For each of the four following statements, indicate whether it is true or false by justifying your answer. One point is awarded for each correct answer with proper justification. An answer without justification is not taken into account. An absence of answer is not penalized.
In the plane with an orthonormal coordinate system, let $S$ denote the set of points $M$ whose affix $z$ satisfies the two conditions: $$|z - 1| = |z - \mathrm{i}| \quad \text{and} \quad |z - 3 - 2\mathrm{i}| \leqslant 2.$$ In the figure below, we have represented the circle with center at the point with coordinates $(3;2)$ and radius 2, and the line with equation $y = x$. This line intersects the circle at two points A and B. Statement 1: the set $S$ is the segment $[AB]$.
Statement 2: the complex number $(\sqrt{3} + \mathrm{i})^{1515}$ is a real number.
For questions 3 and 4, consider the points $\mathrm{E}(2; 1; -3)$, $\mathrm{F}(1; -1; 2)$ and $\mathrm{G}(-1; 3; 1)$ whose coordinates are defined in an orthonormal coordinate system of space. Statement 3: a parametric representation of the line $(EF)$ is given by: $$\left\{\begin{array}{rlr}
x & = & 2t \\
y & = & -3 + 4t, \quad t \in \mathbb{R} \\
z & = 7 - 10t
\end{array}\right.$$
Statement 4: a measure in degrees of the geometric angle $\widehat{\mathrm{FEG}}$, rounded to the nearest degree, is $50°$.
For each of the four following statements, indicate whether it is true or false by justifying your answer.
One point is awarded for each correct answer with proper justification. An answer without justification is not taken into account. An absence of answer is not penalized.
\begin{enumerate}
\item In the plane with an orthonormal coordinate system, let $S$ denote the set of points $M$ whose affix $z$ satisfies the two conditions:
$$|z - 1| = |z - \mathrm{i}| \quad \text{and} \quad |z - 3 - 2\mathrm{i}| \leqslant 2.$$
In the figure below, we have represented the circle with center at the point with coordinates $(3;2)$ and radius 2, and the line with equation $y = x$. This line intersects the circle at two points A and B.
Statement 1: the set $S$ is the segment $[AB]$.
\item Statement 2: the complex number $(\sqrt{3} + \mathrm{i})^{1515}$ is a real number.
\item For questions 3 and 4, consider the points $\mathrm{E}(2; 1; -3)$, $\mathrm{F}(1; -1; 2)$ and $\mathrm{G}(-1; 3; 1)$ whose coordinates are defined in an orthonormal coordinate system of space.
Statement 3: a parametric representation of the line $(EF)$ is given by:
$$\left\{\begin{array}{rlr}
x & = & 2t \\
y & = & -3 + 4t, \quad t \in \mathbb{R} \\
z & = 7 - 10t
\end{array}\right.$$
\item Statement 4: a measure in degrees of the geometric angle $\widehat{\mathrm{FEG}}$, rounded to the nearest degree, is $50°$.
\end{enumerate}