In this exercise, $x$ and $y$ are real numbers greater than 1. In the complex plane equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ), we consider the points $\mathrm { A} , \mathrm { B }$ and C with affixes respectively $z_{\mathrm{A}} = 1 + \mathrm{i}$, $z_{\mathrm{B}} = x + \mathrm{i}$, $z_{\mathrm{C}} = y + \mathrm{i}$.
Problem: we seek the possible values of real numbers $x$ and $y$, greater than 1, for which :
$$\mathrm { OC } = \mathrm { OA } \times \mathrm { OB } \quad \text { and } ( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$$

1. Prove that if $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$, then $y ^ { 2 } = 2 x ^ { 2 } + 1$.
2. Reproduce on your answer sheet and complete the following algorithm so that it displays all couples $( x , y )$ such that : \begin{verbatim} For x going from 1 to ... do For... If... Display x and y End If End For End For \end{verbatim} When this algorithm is executed, it displays the value 2 for variable $x$ and the value 3 for variable $y$.
3. Study of a particular case: in this question only, we take $x = 2$ and $y = 3$. a. Give the modulus and an argument of $z _ { \mathrm { A } }$. b. Show that $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$. c. Show that $z _ { \mathrm { B } } z _ { \mathrm { C } } = 5 z _ { \mathrm { A } }$ and deduce that $( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$.
4. We return to the general case, and we seek whether there exist other values of real numbers $x$ and $y$ such that points $\mathrm { A } , \mathrm { B }$ and C satisfy the two conditions : $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$ and $( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$. Recall that if $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$, then $y ^ { 2 } = 2 x ^ { 2 } + 1$ (question 1.). a. Prove that if $( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$, then $\arg \left[ \frac { ( x + \mathrm { i } ) ( y + \mathrm { i } ) } { 1 + \mathrm { i } } \right] = 0 \bmod 2 \pi$.
   Deduce that under this condition : $x + y - x y + 1 = 0$. b. Prove that if the two conditions are satisfied and moreover $x \neq 1$, then :
   $$y = \sqrt { 2 x ^ { 2 } + 1 } \quad \text { and } y = \frac { x + 1 } { x - 1 }$$
   
5. We define the functions $f$ and $g$ on the interval $] 1 ; + \infty [$ by :
   $$f ( x ) = \sqrt { 2 x ^ { 2 } + 1 } \quad \text { and } g ( x ) = \frac { x + 1 } { x - 1 }$$
   Determine the number of solutions to the initial problem. We may use the function $h$ defined on the interval $] 1 ; + \infty [$ by $h ( x ) = f ( x ) - g ( x )$ and rely on the screenshot of a computer algebra software given below.

11. If the modulus of the complex number $\mathrm { a } + \mathrm { bi } ( \mathrm { a } , \mathrm { b } \in \mathrm { R } )$ is $\sqrt { 3 }$, then $( \mathrm { a } + \mathrm { bi } ) ( \mathrm { a } - \mathrm { bi } ) = $ $\_\_\_\_$ .

3. Let the complex number z satisfy $z ^ { 2 } = 3 + 4 i$ (where $i$ is the imaginary unit), then the modulus of z is $\_\_\_\_$ .

Let $z = \frac { 1 - \mathrm { i } } { 1 + \mathrm { i } } + 2 \mathrm { i }$, then $| z | =$
A. 0
B. $\frac { 1 } { 2 }$
C. 1
D. $\sqrt { 2 }$

Let $z = \frac { 1 - \mathrm { i } } { 1 + \mathrm { i } } + 2 \mathrm { i }$, then $| z | =$
A. 0
B. $\frac { 1 } { 2 }$
C. 1
D. $\sqrt { 2 }$

Given the complex number $z = 2 + \mathrm { i }$, then $z \cdot \bar { z } =$ (A) $\sqrt { 3 }$ (B) $\sqrt { 5 }$ (C) 3 (D) 5

If $z = 1 + 2 \mathrm { i } + \mathrm { i } ^ { 3 }$ , then $| z | =$
A. 0
B. 1
C. $\sqrt { 2 }$
D. 2

If $z = 1 + \mathrm { i }$, then $\left| z ^ { 2 } - 2 z \right| =$
A. 0
B. 1
C. $\sqrt { 2 }$
D. 2

Given that the complex number $z$ satisfies $z = 1 - 2 i$ ($i$ is the imaginary unit), find $| z | =$ $\_\_\_\_$

