Exercise 2 (Candidates who have not followed the specialization course)
The complex plane is referred to the direct orthonormal frame $(\mathrm{O}, \vec{u}, \vec{v})$. Let $R$ be the rotation of the plane with centre $\Omega$, with affix $\omega$ and angle of measure $\theta$. The image by $R$ of a point in the plane is therefore defined as follows:

• $R(\Omega) = \Omega$
• for any point $M$ in the plane, distinct from $\Omega$, the image $M'$ of $M$ is defined by $$\Omega M' = \Omega M \text{ and } (\overrightarrow{\Omega M}, \overrightarrow{\Omega M'}) = \theta \quad [2\pi].$$

We recall that, for points $A$ and $B$ with affixes $a$ and $b$ respectively, $$AB = |b - a| \text{ and } (\vec{u}, \overrightarrow{AB}) = \arg(b - a) \quad [2\pi]$$

1. Show that the affixes $z$ and $z'$ of any point $M$ in the plane and its image $M'$ by the rotation $R$ are related by the relation $$z' - \omega = \mathrm{e}^{\mathrm{i}\theta}(z - \omega).$$
2. We consider the points I and B with affixes $z_{\mathrm{I}} = 1 + \mathrm{i}$ and $z_{\mathrm{B}} = 2 + 2\mathrm{i}$ respectively. Let $R$ be the rotation with centre B and angle of measure $\frac{\pi}{3}$. a. Give the complex form of $R$. b. Let A be the image of I by $R$. Calculate the affix $z_{\mathrm{A}}$ of A. c. Show that O, A and B lie on the same circle with centre I. Deduce that OAB is a right-angled triangle at A. Give a measure of the angle $(\overrightarrow{\mathrm{OA}}, \overrightarrow{\mathrm{OB}})$. d. Deduce a measure of the angle $(\vec{u}, \overrightarrow{\mathrm{OA}})$.
3. Let $T$ be the translation of vector $\overrightarrow{\mathrm{IO}}$. We set $\mathrm{A}' = T(\mathrm{A})$. a. Calculate the affix $z_{\mathrm{A}'}$ of $\mathrm{A}'$. b. What is the nature of the quadrilateral OIAA'? c. Show that $-\frac{\pi}{12}$ is an argument of $z_{\mathrm{A}'}$.

Exercise 2 (For candidates who did not choose the mathematics speciality)
The complex plane is equipped with a direct orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$ (graphical unit: 4 cm). Let A be the point with affixe $z_{\mathrm{A}} = \mathrm{i}$ and B the point with affixe $z_{\mathrm{B}} = \mathrm{e}^{-\mathrm{i}\frac{5\pi}{6}}$.

1. Let $r$ be the rotation with centre O and angle $\frac{2\pi}{3}$. Let C denote the image of B by $r$. a. Determine a complex expression for $r$. b. Show that the affixe of C is $z_{\mathrm{C}} = \mathrm{e}^{-\mathrm{i}\frac{\pi}{6}}$. c. Write $z_{\mathrm{B}}$ and $z_{\mathrm{C}}$ in algebraic form. d. Plot the points A, B and C.
2. Let D be the centroid of points A, B and C with respective coefficients $2, -1$ and $2$. a. Show that the affixe of D is $z_{\mathrm{D}} = \frac{\sqrt{3}}{2} + \frac{1}{2}\mathrm{i}$. Plot point D. b. Show that A, B, C and D lie on the same circle.
3. Let $h$ be the homothety with centre A and ratio 2. Let E denote the image of D by $h$. a. Determine a complex expression for $h$. b. Show that the affixe of E is $z_{\mathrm{E}} = \sqrt{3}$. Plot point E.
4. a. Calculate the ratio $\frac{z_{\mathrm{D}} - z_{\mathrm{C}}}{z_{\mathrm{E}} - z_{\mathrm{C}}}$. Write the result in exponential form. b. Deduce the nature of triangle CDE.

The plane is equipped with an orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). The points $\mathrm{A}$, $\mathrm{B}$ and C have affixes respectively $a = -4$, $b = 2$ and $c = 4$.

1. We consider the three points $\mathrm{A}^{\prime}$, $\mathrm{B}^{\prime}$ and $\mathrm{C}^{\prime}$ with affixes respectively $a^{\prime} = \mathrm{j}a$, $b^{\prime} = \mathrm{j}b$ and $c^{\prime} = \mathrm{j}c$ where j is the complex number $-\frac{1}{2} + \mathrm{i}\frac{\sqrt{3}}{2}$. a. Give the trigonometric form and the exponential form of j. Deduce the algebraic and exponential forms of $a^{\prime}$, $b^{\prime}$ and $c^{\prime}$. b. The points $\mathrm{A}$, $\mathrm{B}$ and C as well as the circles with center O and radii 2, 3 and 4 are represented on the graph provided in the Appendix. Place the points $\mathrm{A}^{\prime}$, $\mathrm{B}^{\prime}$ and $\mathrm{C}^{\prime}$ on this graph.
2. Show that the points $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ are collinear.
3. We denote M the midpoint of segment $[\mathrm{A}^{\prime}\mathrm{C}]$, N the midpoint of segment $[\mathrm{C}^{\prime}\mathrm{C}]$ and P the midpoint of segment $[\mathrm{C}^{\prime}\mathrm{A}]$. Prove that triangle MNP is isosceles.

In the complex plane, the vertices of the quadrilateral formed by the roots of the equation
$$z ^ { 4 } = 16$$
have what area in square units?
A) 8
B) 12
C) 16
D) $4 \sqrt { 3 }$
E) $6 \sqrt { 2 }$