If $\frac { z } { z - 1 } = 1 + \mathrm { i }$ , then $z =$
A. $- 1 - \mathrm { i }$
B. $- 1 + \mathrm { i }$
C. $1 - \mathrm{i}$
D. $1 + \mathrm { i }$

The imaginary part of $(1 + 5\mathrm{i})\mathrm{i}$ is
A. $-1$
B. $0$
C. $1$
D. $6$

The imaginary part of $(1 + 5i)i$ is
A. $-1$
B. $0$
C. $1$
D. $6$

Given $z = 1 + \mathrm{i}$, then $\frac{1}{z-1} = $ ( )
A. $-i$
B. $i$
C. $-1$
D. $1$

We fix a natural number $p$ greater than or equal to 2. We denote $\mathcal { V } _ { p } = \left\{ \mathrm { e } ^ { \frac { 2 \mathrm { i } k \pi } { p } } ; k \in \llbracket 1 , p - 1 \rrbracket \right\}$, the set of $p$-th roots of unity different from 1. Let $a$ and $\lambda$ be non-zero complex numbers. Assume that $\frac { a } { \lambda } \notin \mathcal { V } _ { p }$, which means that either $a = \lambda$ or $\frac { a ^ { p } } { \lambda ^ { p } } \neq 1$. Prove that the complex number $\sum _ { j = 0 } ^ { p - 1 } \lambda ^ { p - 1 - j } a ^ { j }$ is non-zero.

Let $z _ { 1 } , \ldots , z _ { n }$ be complex numbers. Show that if $$\left| z _ { 1 } + \cdots + z _ { n } \right| = \left| z _ { 1 } \right| + \cdots + \left| z _ { n } \right|$$ then the vector $\left( \begin{array} { c } z _ { 1 } \\ \vdots \\ z _ { n } \end{array} \right)$ is collinear with the vector $\left( \begin{array} { c } \left| z _ { 1 } \right| \\ \vdots \\ \left| z _ { n } \right| \end{array} \right)$.

We fix an integer $n \geq 1$ and $A(z) = \sum_{k=0}^{n-1} a_k z^k$ a polynomial such that $a_k \in \{-1, 0, 1\}$ for all $0 \leq k \leq n-1$. If $z \in \mathbb{C}$ satisfies $|z| = 1$, show that $|A(z)| \leq n$.

We fix an integer $n \geq 1$ and $A(z) = \sum_{k=0}^{n-1} a_k z^k$ a polynomial such that $a_k \in \{-1, 0, 1\}$ for all $0 \leq k \leq n-1$. We also fix an integer $L \geq 1$. We assume in this question that $a_0 = 1$, and we set, for all $z \in \mathbb{C}$, $$F(z) = \prod_{j=0}^{L-1} A\left(z e^{\frac{2i\pi j}{L}}\right)$$
a. Show that there exists $z_0 \in \mathbb{C}$ such that $|z_0| = 1$ and $|F(z_0)| \geq 1$.
b. Show that $|F(z_0)| \leq n^{L-1} \cdot \sup_{\theta \in \left[-\frac{\pi}{L}, \frac{\pi}{L}\right]} \left|A\left(e^{i\theta}\right)\right|$.

Prove Theorem 2: Let $n \geq 1$ be an integer and $A(z) = \sum_{k=0}^{n-1} a_k z^k$ a nonzero polynomial such that $a_k \in \{-1, 0, 1\}$ for all $0 \leq k \leq n-1$. Then for all integer $L \geq 1$ we have $$\sup_{\theta \in \left[-\frac{\pi}{L}, \frac{\pi}{L}\right]} \left|A\left(e^{i\theta}\right)\right| \geq \frac{1}{n^{L-1}}$$

Let $n \in \mathbf { N } ^ { * }$ as well as complex numbers $z _ { 1 } , \ldots , z _ { n } , u _ { 1 } , \ldots , u _ { n }$ all of modulus at most 1. Show that
$$\left| \prod _ { k = 1 } ^ { n } z _ { k } - \prod _ { k = 1 } ^ { n } u _ { k } \right| \leq \sum _ { k = 1 } ^ { n } \left| z _ { k } - u _ { k } \right|$$

a) Show that for all $x, y, z, t \in \mathbb{R}$ we have $$N(xE + yI + zJ + tK) = x^2 + y^2 + z^2 + t^2.$$ b) Show that for all $U \in \mathbb{H}^{\mathrm{im}}$ we have $U^2 = -N(U)E$ and that $$\mathbb{H}^{\mathrm{im}} = \left\{ U \in \mathbb{H} \mid U^2 \in \left]-\infty, 0\right] E \right\}.$$

We denote $\mathbb{U}$ the multiplicative group of complex numbers of modulus 1. Show that there exist a function $\beta \in \mathcal{E}$ and a constant $C \in ]0,1[$ such that, for all $\zeta \in \mathbb{U}$, $$\mathrm{e}^{\zeta} - 1 = \zeta(1 + \zeta \beta(\zeta)) \quad \text{and} \quad |\beta(\zeta)| \leqslant C.$$

