For each of the four propositions below, indicate whether it is true or false and justify your chosen answer. One point is awarded for each correct answer with proper justification. An answer without justification is not taken into account. No answer is not penalized.
Proposition 1: In the plane with an orthonormal coordinate system, the set of points $M$ whose affix $z$ satisfies the equality $| z - \mathrm { i } | = | z + 1 |$ is a line.
Proposition 2: The complex number $( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 4 }$ is a real number.
Let ABCDEFGH be a cube. Proposition 3: The lines (EC) and (BG) are orthogonal.
Space is equipped with an orthonormal coordinate system ($\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$). Let the plane $\mathscr { P }$ with Cartesian equation $x + y + 3z + 4 = 0$. We denote S the point with coordinates $( 1 , - 2 , - 2 )$. Proposition 4: The line passing through S and perpendicular to the plane $\mathscr { P }$ has parametric representation $\left\{ \begin{array} { l } x = 2 + t \\ y = - 1 + t \\ z = 1 + 3 t \end{array} , t \in \mathbf { R } \right.$.
For each of the four propositions below, indicate whether it is true or false and justify your chosen answer.\\
One point is awarded for each correct answer with proper justification. An answer without justification is not taken into account. No answer is not penalized.
\begin{enumerate}
\item Proposition 1: In the plane with an orthonormal coordinate system, the set of points $M$ whose affix $z$ satisfies the equality $| z - \mathrm { i } | = | z + 1 |$ is a line.
\item Proposition 2: The complex number $( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 4 }$ is a real number.
\item Let ABCDEFGH be a cube.\\
Proposition 3: The lines (EC) and (BG) are orthogonal.
\item Space is equipped with an orthonormal coordinate system ($\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$). Let the plane $\mathscr { P }$ with Cartesian equation $x + y + 3z + 4 = 0$. We denote S the point with coordinates $( 1 , - 2 , - 2 )$.\\
Proposition 4: The line passing through S and perpendicular to the plane $\mathscr { P }$ has parametric representation $\left\{ \begin{array} { l } x = 2 + t \\ y = - 1 + t \\ z = 1 + 3 t \end{array} , t \in \mathbf { R } \right.$.
\end{enumerate}