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.

1. 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 $[AB]$.
   
2. Statement 2: the complex number $(\sqrt{3} + \mathrm{i})^{1515}$ is a real number.
   
3. 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: $$\left\{\begin{array}{rlr} x & = & 2t \\ y & = & -3 + 4t, \quad t \in \mathbb{R} \\ z & = 7 - 10t \end{array}\right.$$
   
4. Statement 4: a measure in degrees of the geometric angle $\widehat{\mathrm{FEG}}$, rounded to the nearest degree, is $50°$.

Let $A , B , C$ be three sets of complex numbers as defined below $$\begin{aligned} & A = \{ z : \operatorname { Im } z \geq 1 \} \\ & B = \{ z : | z - 2 - i | = 3 \} \\ & C = \{ z : \operatorname { Re } ( ( 1 - i ) z ) = \sqrt { 2 } \} \end{aligned}$$ The number of elements in the set $A \cap B \cap C$ is
(A) 0
(B) 1
(C) 2
(D) $\infty$

Let $S = S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$, where $$S _ { 1 } = \{ z \in \mathbb { C } : | \mathrm { z } | < 4 \} , \quad S _ { 2 } = \left\{ z \in \mathbb { C } : \operatorname { Im } \left[ \frac { z - 1 + \sqrt { 3 } i } { 1 - \sqrt { 3 } i } \right] > 0 \right\}$$ and $S _ { 3 } = \{ z \in \mathbb { C } : \operatorname { Re } z > 0 \}$.
Area of $S =$
(A) $\frac { 10 \pi } { 3 }$
(B) $\frac { 20 \pi } { 3 }$
(C) $\frac { 16 \pi } { 3 }$
(D) $\frac { 32 \pi } { 3 }$

Let $z$ be a complex number such that $\left| \frac { z - i } { z + 2 i } \right| = 1$ and $| z | = \frac { 5 } { 2 }$. Then, the value of $| z + 3 i |$ is
(1) $\sqrt { 10 }$
(2) $\frac { 7 } { 2 }$
(3) $\frac { 15 } { 4 }$
(4) $2 \sqrt { 3 }$

Let $S _ { 1 } , S _ { 2 }$ and $S _ { 3 }$ be three sets defined as $S _ { 1 } = \{ z \in \mathbb { C } : | z - 1 | \leq \sqrt { 2 } \}$, $S _ { 2 } = \{ z \in \mathbb { C } : \operatorname { Re } ( ( 1 - i ) z ) \geq 1 \}$ and $S _ { 3 } = \{ z \in \mathbb { C } : \operatorname { Im } ( z ) \leq 1 \}$. Then, the set $S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$
(1) is a singleton
(2) has exactly two elements
(3) has infinitely many elements
(4) has exactly three elements

Let $A = \{ z \in \mathrm { C } : 1 \leqslant | z - ( 1 + i ) | \leqslant 2 \}$ and $B = \{ z \in A : | z - ( 1 - i ) | = 1 \}$. Then, $B$
(1) is an empty set
(2) contains exactly two elements
(3) contains exactly three elements
(4) is an infinite set

If $z = x + i y$ satisfies $|z - 2| = 0$ and $|z - i| - |z + 5i| = 0$, then
(1) $x + 2 y - 4 = 0$
(2) $x ^ { 2 } + y - 4 = 0$
(3) $x + 2 y + 4 = 0$
(4) $x ^ { 2 } - y + 3 = 0$

Let $S = \{z = x + iy : |z - 1 + i| \geq |z|, |z| < 2, |z + i| = |z - 1|\}$. Then the set of all values of $x$, for which $w = 2x + iy \in S$ for some $y \in \mathbb{R}$, is
(1) $\left(-\sqrt{2}, \frac{1}{2\sqrt{2}}\right)$
(2) $\left(-\frac{1}{\sqrt{2}}, \frac{1}{4}\right)$
(3) $\left(-\sqrt{2}, \frac{1}{2}\right)$
(4) $\left(-\frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}\right)$

Let $A = \left\{ z \in C : \left| \frac { z + 1 } { z - 1 } \right| < 1 \right\}$ and $B = \left\{ z \in C : \arg \left( \frac { z - 1 } { z + 1 } \right) = \frac { 2 \pi } { 3 } \right\}$. Then $A \cap B$ is
(1) a portion of a circle centred at $\left( 0 , - \frac { 1 } { \sqrt { 3 } } \right)$ that lies in the second and third quadrants only
(2) a portion of a circle centred at $\left( 0 , - \frac { 1 } { \sqrt { 3 } } \right)$ that lies in the second quadrant only
(3) an empty set
(4) a portion of a circle of radius $\frac { 2 } { \sqrt { 3 } }$ that lies in the third quadrant only

The number of points of intersection $| z - ( 4 + 3 i ) | = 2$ and $| z | + | z - 4 | = 6 , z \in C$ is
(1) 1
(2) 2
(3) 3
(4) 4

For $n \in N$, let $S _ { n } = \left\{ z \in C : \left| z - 3 + 2i \right| = \frac { n } { 4 } \right\}$ and $T _ { n } = \left\{ z \in C : \left| z - 2 + 3i \right| = \frac { 1 } { n } \right\}$. Then the number of elements in the set $\left\{ n \in N : S _ { n } \cap T _ { n } = \phi \right\}$ is
(1) 0
(2) 2
(3) 3
(4) 4

