Exercise 3 -- Candidates who have NOT followed the specialization course

The complex plane is equipped with a direct orthonormal coordinate system. We consider the equation $$(E): \quad z^2 - 2z\sqrt{3} + 4 = 0$$

1. Solve the equation $(E)$ in the set $\mathbb{C}$ of complex numbers.
2. We consider the sequence $(M_n)$ of points with affixes $z_n = 2^n \mathrm{e}^{\mathrm{i}(-1)^n \frac{\pi}{6}}$, defined for $n \geqslant 1$. a. Verify that $z_1$ is a solution of $(E)$. b. Write $z_2$ and $z_3$ in algebraic form. c. Plot the points $M_1, M_2, M_3$ and $M_4$ on the figure provided in the appendix and draw, on the figure provided in the appendix, the segments $[M_1, M_2]$, $[M_2, M_3]$ and $[M_3, M_4]$.
3. Show that, for every integer $n \geqslant 1$, $z_n = 2^n\left(\frac{\sqrt{3}}{2} + \frac{(-1)^n \mathrm{i}}{2}\right)$.
4. Calculate the lengths $M_1M_2$ and $M_2M_3$.

For the rest of the exercise, we admit that, for every integer $n \geqslant 1$, $M_nM_{n+1} = 2^n\sqrt{3}$.
5. We denote $\ell^n = M_1M_2 + M_2M_3 + \cdots + M_nM_{n+1}$. a. Show that, for every integer $n \geqslant 1$, $\ell^n = 2\sqrt{3}(2^n - 1)$. b. Determine the smallest integer $n$ such that $\ell^n \geqslant 1000$.

We denote by (E) the equation $$z ^ { 4 } + 4 z ^ { 2 } + 16 = 0$$ of unknown complex number $z$.

1. Solve in $\mathbb { C }$ the equation $Z ^ { 2 } + 4 Z + 16 = 0$. Write the solutions of this equation in exponential form.
2. We denote by $a$ the complex number whose modulus is equal to 2 and one of whose arguments is equal to $\frac { \pi } { 3 }$. Calculate $a ^ { 2 }$ in algebraic form. Deduce the solutions in $\mathbb { C }$ of the equation $z ^ { 2 } = - 2 + 2 \mathrm { i } \sqrt { 3 }$. Write the solutions in algebraic form.
3. Organized presentation of knowledge We assume it is known that for every complex number $z = x + \mathrm { i } y$ where $x \in \mathbb { R }$ and $y \in \mathbb { R }$, the conjugate of $z$ is the complex number $\bar{z}$ defined by $\bar{z} = x - \mathrm { i } y$. Prove that:
   • For all complex numbers $z _ { 1 }$ and $z _ { 2 } , \overline { z _ { 1 } z _ { 2 } } = \overline { z _ { 1 } } \cdot \overline { z _ { 2 } }$.
   • For every complex number $z$ and every non-zero natural integer $n , \overline { z ^ { n } } = ( \bar { z } ) ^ { n }$.
4. Prove that if $z$ is a solution of equation (E) then its conjugate $\bar { z }$ is also a solution of (E). Deduce the solutions in $\mathbb { C }$ of equation (E). We will assume that (E) has at most four solutions.

The complex plane is given an orthonormal direct coordinate system $(\mathrm{O}; \vec{u}, \vec{v})$. We consider the point A with affixe 4, the point B with affixe $4\mathrm{i}$ and the points C and D such that ABCD is a square with centre O. For any non-zero natural number $n$, we call $M_n$ the point with affixe $z_n = (1 + \mathrm{i})^n$.

1. Write the number $1 + \mathrm{i}$ in exponential form.
2. Show that there exists a natural number $n_0$, which we will determine, such that, for any integer $n \geqslant n_0$, the point $M_n$ is outside the square ABCD.

The complex plane is equipped with a direct orthonormal coordinate system. Consider the equation
$$( E ) : \quad z ^ { 4 } + 2 z ^ { 3 } - z - 2 = 0$$
with unknown complex number $z$.

1. Give an integer solution of ( $E$ ).
2. Prove that, for every complex number $z$, $$z ^ { 4 } + 2 z ^ { 3 } - z - 2 = \left( z ^ { 2 } + z - 2 \right) \left( z ^ { 2 } + z + 1 \right) .$$
3. Solve equation ( $E$ ) in the set of complex numbers.
4. The solutions of equation ( $E$ ) are the affixes of four points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ in the complex plane such that ABCD is a non-crossed quadrilateral. Is quadrilateral ABCD a rhombus? Justify.

The complex plane is given an orthonormal direct coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). The unit of length is one centimetre.