$z _ { 1 } , z _ { 2 }$ are two complex numbers with $z _ { 2 } \neq 0$ and $z _ { 1 } \neq z _ { 2 }$ and satisfying $\left| \frac { z _ { 1 } + z _ { 2 } } { z _ { 1 } - z _ { 2 } } \right| = 1$. Then $\frac { z _ { 1 } } { z _ { 2 } }$ is
(A) real and negative
(B) real and positive
(C) purely imaginary
(D) none of the above need to be true always

Let $z$ be a non-zero complex number such that $\frac { z } { 1 + z }$ is purely imaginary. Then
(A) $z$ is neither real nor purely imaginary
(B) $z$ is real
(C) $z$ is purely imaginary
(D) none of the above

Let $\omega$ denote a cube root of unity which is not equal to 1. Then the number of distinct elements in the set $$\left\{ \left( 1 + \omega + \omega ^ { 2 } + \cdots + \omega ^ { n } \right) ^ { m } \quad : \quad m , n = 1, 2, 3, \cdots \right\}$$ is
(A) 4
(B) 5
(C) 7
(D) infinite.

Let $\mathbb{C}$ denote the set of all complex numbers. Define $$\begin{aligned} & A = \{ ( z , w ) \mid z , w \in \mathbb{C} \text{ and } | z | = | w | \} \\ & B = \left\{ ( z , w ) \mid z , w \in \mathbb{C} , \text{ and } z ^ { 2 } = w ^ { 2 } \right\} \end{aligned}$$ Then,
(A) $A = B$
(B) $A \subset B$ and $A \neq B$
(C) $B \subset A$ and $B \neq A$
(D) none of the above

Let $\mathbb { C }$ denote the set of all complex numbers. Define $$\begin{aligned} & A = \{ ( z ,w ) \mid z , w \in \mathbb { C } \text { and } | z | = | w | \} \\ & B = \left\{ ( z , w ) \mid z , w \in \mathbb { C } , \text { and } z ^ { 2 } = w ^ { 2 } \right\} \end{aligned}$$ Then,
(a) $A = B$
(b) $A \subset B$ and $A \neq B$
(c) $B \subset A$ and $B \neq A$
(d) none of the above.

Let $\mathbb { C }$ denote the set of all complex numbers. Define $$\begin{aligned} & A = \{ ( z ,w ) \mid z , w \in \mathbb { C } \text { and } | z | = | w | \} \\ & B = \left\{ ( z , w ) \mid z , w \in \mathbb { C } , \text { and } z ^ { 2 } = w ^ { 2 } \right\} \end{aligned}$$ Then,
(a) $A = B$
(b) $A \subset B$ and $A \neq B$
(c) $B \subset A$ and $B \neq A$
(d) none of the above.

Let $z$ be a non-zero complex number such that $\frac { z } { 1 + z }$ is purely imaginary. Then
(a) $z$ is neither real nor purely imaginary
(b) $z$ is real
(c) $z$ is purely imaginary
(d) none of the above.

Let $z$ be a non-zero complex number such that $\frac { z } { 1 + z }$ is purely imaginary. Then
(a) $z$ is neither real nor purely imaginary
(b) $z$ is real
(c) $z$ is purely imaginary
(d) none of the above.

$z _ { 1 } , z _ { 2 }$ are two complex numbers with $z _ { 2 } \neq 0$ and $z _ { 1 } \neq z _ { 2 }$ and satisfying $\left| \frac { z _ { 1 } + z _ { 2 } } { z _ { 1 } - z _ { 2 } } \right| = 1$. Then $\frac { z _ { 1 } } { z _ { 2 } }$ is
(A) real and negative
(B) real and positive
(C) purely imaginary
(D) none of the above need to be true always

$z _ { 1 } , z _ { 2 }$ are two complex numbers with $z _ { 2 } \neq 0$ and $z _ { 1 } \neq z _ { 2 }$ and satisfying $\left| \frac { z _ { 1 } + z _ { 2 } } { z _ { 1 } - z _ { 2 } } \right| = 1$. Then $\frac { z _ { 1 } } { z _ { 2 } }$ is
(A) real and negative
(B) real and positive
(C) purely imaginary
(D) none of the above need to be true always

Let $\omega$ denote a cube root of unity which is not equal to 1. Then the number of distinct elements in the set $$\left\{ \left( 1 + \omega + \omega ^ { 2 } + \cdots + \omega ^ { n } \right) ^ { m } \quad : \quad m , n = 1, 2, 3, \cdots \right\}$$ is
(A) 4
(B) 5
(C) 7
(D) infinite

Let $\omega$ denote a cube root of unity which is not equal to 1. Then the number of distinct elements in the set $$\left\{ \left( 1 + \omega + \omega ^ { 2 } + \cdots + \omega ^ { n } \right) ^ { m } \quad : \quad m , n = 1, 2, 3, \cdots \right\}$$ is
(A) 4
(B) 5
(C) 7
(D) infinite

Let $\mathbb{C}$ denote the set of all complex numbers. Define $$\begin{aligned} & A = \{ ( z , w ) \mid z , w \in \mathbb{C} \text{ and } | z | = | w | \} \\ & B = \left\{ ( z , w ) \mid z , w \in \mathbb{C} , \text{ and } z ^ { 2 } = w ^ { 2 } \right\} \end{aligned}$$ Then,
(A) $A = B$
(B) $A \subset B$ and $A \neq B$
(C) $B \subset A$ and $B \neq A$
(D) none of the above