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 answer without justification earns 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 every real $x \in ] - \frac { \pi } { 2 } ; \frac { \pi } { 2 } \left[ \right.$, 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.
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 answer without justification earns 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 every real $x \in ] - \frac { \pi } { 2 } ; \frac { \pi } { 2 } \left[ \right.$, 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.