1. Solve in $\mathbb{C}$ the equation $\left(z^{2} - 2z + 4\right)\left(z^{2} + 4\right) = 0$.
2. We consider the points A and B with complex numbers $z_{\mathrm{A}} = 1 + \mathrm{i}\sqrt{3}$ and $z_{\mathrm{B}} = 2\mathrm{i}$ respectively. a. Write $z_{\mathrm{A}}$ and $z_{\mathrm{B}}$ in exponential form and justify that the points A and B lie on a circle with centre O, whose radius you will specify. b. Draw a figure and place the points A and B. c. Determine a measure of the angle $(\overrightarrow{\mathrm{OA}}, \overrightarrow{\mathrm{OB}})$.
3. We denote by F the point with complex number $z_{\mathrm{F}} = z_{\mathrm{A}} + z_{\mathrm{B}}$. a. Place the point F on the previous figure. Show that OAFB is a rhombus. b. Deduce a measure of the angle $(\overrightarrow{\mathrm{OA}}, \overrightarrow{\mathrm{OF}})$ then of the angle $(\vec{u}, \overrightarrow{\mathrm{OF}})$. c. Calculate the modulus of $z_{\mathrm{F}}$ and deduce the expression of $z_{\mathrm{F}}$ in trigonometric form. d. Deduce the exact value of: $$\cos\left(\frac{5\pi}{12}\right)$$
4. Two calculator models from different manufacturers give for one: $$\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{2 - \sqrt{3}}}{2}$$ and for the other: $$\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6} - \sqrt{2}}{4}$$ Are these results contradictory? Justify your answer.

Consider the equation $( E ) : z ^ { 3 } = 4 z ^ { 2 } - 8 z + 8$ with unknown complex number $z$.
a. Prove that, for all complex numbers $z$, $$z ^ { 3 } - 4 z ^ { 2 } + 8 z - 8 = ( z - 2 ) \left( z ^ { 2 } - 2 z + 4 \right) .$$
b. Solve equation ( $E$ ).
c. Write the solutions of equation ( $E$ ) in exponential form.
The complex plane is equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). Let $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D be the four points with respective affixes $$z _ { \mathrm { A } } = 1 + \mathrm { i } \sqrt { 3 } \quad z _ { \mathrm { B } } = 2 \quad z _ { \mathrm { C } } = 1 - \mathrm { i } \sqrt { 3 } \quad z _ { \mathrm { D } } = 1 .$$
2. What is the nature of quadrilateral OABC? Justify.
3. Let M be the point with affix $z _ { \mathrm { M } } = \frac { 7 } { 4 } + \mathrm { i } \frac { \sqrt { 3 } } { 4 }$.
a. Prove that points $\mathrm { A } , \mathrm { M }$ and B are collinear.
b. Prove that triangle DMB is right-angled.

Consider the equation $$z^{2018} = 2018^{2018} + i$$ where $i = \sqrt{-1}$.
(a) How many complex solutions does this equation have?
(b) How many solutions lie in the first quadrant?
(c) How many solutions lie in the second quadrant?

Consider the complex polynomial $P ( x ) = x ^ { 6 } + i x ^ { 4 } + 1$. (Here $i$ denotes a square-root of $-1$.) Pick the correct statement(s) from below.
(A) $P$ has at least one real zero.
(B) $P$ has no real zeros.
(C) $P$ has at least three zeros of the form $\alpha + i \beta$ with $\beta < 0$.
(D) $P$ has exactly three zeros $\alpha + i \beta$ with $\beta > 0$.

Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Solve using $R$ the equation $Z^2 = z$, with unknown $Z \in \mathbb{C}$.

Let us consider the solutions to the equation in the complex number $z$
$$z ^ { 4 } = - 324 \quad \cdots (1)$$
and the solutions to the equation in the complex number $z$
$$z ^ { 4 } = t ^ { 4 } \quad \cdots (2)$$
where $t$ is a positive real number.
(1) To find the solutions to (1), let us set
$$z = r ( \cos \theta + i \sin \theta ) \quad ( r > 0,0 < \theta \leqq 2 \pi )$$
Then
$$z ^ { 4 } = r ^ { \mathbf { M } } ( \cos \mathbf { N } \theta + i \sin \mathbf { N } \theta ) .$$
The values of $r$ and $\theta$ such that this expression is equal to $-324$ are
$$\begin{aligned} & r = \mathbf { O } \sqrt { \mathbf { P } } , \\ & \theta = \frac { \mathbf { Q } } { \mathbf { R } } \pi , \frac { \mathbf { S } } { \mathbf { R } } \pi , \frac { \mathbf { T } } { \mathbf { R } } \pi , \frac { \mathbf { U } } { \mathbf { R } } \pi , \end{aligned}$$
where $\mathbf { Q } < \mathbf { S } < \mathbf { T } < \mathbf { U }$.
(2) There are $\mathbf { V }$ solutions to equation (2), and these solutions are dependent on $t$. Now, consider any one of the solutions to (1) and any one of the solutions to (2), and let $d$ be the distance on the complex number plane between these two solutions. Then, over the interval $0 < t \leqq 4$, the minimum value of $d$ is $\mathbf { W }$ and the maximum value is $\sqrt { \mathbf { X Y } }$.