The area (in sq. units) of the region $S = \{ z \in \mathbb { C } : | z - 1 | \leq 2 ; ( z + \bar { z } ) + i ( z - \bar { z } ) \leq 2 , \operatorname { Im } ( z ) \geq 0 \}$ is
(1) $\frac { 7 \pi } { 3 }$
(2) $\frac { 7 \pi } { 4 }$
(3) $\frac { 17 \pi } { 8 }$
(4) $\frac { 3 \pi } { 2 }$

Let $S _ { 1 } = \{ z \in C : | z | \leq 5 \} , S _ { 2 } = \left\{ z \in C : \operatorname { Im } \left( \frac { z + 1 - \sqrt { 3 } i } { 1 - \sqrt { 3 } i } \right) \geq 0 \right\}$ and $S _ { 3 } = \{ z \in C : \operatorname { Re } ( z ) \geq 0 \}$. Then the area of the region $S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$ is :
(1) $\frac { 125 \pi } { 12 }$
(2) $\frac { 125 \pi } { 4 }$
(3) $\frac { 125 \pi } { 24 }$
(4) $\frac { 125 \pi } { 6 }$

Q61. The area (in sq. units) of the region $S = \{ z \in \mathbb { C } : | z - 1 | \leq 2 ; ( z + \bar { z } ) + i ( z - \bar { z } ) \leq 2 , \operatorname { Im } ( z ) \geq 0 \}$ is
(1) $\frac { 7 \pi } { 3 }$
(2) $\frac { 7 \pi } { 4 }$
(3) $\frac { 17 \pi } { 8 }$
(4) $\frac { 3 \pi } { 2 }$

Q61. Let $S _ { 1 } = \{ z \in C : | z | \leq 5 \} , S _ { 2 } = \left\{ z \in C : \operatorname { Im } \left( \frac { z + 1 - \sqrt { 3 } i } { 1 - \sqrt { 3 } i } \right) \geq 0 \right\}$ and $S _ { 3 } = \{ z \in C : \operatorname { Re } ( z ) \geq 0 \}$. Then the area of the region $S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$ is :
(1) $\frac { 125 \pi } { 12 }$
(2) $\frac { 125 \pi } { 4 }$
(3) $\frac { 125 \pi } { 24 }$
(4) $\frac { 125 \pi } { 6 }$

Let $\alpha$ and $\beta$ be the two solutions of the quadratic equation $x ^ { 2 } + \sqrt { 3 } x + 1 = 0$, where $0 < \arg \alpha < \arg \beta < 2 \pi$. Consider the complex numbers $z$ satisfying the following three conditions:
$$\begin{cases} \arg \dfrac { \alpha - z } { \beta - z } = \dfrac { \pi } { 2 } & \ldots\ldots\ldots (1)\\ ( 1 + i ) z + ( 1 - i ) \bar { z } + k = 0 & \ldots\ldots\ldots (2)\\ \dfrac { \pi } { 2 } < \arg z < \pi , & \ldots\ldots\ldots (3) \end{cases}$$
where $k$ is a real number.
Let us denote the points on the complex number plane which express $\alpha$, $\beta$ and $z$ by $\mathrm{ A }$, $\mathrm{ B }$ and P.
(1) The arguments of $\alpha$ and $\beta$ are
$$\arg \alpha = \frac { \mathbf { A } } { \mathbf { B } } \pi \quad \text{ and } \quad \arg \beta = \frac { \mathbf { C } } { \mathbf { D } } \pi .$$
(2) For each of $\mathbf { E }$ $\sim$ $\mathbf { Q }$ in the following sentences, choose the correct answer from among (0) $\sim$ (9) below.
Since $\mathbf { E } = \dfrac { \pi } { 2 }$ from (1), the point P is located on the circumference of the circle with the center $-\dfrac{ \sqrt{\mathbf{F}} }{ \mathbf{G} }$ and the radius $\dfrac { \mathbf { H } } { \mathbf { I } }$.
On the other hand, from (2), the point P is on the straight line which has the slope $\mathbf{J}$ and passes through a certain point.
From these, we see that when $n$ is the number of complex numbers $z$ which simultaneously satisfy (1), (2) and (3), the maximum value of $n$ is $\mathbf { M }$, and in this case the range of values of $k$ is
$$\mathbf { N } + \sqrt { \mathbf { O } } < k < \sqrt { \mathbf { P } } + \sqrt { \mathbf { Q } }$$
where $\mathbf { P } < \mathbf { Q }$.
(0) 0
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5 (6) 6 (7) $\angle \mathrm{ PAB }$ (8) $\angle \mathrm{ PBA }$ (9) $\angle \mathrm{ APB }$

In the complex plane, a complex number $z$ is in the first quadrant and satisfies $|z| = 1$ and $\left|\frac{-3+4i}{5} - z^3\right| = \left|\frac{-3+4i}{5} - z\right|$, where $i = \sqrt{-1}$. If the real part of $z$ is $a$ and the imaginary part is $b$, then $a = \dfrac{\sqrt{\phantom{0}}}{\sqrt{\phantom{0}}}$ and $b = \dfrac{\sqrt{\phantom{0}}}{\sqrt{\phantom{0}}}$. (Express in simplest radical form)

Let z be a complex number such that
$$\begin{aligned} & | z - 1 | = | z - 2 | \\ & | z | = \sqrt { 3 } \end{aligned}$$
What is the value of $| z - 3 |$?
A) 2
B) $\sqrt { 2 }$
C) $\sqrt { 3 }$
D) $1 + \sqrt { 2 }$
E) $\sqrt { 3 } - 1$