The complex plane is equipped with a direct orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). In what follows, $z$ denotes a complex number.
For each of the statements below, indicate on your answer sheet whether it is true or false. Justify. Any unjustified answer receives no points.
Statement 1: The equation $z - \mathrm{i} = \mathrm{i}(z + 1)$ has solution $\sqrt{2}\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$.
Statement 2: For all real $x \in ]-\frac{\pi}{2}; \frac{\pi}{2}[$, the complex number $1 + \mathrm{e}^{2\mathrm{i}x}$ has exponential form $2\cos x\, \mathrm{e}^{-\mathrm{i}x}$.
Statement 3: A point M with affix $z$ such that $|z - \mathrm{i}| = |z + 1|$ belongs to the line with equation $y = -x$.
Statement 4: The equation $z^5 + z - \mathrm{i} + 1 = 0$ has a real solution.