Exercise 4 (5 points) -- Candidates who have chosen the specialization option
A software allows transforming a rectangular element of a photograph. Thus, the initial rectangle OEFG is transformed into a rectangle $OE'F'G'$, called the image of OEFG.
The purpose of this exercise is to study the rectangle obtained after several successive transformations.
Part A
The plane is referred to an orthonormal coordinate system $(O, \vec{\imath}, \vec{\jmath})$. The points E, F and G have coordinates respectively $(2; 2)$, $(-1; 5)$ and $(-3; 3)$. The software transformation associates with any point $M(x; y)$ of the plane the point $M'(x'; y')$, image of point $M$ such that: $$\left\{\begin{aligned} x' &= \frac{5}{4}x + \frac{3}{4}y \\ y' &= \frac{3}{4}x + \frac{5}{4}y \end{aligned}\right.$$

1. a. Calculate the coordinates of points $E'$, $F'$ and $G'$, images of points E, F and G by this transformation. b. Compare the lengths OE and $OE'$ on one hand, OG and $OG'$ on the other hand.

Give the square matrix of order 2, denoted $A$, such that: $\binom{x'}{y'} = A\binom{x}{y}$.
Part B
In this part, we study the coordinates of the successive images of vertex F of rectangle OEFG when the software transformation is applied multiple times.

1. Consider the algorithm intended to display the coordinates of these successive images.
2. a. Prove that, for every natural integer $n$, the point $E_{n}$ is located on the line with equation $y = x$. One may use the fact that, for every natural integer $n$, the coordinates $(x_{n}; y_{n})$ of point $E_{n}$ satisfy: $$\binom{x_{n}}{y_{n}} = A^{n}\binom{2}{2}$$ b. Prove that the length $\mathrm{O}E_{n}$ tends towards $+\infty$ when $n$ tends towards $+\infty$.

Let $A$ be a $2 \times 2$ orthogonal matrix such that $\operatorname{det}(A) = -1$. Show that $A$ represents reflection about a line in $\mathbb{R}^2$.

On the coordinate plane, three points $\mathrm { A } ( 3,0 ) , \mathrm { B } ( 3,3 ) , \mathrm { C } ( 0,3 )$ are mapped to $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime }$ respectively by the linear transformation represented by the matrix $\left( \begin{array} { l l } k & 0 \\ 0 & k \end{array} \right) ( k > 1 )$. When the area of the common part of the interior of triangle ABC and the interior of triangle $\mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }$ is $\frac { 1 } { 2 }$, what is the value of $k$? [3 points]
(1) $\frac { 3 } { 2 }$
(2) $\frac { 5 } { 3 }$
(3) $\frac { 7 } { 4 }$
(4) $\frac { 9 } { 5 }$
(5) $\frac { 11 } { 6 }$

On the coordinate plane, a rotation transformation $f$ centered at the origin maps the point $( 1,0 )$ to a point $\left( \frac { \sqrt { 3 } } { 2 } , a \right)$ in the first quadrant. When the sum of all components of the matrix representing the rotation transformation $f$ is $b$, what is the value of $a ^ { 2 } + b ^ { 2 }$? [3 points]
(1) $\frac { 31 } { 12 }$
(2) $\frac { 11 } { 4 }$
(3) $\frac { 35 } { 12 }$
(4) $\frac { 37 } { 12 }$
(5) $\frac { 13 } { 4 }$

In the coordinate plane, let $f$ be the rotation transformation that rotates by $\frac { \pi } { 3 }$ about the origin, and let $g$ be the reflection transformation about the line $y = x$. When the line $x + 2 y + 5 = 0$ is mapped to the line $a x + b y + 5 = 0$ by the composite transformation $g ^ { - 1 } \circ f \circ g$, what is the value of $a + 2 b$? (Given that $a , b$ are constants.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) 1
(3) $\frac { 3 } { 2 }$
(4) 2
(5) $\frac { 5 } { 2 }$

