\section*{Exercise 4 — Candidates who have not chosen the specialty course}
The plane is equipped with the direct orthonormal coordinate system $(\mathrm{O}; \vec{u}, \vec{v})$.\\
We are given the complex number $\mathrm{j} = -\frac{1}{2} + \mathrm{i}\frac{\sqrt{3}}{2}$.\\
The purpose of this exercise is to study some properties of the number j and to highlight a connection between this number and equilateral triangles.

\section*{Part A: properties of the number j}
\begin{enumerate}
  \item a. Solve in the set $\mathbb{C}$ of complex numbers the equation
$$z^{2} + z + 1 = 0$$
b. Verify that the complex number j is a solution of this equation.\\
  \item Determine the modulus and an argument of the complex number j, then give its exponential form.\\
  \item Prove the following equalities:\\
a. $j^{3} = 1$;\\
b. $\mathrm{j}^{2} = -1 - \mathrm{j}$.\\
  \item Let $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ be the images respectively of the complex numbers $1, \mathrm{j}$ and $\mathrm{j}^{2}$ in the plane.

What is the nature of triangle PQR? Justify the answer.
\end{enumerate}

\section*{Part B}
Let $a, b, c$ be three complex numbers satisfying the equality $a + \mathrm{j}b + \mathrm{j}^{2}c = 0$.\\
Let $\mathrm{A}, \mathrm{B}, \mathrm{C}$ denote the images respectively of the numbers $a, b, c$ in the plane.

\begin{enumerate}
  \item Using question A-3.b., prove the equality: $a - c = \mathrm{j}(c - b)$.
  \item Deduce that $\mathrm{AC} = \mathrm{BC}$.
  \item Prove the equality: $a - b = \mathrm{j}^{2}(b - c)$.
  \item Deduce that triangle ABC is equilateral.
\end{enumerate}