABG).''
\end{enumerate}