Let $A$ be the matrix representing the linear transformation $f : ( x , y ) \rightarrow ( 2 x - y , x - 2 y )$. When the point $( 5 , - 1 )$ is mapped to the point $( a , b )$ by the linear transformation represented by the matrix $A ^ { 4 }$, find the value of $a + b$. [3 points]

As shown in the figure, there is a line $l : x - y - 1 = 0$ and a hyperbola $C : x ^ { 2 } - 2 y ^ { 2 } = 1$ with one focus at point $\mathrm { F } ( c , 0 )$ (where $c < 0$).
Under a rotation transformation by angle $\theta$ about the origin, the line $l$ is mapped to a line passing through the focus F of the hyperbola $C$. What is the value of $\sin 2 \theta$? [4 points]
(1) $- \frac { 2 } { 3 }$
(2) $- \frac { 5 } { 9 }$
(3) $- \frac { 4 } { 9 }$
(4) $- \frac { 1 } { 3 }$
(5) $- \frac { 2 } { 9 }$

Let the matrices representing two linear transformations $f , g$ be $\left( \begin{array} { l l } 2 & 1 \\ 4 & 2 \end{array} \right) , \left( \begin{array} { r r } 2 & 0 \\ 1 & - 1 \end{array} \right)$ respectively. When the point $( 1,2 )$ is mapped to the point $( a , 6 )$ by the composite transformation $f \circ g$, what is the value of $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

When point $\mathrm { P } ( 2 , - 1 )$ is mapped to point Q by the linear transformation represented by the matrix $\left( \begin{array} { r r } 1 & 2 \\ - 2 & 1 \end{array} \right)$, what is the slope of line PQ? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Suppose in this question that $n = 3$. What geometric interpretation can be given to $R_{p,q}(\theta)$?

We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$.
(a) Let $s _ { 1 } : E \longrightarrow E$ be the orthogonal reflection with respect to the line $\mathbb { R } \omega _ { 1 }$. Show that the matrix of $s _ { 1 }$ in the basis $\mathcal { C }$ is $\left( \begin{array} { c c } 1 & 1 \\ 0 & - 1 \end{array} \right)$.
(b) Let $s _ { 2 } : E \longrightarrow E$ be the orthogonal reflection with respect to the line $\mathbb { R } \omega _ { 2 }$. Show that the matrix of $s _ { 2 }$ in the basis $\mathcal { C }$ is $\left( \begin{array} { c c } - 1 & 0 \\ 1 & 1 \end{array} \right)$.

We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$. Let $s_1$ be the orthogonal reflection with respect to $\mathbb{R}\omega_1$ and $s_2$ the orthogonal reflection with respect to $\mathbb{R}\omega_2$. Let $H$ be the set of vectors $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathbb { R } ^ { 3 }$ such that $m _ { 1 } + m _ { 2 } + m _ { 3 } = 0$. We denote by $H ^ { + }$ the subset of $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$ such that $m _ { 1 } \geqslant m _ { 2 } \geqslant m _ { 3 }$. We consider the application $$\begin{array} { c c c c } \varphi : & H & \longrightarrow & E \\ & \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) & \longmapsto & \left( m _ { 1 } - m _ { 2 } \right) \omega _ { 1 } + \left( m _ { 2 } - m _ { 3 } \right) \omega _ { 2 } \end{array}$$ (a) Show that $\varphi$ is a linear isomorphism. Describe $\varphi \left( H ^ { + } \right)$.
(b) Show that, for all $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$, we have $$s _ { 1 } \circ \varphi \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) = \varphi \left( m _ { 1 } , m _ { 3 } , m _ { 2 } \right) \quad \text { and } \quad s _ { 2 } \circ \varphi \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) = \varphi \left( m _ { 2 } , m _ { 1 } , m _ { 3 } \right) .$$ (c) Let $\widehat { \lambda } = \left( \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \right) \in H$ such that $\lambda _ { 1 } > \lambda _ { 2 } > \lambda _ { 3 }$. We denote by $\mathcal { Q } _ { \widehat { \lambda } }$ the set of $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H ^ { + }$ such that $m _ { 1 } \leqslant \lambda _ { 1 }$ and $m _ { 1 } + m _ { 2 } \leqslant \lambda _ { 1 } + \lambda _ { 2 }$. Show that $\varphi \left( \mathcal { Q } _ { \widehat { \lambda } } \right)$ is a quadrilateral whose vertices will be described.

