\textbf{Exercise 3 -- Common to all candidates}

The four questions in this exercise are independent.\\
For each question, a statement is proposed. Indicate whether each of them is true or false, by justifying your answer. An unjustified answer earns no points.\\
In questions 1. and 2., the plane is referred to the direct orthonormal coordinate system $(O, \vec{u}, \vec{v})$. Consider the points A, B, C, D and E with complex numbers respectively:
$$a = 2 + 2\mathrm{i}, \quad b = -\sqrt{3} + \mathrm{i}, \quad c = 1 + \mathrm{i}\sqrt{3}, \quad d = -1 + \frac{\sqrt{3}}{2}\mathrm{i} \quad \text{and} \quad e = -1 + (2 + \sqrt{3})\mathrm{i}.$$

\begin{enumerate}
  \item Statement 1: the points A, B and C are collinear.
  \item Statement 2: the points B, C and D belong to the same circle with center E.
  \item In this question, space is equipped with a coordinate system $(O, \vec{\imath}, \vec{\jmath}, \vec{k})$.\\
Consider the points $I(1; 0; 0)$, $J(0; 1; 0)$ and $K(0; 0; 1)$.\\
Statement 3: the line $\mathscr{D}$ with parametric representation $\left\{\begin{aligned} x &= 2 - t \\ y &= 6 - 2t \\ z &= -2 + t \end{aligned}\right.$ where $t \in \mathbb{R}$, intersects the plane (IJK) at point $E\left(-\frac{1}{2}; 1; \frac{1}{2}\right)$.
  \item In the cube ABCDEFGH, the point T is the midpoint of segment $[HF]$.\\
Statement 4: the lines (AT) and (EC) are orthogonal.
\end{enumerate}