Exercise 4 (5 points)

Candidates who have not followed the specialization course

A radio-controlled scooter moves in a straight line at the constant speed of $1\,\mathrm{m.s}^{-1}$. It is pursued by a dog that moves at the same speed. We represent the situation from above in an orthonormal coordinate system of the plane with unit 1 meter. The origin of this coordinate system is the initial position of the dog. The scooter is represented by a point belonging to the line with equation $x = 5$. It moves on this line in the direction of increasing ordinates.

Part A - Modeling using a sequence

At the initial instant, the scooter is represented by the point $S_0$. The dog pursuing it is represented by the point $M_0$. We consider that at each second, the dog instantly orients itself in the direction of the scooter and moves in a straight line over a distance of 1 meter. We then model the trajectories of the dog and the scooter by two sequences of points denoted $(M_n)$ and $(S_n)$. After $n$ seconds, the coordinates of point $S_n$ are $(5; n)$. We denote $(x_n; y_n)$ the coordinates of point $M_n$.

1. Construct on graph $\mathrm{n}^\circ 1$ given in the appendix the points $M_2$ and $M_3$.
2. We denote $d_n$ the distance between the dog and the scooter $n$ seconds after the start of the pursuit, $d_n = M_nS_n$. Calculate $d_0$ and $d_1$.
3. Justify that the point $M_2$ has coordinates $\left(1 + \frac{4}{\sqrt{17}}; \frac{1}{\sqrt{17}}\right)$.
4. We admit that, for every natural integer $n$: $$\left\{\begin{array}{l} x_{n+1} = x_n + \dfrac{5 - x_n}{d_n} \\[6pt] y_{n+1} = y_n + \dfrac{n - y_n}{d_n} \end{array}\right.$$ a. The table below, obtained using a spreadsheet, gives the coordinates of points $M_n$ and $S_n$ as well as the distance $d_n$ as a function of $n$. What formulas should be written in cells C5 and F5 and copied downward to fill columns C and F?
   

   	A	B	C	D	E	F
   1	$n$	\multicolumn{2}{|c|}{$M_n$}	\multicolumn{2}{|c|}{$S_n$}	$d_n$
   2		$x_n$	$y_n$	5	n
   3	0	0	0	5	0	5
   4	1	1	0	5	1	4.12310563
   5	2	1.9701425	$\cdots$	$\cdots$	$\cdots$	$\cdots$

   
   

A group of junior scouts, in an activity at the city park where they live, set up a tent as shown in the photo in Figure 1. Figure 2 shows the diagram of this tent's structure, in the form of a right prism, in which metal rods were used.
After assembling the rods, one of the scouts observed an insect moving on them, starting from vertex $A$ toward vertex $B$, from there toward vertex $E$ and, finally, made the journey from vertex $E$ to $C$. Consider that all these movements were made by the shortest distance path between the points.
The projection of the insect's displacement on the plane containing the base $ABCD$ is given by (see answer options with figures).

In a computer game, a cube is initially positioned as indicated in the figure.
Each displacement made by this cube always occurs in one of the directions defined by the three coordinate axes. When moving from the initial position, this cube moved 3 units closer to the $yz$ plane, moved 5 units away from the $xz$ plane, and moved 4 units closer to the $xy$ plane.
The figure that presents the orthogonal projections of this cube onto the three coordinate planes, after performing the described movements, is
(A), (B), (C), (D), or (E) as indicated in the figures.

In sailing competitions, athletes can use an anemometer to measure wind speed and direction. The measured result is called apparent wind speed in nautical science. The vector corresponding to apparent wind speed is the sum of the vector corresponding to true wind speed and the vector corresponding to ship's wind speed, where the vector corresponding to ship's wind speed has the same magnitude as the vector corresponding to ship's velocity but opposite direction. Figure 1 shows the correspondence between part of the wind force levels, names, and wind speeds. An athlete measured the vector corresponding to apparent wind speed and the vector corresponding to ship's velocity as shown in Figure 2 (the magnitude of wind speed and the magnitude of the vector are the same, unit: $\mathrm{m/s}$). Then the true wind is

Level	Wind Speed $\mathrm{m/s}$	Name
2	$1.1 \sim 3.3$	Light Breeze
3	$3.4 \sim 5.4$	Gentle Breeze
4	$5.5 \sim 7.9$	Moderate Wind
5	$8.0 \sim 10.1$	Fresh Wind

A. Light Breeze
B. Gentle Breeze
C. Moderate Wind
D. Fresh Wind

In sailing competitions, athletes can use anemometers to measure wind speed and direction. The measured result is called apparent wind speed in nautical science. The vector corresponding to apparent wind speed is the sum of the vector corresponding to true wind speed and the vector corresponding to ship's wind speed, where the vector corresponding to ship's wind speed has the same magnitude as the vector corresponding to ship's speed but opposite direction. Figure 1 shows the correspondence between part of the wind force levels, names, and wind speeds. A sailor measured the vectors corresponding to apparent wind speed and ship's speed at a certain moment as shown in Figure 2 (the magnitude of wind speed and the magnitude of the vector are the same, unit: m/s). Then the true wind is

Level	Wind Speed	Name
2	$1.1 \sim 3.3$	Light breeze
3	$3.4 \sim 5.4$	Gentle breeze
4	$5.5 \sim 7.9$	Moderate breeze
5	$8.0 \sim 10.1$	Fresh breeze

A. Light breeze
B. Gentle breeze
C. Moderate breeze
D. Fresh breeze

A uniform chain of length 2 m is kept on a table such that a length of 60 cm hangs freely from the edge of the table. The total mass of the chain is 4 kg . What is the work done in pulling the entire chain on the table?
(1) 7.2 J
(2) 3.6 J
(3) 120 J
(4) 1200 J

A particle is moving eastwards with a velocity of $5 \mathrm{~m}/\mathrm{s}$ in 10 seconds the velocity changes to $5 \mathrm{~m}/\mathrm{s}$ northwards. The average acceleration in this time is
(1) $\frac{1}{\sqrt{2}} \mathrm{~m}/\mathrm{s}^2$ towards north-east
(2) $\frac{1}{2} \mathrm{~m}/\mathrm{s}^2$ towards north.
(3) zero
(4) $\frac{1}{\sqrt{2}} \mathrm{~m}/\mathrm{s}^2$ towards north-west

On a coordinate plane, a particle starts from point $( - 3 , - 2 )$ and moves 5 units in the direction of vector $( a , 1 )$ and arrives exactly at the $x$-axis, where $a$ is a positive real number. What is the value of $a$?
(1) $\frac { \sqrt { 13 } } { 2 }$
(2) 2
(3) $\sqrt { 5 }$
(4) $\frac { \sqrt { 21 } } { 2 }$
(5) $2 \sqrt { 6 }$

In the rectangular coordinate plane, a point $P ( a , b )$ is rotated counterclockwise by $90 ^ { \circ }$ around the origin, and then the resulting point is translated 3 units in the positive direction along the x-axis and 1 unit in the positive direction along the y-axis, yielding the point $P ( a , b )$ again. Accordingly, what is the product $\mathbf { a } \cdot \mathbf { b }$?
A) 0
B) 1
C) 2
D) 3
E) 4