We assume $n = 2$. We identify elements $x = \binom { x _ { 1 } } { x _ { 2 } } \in \mathbb { R } ^ { 2 }$ with vectors $\vec { x } = x _ { 1 } \vec { i } + x _ { 2 } \vec { j }$ of the Euclidean plane relative to an orthonormal frame $(\Omega , \vec { i } , \vec { j })$.
Let $O$ be the matrix of a reflection relative to a line passing through $\Omega$ and directed by a vector $\vec { v } _ { + }$. Determine a vector $x \in \mathbb { R } ^ { 2 }$ strictly positive and a sign diagonal matrix $S \in M _ { 2 } ( \mathbb { R } )$ such that $O x = S x$. Hint: begin by treating the case where $\vec { v } _ { + } \in \{ \vec { i } , \vec { j } \}$.

We assume $n = 2$. We identify elements $x = \binom { x _ { 1 } } { x _ { 2 } } \in \mathbb { R } ^ { 2 }$ with vectors $\vec { x } = x _ { 1 } \vec { i } + x _ { 2 } \vec { j }$ of the Euclidean plane relative to an orthonormal frame $(\Omega , \vec { i } , \vec { j })$.
Let $O$ be the matrix of a rotation with center $\Omega$ and angle $\theta \in ] - \pi , \pi ]$ nonzero. Using a drawing, find two vectors $x _ { + }$ and $x _ { - }$ such that $$O x _ { + } = \operatorname { diag } ( 1 , - 1 ) x _ { + } \text { and } O x _ { - } = \operatorname { diag } ( - 1,1 ) x _ { - }$$ Then discuss according to the sign of $\theta$, which of $x _ { + }$ and $x _ { - }$ is strictly positive.

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\widetilde{\mathcal{B}} = (z_1, z_2, y_1, -y_2)$ be the basis constructed in questions 16--18, where $\operatorname{Mat}_{\widetilde{\mathcal{B}}}(\omega) = J_4$.
Show that there exist $r > 0$ and $\theta \in \mathbb { R } \backslash \pi \mathbb { Z }$ such that $$\operatorname { Mat } _ { \widetilde { \mathcal { B } } } ( u ) = r \left( \begin{array} { c c } R _ { \theta } & 0 \\ 0 & R _ { - \theta } \end{array} \right)$$ where $R _ { \theta } = \left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right)$, and conclude that $$\operatorname { Mat } _ { \widetilde { \mathcal { B } } } ( \omega ) = J _ { 4 } \quad \text{and} \quad \operatorname { Mat } _ { \widetilde { \mathcal { B } } } \left( \omega _ { 1 } \right) = r \left( \begin{array} { c c } 0 & - R _ { - \theta } \\ R _ { \theta } & 0 \end{array} \right).$$

Let a $2 \times 2$ real matrix $A$ represent a reflection transformation of the coordinate plane and satisfy $A^{3} = \left[\begin{array}{cc} 0 & -1 \\ -1 & 0 \end{array}\right]$; let a $2 \times 2$ real matrix $B$ represent a rotation transformation (centered at the origin) of the coordinate plane and satisfy $B^{3} = \left[\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right]$. Select the correct options.
(1) There are exactly three possible matrices for $A$
(2) There are exactly three possible matrices for $B$
(3) $AB = BA$
(4) The $2 \times 2$ matrix $AB$ represents a rotation transformation of the coordinate plane
(5) $BABA = \left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]$