Given $z = - 1 - \mathrm { i }$, then $| z | =$
A. 0
B. 1
C. $\sqrt { 2 }$
D. 2

$z _ { 1 } , z _ { 2 }$ 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

$z _ { 1 } , z _ { 2 }$ 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

For each natural number $k$, choose a complex number $z _ { k }$ with $\left| z _ { k } \right| = 1$ and denote by $a _ { k }$ the area of the triangle formed by $z _ { k } , i z _ { k } , z _ { k } + i z _ { k }$. Then, which of the following is true for the series below?
$$\sum _ { k = 1 } ^ { \infty } \left( a _ { k } \right) ^ { k }$$
(A) It converges only if every $z _ { k }$ lies in the same quadrant.
(B) It always diverges.
(C) It always converges.
(D) none of the above.

The number of complex roots of the polynomial $z ^ { 5 } - z ^ { 4 } - 1$ which have modulus 1 is
(A) 0
(B) 1
(C) 2
(D) more than 2 .

$\left| z _ { 1 } + z _ { 2 } \right| ^ { 2 } + \left| z _ { 1 } - z _ { 2 } \right| ^ { 2 }$ is equal to
(1) $2 \left( \left| z _ { 1 } \right| + \left| z _ { 2 } \right| \right)$
(2) $2 \left( \left| z _ { 1 } \right| ^ { 2 } + \left| z _ { 2 } \right| ^ { 2 } \right)$
(3) $\left| z _ { 1 } \right| \left| z _ { 2 } \right|$
(4) $\left| z _ { 1 } \right| ^ { 2 } + \left| z _ { 2 } \right| ^ { 2 }$

If $Z _ { 1 } \neq 0$ and $Z _ { 2 }$ be two complex numbers such that $\frac { Z _ { 2 } } { Z _ { 1 } }$ is a purely imaginary number, then $\left| \frac { 2 Z _ { 1 } + 3 Z _ { 2 } } { 2 Z _ { 1 } - 3 Z _ { 2 } } \right|$ is equal to:
(1) 2
(2) 5
(3) 3
(4) 1

Let $z _ { 1 }$ and $z _ { 2 }$ be any two non-zero complex numbers such that $3 \left| z _ { 1 } \right| = 4 \left| z _ { 2 } \right|$. If $z = \frac { 3 z _ { 1 } } { 2 z _ { 2 } } + \frac { 2 z _ { 2 } } { 3 z _ { 1 } }$ then maximum value of $| z |$ is
(1) $\frac { 7 } { 2 }$
(2) $\frac { 9 } { 2 }$
(3) $\frac { 5 } { 2 }$
(4) $\frac { 1 } { 2 } \sqrt { \frac { 17 } { 2 } }$

Let $z _ { 1 }$ and $z _ { 2 }$ be two complex numbers satisfying $\left| z _ { 1 } \right| = 9$ and $\left| z _ { 2 } - 3 - 4 i \right| = 4$. Then the minimum value of $\left| z _ { 1 } - z _ { 2 } \right|$ is :
(1) 2
(2) $\sqrt { 2 }$
(3) 0
(4) 1

If $z \neq 0$ be a complex number such that $z - \frac{1}{z} = 2$, then the maximum value of $|z|$ is
(1) $\sqrt{2}$
(2) 1
(3) $\sqrt{2} - 1$
(4) $\sqrt{2} + 1$

If $\mathrm { S } = \mathrm { z } \in \mathrm { C } : | \mathrm { z } - \mathrm { i } | = | \mathrm { z } + \mathrm { i } | = | \mathrm { z } - 1 |$, then, $\mathrm { n } ( \mathrm { S } )$ is:
(1) 1
(2) 0
(3) 3
(4) 2

Let the complex numbers $\alpha$ and $\frac { 1 } { \alpha }$ lie on the circles $\mathrm { z } - \mathrm { z } _ { 0 } { } ^ { 2 } = 4$ and $\mathrm { z } - \mathrm { z } _ { 0 } { } ^ { 2 } = 16$ respectively, where $\mathrm { z } _ { 0 } = 1 + \mathrm { i }$. Then, the value of $100 | \alpha | ^ { 2 }$ is $\_\_\_\_$ .

Let $z$ be a complex number and
$$z \cdot | \operatorname { Re } ( z ) | = - 4 + 3 i$$
Accordingly, what is $| \mathbf { z } |$?
A) $\frac { 5 } { 2 }$
B) $\frac { 7 } { 2 }$
C) $\frac { 9 } { 2 }$
D) $\frac { 8 } { 3 }$
E) $\frac { 10 } { 3 }$