Space is referred to an orthonormal coordinate system $( O ; \vec { \imath } , \vec { \jmath } , \vec { k } )$.\nWe consider:

\begin{itemize}
  \item $\alpha$ any real number;
  \item the points $A ( 1 ; 1 ; 0 ) , B ( 2 ; 1 ; 0 )$ and $C ( \alpha ; 3 ; \alpha )$;
  \item (d) the line with parametric representation: $\left\{ \begin{array} { l } x = 1 + t \\ y = 2 t \\ z = - t \end{array} \quad , t \in \mathbf { R } \right.$.
\end{itemize}

For each of the following statements, specify whether it is true or false, then justify the answer given. An answer without justification will not be taken into account.

\section*{Statement 1:}
For all values of $\alpha$, the points $A , B$ and $C$ define a plane and a normal vector to this plane is $\vec { \jmath } \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$.

\section*{Statement 2:}
There exists exactly one value of the real number $\alpha$ such that the lines $( A C )$ and $d$ are parallel.

\section*{Statement 3 :}
A measure of the angle $\widehat { O A B }$ is $135 ^ { \circ }$.

\section*{Statement 4 :}
The orthogonal projection of point $A$ onto the line $( d )$ is the point $H$ with coordinates: $H ( 1 ; 2 ; 2 )$.

\section*{Statement 5 :}
The sphere with center $O$ and radius 1 meets the line ( $d$ ) at two distinct points.\nWe recall that the sphere with center $\Omega$ and radius $r$ is the set of points in space at distance $r$ from $\Omega$.