Suppose a $2 \times 2$ matrix $\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$ representing a linear transformation maps three points $O(0,0), A(1,0), B(0,1)$ on the coordinate plane to $O(0,0), A'(3, \sqrt{3}), B'(-\sqrt{3}, 3)$ respectively, and maps a point $C(x, y)$ at distance 1 from the origin to point $C'(x', y')$. Select the correct options.
(1) The determinant $\left|\begin{array}{ll} a & b \\ c & d \end{array}\right| = 6$
(2) $\overline{OC'} = 2\sqrt{3}$
(3) The angle between $\overrightarrow{OC}$ and $\overrightarrow{OC'}$ is $60^\circ$
(4) It is possible that $y = y'$
(5) If $x < y$ then $x' < y'$

On the coordinate plane, let $\Gamma$ be an ellipse with center at the origin and major axis on the $y$-axis. It is known that a linear transformation of counterclockwise rotation by angle $\theta$ about the origin (where $0 < \theta < \pi$) transforms $\Gamma$ to a new ellipse $\Gamma ^ { \prime } : 40 x ^ { 2 } + 4 \sqrt { 5 } x y + 41 y ^ { 2 } = 180$. The point $\left( - \frac { 5 } { 3 } , \frac { 2 \sqrt { 5 } } { 3 } \right)$ is one of the two points on $\Gamma ^ { \prime }$ farthest from the origin.
It is known that a point $P$ on $\Gamma$ is transformed by this rotation to a point $P ^ { \prime }$ that falls on the $x$-axis, and the $x$-coordinate of $P ^ { \prime }$ is positive. Find the coordinates of point $P$.

Three points $A(1,0), B(0,1), C(-1,0)$ are given on the coordinate plane. Let $\Gamma$ be the graph obtained by transforming $\triangle ABC$ by the matrix $T = \left[\begin{array}{ll} 3 & 0 \\ a & 1 \end{array}\right]$, where $a$ is a real number. Select the correct options.
(1) If $a = 0$, then $\Gamma$ is an isosceles right triangle
(2) At least two points on the sides of $\triangle ABC$ have unchanged coordinates after transformation by $T$
(3) $\Gamma$ must have part of it in the fourth quadrant
(4) There exists a figure $\Omega$ on the plane such that after transformation by $T$ it becomes $\triangle ABC$
(5) The area of $\Gamma$ is a constant value

Let $A$ be the rotation matrix that rotates counterclockwise by angle $\theta$ about the origin, and let $B$ be the reflection matrix with the $x$-axis as the axis of reflection (axis of symmetry). Let $A = \left[\begin{array}{ll} a_{1} & a_{2} \\ a_{3} & a_{4} \end{array}\right]$ and $BA = \left[\begin{array}{ll} c_{1} & c_{2} \\ c_{3} & c_{4} \end{array}\right]$.
Given that $a_{1} + a_{2} + a_{3} + a_{4} = 2(c_{1} + c_{2} + c_{3} + c_{4})$, then $\tan\theta =$ \hspace{2cm}. (Express as a fraction in lowest terms)

Let $A = \left[ \begin{array} { l l } a _ { 1 } & a _ { 2 } \\ a _ { 3 } & a _ { 4 } \end{array} \right]$ and $B = \left[ \begin{array} { l l } b _ { 1 } & b _ { 2 } \\ b _ { 3 } & b _ { 4 } \end{array} \right]$ both be rotation matrices on the coordinate plane with center at the origin $O$, rotating counterclockwise by an acute angle, and satisfying $A ^ { 2 } = B ^ { 3 } = \left[ \begin{array} { l l } 0 & c \\ 1 & d \end{array} \right]$, where $c$ and $d$ are real numbers.
Let point $P ( 1,1 )$ be transformed by $A ^ { 3 }$ to point $Q$, and point $Q$ be transformed by $B ^ { 4 }$ to point $R$.
What is the value of $c$? (Single choice question, 3 points)
(1) 0
(2) $- 1$
(3) 1
(4) $- \frac { 1 } { 2 }$
(5) $\frac { 1 } { 2 }$

