Exercise 4

Common to all candidates

The plane is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). We consider the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D distinct with complex numbers $z _ { \mathrm { A } } , z _ { \mathrm { B } } , z _ { \mathrm { C } }$ and $z _ { \mathrm { D } }$ such that:
$$\left\{ \begin{array} { l } z _ { \mathrm { A } } + z _ { \mathrm { C } } = z _ { \mathrm { B } } + z _ { \mathrm { D } } \\ z _ { \mathrm { A } } + \mathrm { i } z _ { \mathrm { B } } = z _ { \mathrm { C } } + \mathrm { i } z _ { \mathrm { D } } \end{array} \right.$$
Prove that the quadrilateral ABCD is a square.

Let $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$.
Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation.
$\{Q \mid \mathbf{u} \cdot \mathbf{v} = -2024\sqrt{\mathbf{v} \cdot \mathbf{v}}\}$. [2 points]
Options:

1. The empty set
2. A singleton set
3. A line
4. A pair of lines
5. A circle
6. A plane perpendicular to $\mathbf{u}$
7. A plane parallel to $\mathbf{u}$
8. An infinite cone
9. A finite cone
10. A sphere
11. None of the above

Let $z = x + i y$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation
$$z \bar { z } ^ { 3 } + \bar { z } z ^ { 3 } = 350$$
is
(A) 48
(B) 32
(C) 40
(D) 80

The area of the triangle with vertices $P ( z ) , Q ( i z )$ and $R ( z + i z )$ is
(1) 1
(2) $\frac { 1 } { 2 } | z | ^ { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 2 } | z + i z | ^ { 2 }$

Let $z _ { 1 } = 2 + 3 i$ and $z _ { 2 } = 3 + 4 i$. The set $\mathrm { S } = \left\{ \mathrm { z } \in \mathrm { C } : \left| \mathrm { z } - \mathrm { z } _ { 1 } \right| ^ { 2 } - \left| \mathrm { z } - \mathrm { z } _ { 2 } \right| ^ { 2 } = \left| \mathrm { z } _ { 1 } - \mathrm { z } _ { 2 } \right| ^ { 2 } \right\}$ represents a
(1) straight line with sum of its intercepts on the coordinate axes equals 14
(2) hyperbola with the length of the transverse axis 7
(3) straight line with the sum of its intercepts on the coordinate axes equals $-18$
(4) hyperbola with eccentricity 2

For all $z \in C$ on the curve $C_1 : |z| = 4$, let the locus of the point $z + \frac{1}{z}$ be the curve $C_2$. Then
(1) the curves $C_1$ and $C_2$ intersect at 4 points
(2) the curves $C_1$ lies inside $C_2$
(3) the curves $C_1$ and $C_2$ intersect at 2 points
(4) the curves $C_2$ lies inside $C_1$

(Course 2) Q2 Let $\alpha , \beta$ and $\gamma$ be three complex numbers representing three different points A, B and C on a complex plane. Also, $\alpha , \beta$ and $\gamma$ satisfy
$$\begin{aligned} & ( \gamma - \alpha ) ^ { 2 } + ( \gamma - \alpha ) ( \beta - \alpha ) + ( \beta - \alpha ) ^ { 2 } = 0 \quad \cdots (1)\\ & | \beta - 2 \alpha + \gamma | = 3 \quad \cdots (2) \end{aligned}$$
We are to find the area of the triangle ABC.
Since from (1)
$$\frac { \gamma - \alpha } { \beta - \alpha } = \frac { - \mathbf { M } \pm \sqrt { \mathbf { N } } i } { \mathbf { O } } ,$$
we have
$$\left| \frac { \gamma - \alpha } { \beta - \alpha } \right| = \mathbf { P } , \quad \arg \frac { \gamma - \alpha } { \beta - \alpha } = \pm \frac { \mathbf { Q } } { \mathbf { R } } \pi ,$$
where $- \pi < \arg \frac { \gamma - \alpha } { \beta - \alpha } < \pi$. Also, since
$$\beta - 2 \alpha + \gamma = ( \beta - \alpha ) \cdot \frac { \mathbf { S } \pm \sqrt { \mathbf { T } } } { \mathbf { U } } ,$$
we have from (2) that
$$| \beta - \alpha | = \mathbf { V } .$$

Let $z _ { 1 }$、$z _ { 2 }$、$z _ { 3 }$、$z _ { 4 }$ be four distinct complex numbers whose corresponding points on the complex plane can be connected in order to form a parallelogram. Which of the following options must be real numbers?
(1) $\left( z _ { 1 } - z _ { 3 } \right) \left( z _ { 2 } - z _ { 4 } \right)$
(2) $z _ { 1 } - z _ { 2 } + z _ { 3 } - z _ { 4 }$
(3) $z _ { 1 } + z _ { 2 } + z _ { 3 } + z _ { 4 }$
(4) $\frac { z _ { 1 } - z _ { 2 } } { z _ { 3 } - z _ { 4 } }$
(5) $\left( \frac { z _ { 2 } - z _ { 4 } } { z _ { 1 } - z _ { 3 } } \right) ^ { 2 }$