Let $A = \left[ \begin{array} { l l } a _ { 1 } & a _ { 2 } \\ a _ { 3 } & a _ { 4 } \end{array} \right]$ and $B = \left[ \begin{array} { l l } b _ { 1 } & b _ { 2 } \\ b _ { 3 } & b _ { 4 } \end{array} \right]$ both be rotation matrices on the coordinate plane with center at the origin $O$, rotating counterclockwise by an acute angle, and satisfying $A ^ { 2 } = B ^ { 3 } = \left[ \begin{array} { l l } 0 & c \\ 1 & d \end{array} \right]$, where $c$ and $d$ are real numbers.
Let point $P ( 1,1 )$ be transformed by $A ^ { 3 }$ to point $Q$, and point $Q$ be transformed by $B ^ { 4 }$ to point $R$.
Find the coordinates of point $Q$ and the angle between $\overrightarrow { O R }$ and the vector $(1, 0)$. (Non-multiple choice question, 6 points)

Let $A = \left[ \begin{array} { l l } a _ { 1 } & a _ { 2 } \\ a _ { 3 } & a _ { 4 } \end{array} \right]$ and $B = \left[ \begin{array} { l l } b _ { 1 } & b _ { 2 } \\ b _ { 3 } & b _ { 4 } \end{array} \right]$ both be rotation matrices on the coordinate plane with center at the origin $O$, rotating counterclockwise by an acute angle, and satisfying $A ^ { 2 } = B ^ { 3 } = \left[ \begin{array} { l l } 0 & c \\ 1 & d \end{array} \right]$, where $c$ and $d$ are real numbers.
Let point $P ( 1,1 )$ be transformed by $A ^ { 3 }$ to point $Q$, and point $Q$ be transformed by $B ^ { 4 }$ to point $R$.
Let $L$ be the line passing through point $P$ and parallel to line $OQ$. Let point $S$ be the intersection of $L$ and line $OR$. Find $\angle O S P$ and the coordinates of point $S$. (Non-multiple choice question, 6 points)

Consider the following multiple conditions on $x , y , z \in \mathbb { R }$.
$$\left\{ \begin{array} { c c c c c } 0 & < & z - x y & < & 1 \\ 0 & < & z - ( x + y ) ^ { 2 } & < & - x y \end{array} \right.$$
Let $\Omega$ be the set of points $( x , y )$ for which at least one $z$ exists satisfying the above conditions. Note that the set $\Omega$ can be seen in the three-dimensional Cartesian coordinate system as the orthogonal projection of points $( x , y , z )$ satisfying the above conditions onto the $x y$-plane. Answer the following questions.
(1) Find the inequalities on $x$ and $y$ representing $\Omega$.
(2) Draw a figure of $\Omega$ in the $x y$-plane. If the boundary of $\Omega$ intersects with the $x$-axis or the $y$-axis, write down the coordinates at each intersection.
(3) The curved segments of the boundary of $\Omega$ correspond to the linear transformation of arcs of the unit circle with a matrix $\mathbf { X }$. Find one such $\mathbf { X }$. Note that the point $( 1,0 )$ on the unit circle must be transformed to a point where the curvature is maximized in the curved segments.
(4) Calculate the determinant of $\mathbf { X }$ found in (3).
(5) Calculate the area of the set $\Omega$. Note that the absolute value of the determinant of a matrix is the area scale factor of the transformation with that matrix.

The reflection of the right triangle ABC given in the rectangular coordinate plane with respect to the y-axis is taken, and the triangle $A'B'C'$ is obtained such that A is paired with $A'$, B with $B'$, and C with $C'$ as symmetric point pairs. This obtained triangle is then rotated $90^{\circ}$ clockwise around point $A'$.
As a result of this rotation, what are the coordinates of the B'' point corresponding to $\mathrm{B}'$?
A) $(0, 3)$ B) $(2, 4)$ C) $(3, 5)$ D) $(4, 6)$ E) $(5, 4)$