4. Aviation regulations stipulate that two aircraft on approach must be at least 3 nautical miles apart at all times (1 nautical mile equals 1852 m). If aircraft Alpha and Beta are respectively at $E$ and $F$ at the same instant, is their safety distance respected?

Exercise 3. (5 points)

The plane is equipped with an orthonormal coordinate system. For every natural number $n$, we consider the function $f_n$ defined on $[0; +\infty[$ by:
$$f_0(x) = \mathrm{e}^{-x} \text{ and, for } n \geq 1, f_n(x) = x^n \mathrm{e}^{-x}.$$
For every natural number $n$, we denote $C_n$ the representative curve of function $f_n$. Parts A and B are independent.

Part A: Study of functions $\boldsymbol{f}_{\boldsymbol{n}}$ for $\boldsymbol{n} \geq \mathbf{1}$

We consider a natural number $n \geq 1$.

1. a. We admit that function $f_n$ is differentiable on $[0; +\infty[$.

Show that for all $x \geq 0$,
$$f_n'(x) = (n - x)x^{n-1}\mathrm{e}^{-x}$$
b. Justify all elements of the table below:

$x$	0		$n$		$+\infty$
$f_n'(x)$		+	0	-
			$\left(\frac{n}{\mathrm{e}}\right)^n$
$f(x)$
	0			0

1. Justify by calculation that point $A\left(1; \mathrm{e}^{-1}\right)$ belongs to curve $C_n$.

Part B: Study of integrals $\int_0^1 \boldsymbol{f}_{\boldsymbol{n}}(\boldsymbol{x})\mathrm{d}\boldsymbol{x}$ for $\boldsymbol{n} \geq \mathbf{0}$

In this part, we study functions $f_n$ on $[0; 1]$ and we consider the sequence $(I_n)$ defined for every natural number $n$ by:
$$I_n = \int_0^1 f_n(x)\mathrm{d}x = \int_0^1 x^n \mathrm{e}^{-x}\mathrm{d}x$$

1. On the graph in APPENDIX (page 9/9), curves $C_0, C_1, C_2, C_{10}$ and $C_{100}$ are represented. a. Give a graphical interpretation of $I_n$. b. By reading this graph, what conjecture can be made about the limit of sequence $(I_n)$?
2. Calculate $I_0$.
3. a. Let $n$ be a natural number.

Prove that for all $x \in [0; 1]$,
$$0 \leq x^{n+1} \leq x^n.$$
b. Deduce that for every natural number $n$, we have:
$$0 \leq I_{n+1} \leq I_n$$

1. Prove that sequence $(I_n)$ is convergent, towards a limit greater than or equal to zero that we denote $\ell$.
2. Using integration by parts, prove that for every natural number $n$ we have:

$$I_{n+1} = (n+1)I_n - \frac{1}{\mathrm{e}}$$

1. a. Prove that if $\ell > 0$, the equality from question 5 leads to a contradiction. b. Prove that $\ell = 0$. You may use question 6.a.

Below is the script of the mystere function, written in Python. The constant e has been imported.
\begin{verbatim} def mystere(n): I = 1 - 1/e L = [I] for i in range(n): I = (i + 1)*I - 1/e L.append(I) return L \end{verbatim}

1. What does mystere(100) return in the context of the exercise?

Exercise 4. (5 points)

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. In this exercise, the questions are independent of one another.

1. We consider the differential equation (E):

$$y' = \frac{1}{2}y + 4.$$
Statement 1: The solutions of (E) are the functions $f$ defined on $\mathbb{R}$ by:
$$f(x) = k\mathrm{e}^{\frac{1}{2}x} - 8, \text{ with } k \in \mathbb{R}$$

1. In a final year class, there are 18 girls and 14 boys.

A volleyball team is formed by randomly choosing 3 girls and 3 boys. Statement 2: There are 297024 possibilities for forming such a team.

A sphere with center $\mathrm { C } ( 0,1,1 )$ and radius $2 \sqrt { 2 }$ intersects the line $\frac { x } { 2 } = y = - z$ at two points A and B. Let $S$ be the area of triangle CAB. Find the value of $S ^ { 2 }$.

On plane $\alpha$, there is a right isosceles triangle ABC with $\angle \mathrm { A } = 90 ^ { \circ }$ and $\overline { \mathrm { BC } } = 6$. A point P outside plane $\alpha$ is at a distance of 4 from the plane, and the foot of the perpendicular from P to plane $\alpha$ is point A. What is the distance from point P to line BC? [3 points]
(1) $3 \sqrt { 2 }$
(2) 5
(3) $3 \sqrt { 3 }$
(4) $4 \sqrt { 2 }$
(5) 6

Let $l$ be the line passing through two distinct points $\mathrm { A } , \mathrm { B }$ on plane $\alpha$, and let H be the foot of the perpendicular from point P (not on plane $\alpha$) to plane $\alpha$. When $\overline { \mathrm { AB } } = \overline { \mathrm { PA } } = \overline { \mathrm { PB } } = 6 , \overline { \mathrm { PH } } = 4$, what is the distance between point H and line $l$? [3 points]
(1) $\sqrt { 11 }$
(2) $2 \sqrt { 3 }$
(3) $\sqrt { 13 }$
(4) $\sqrt { 14 }$
(5) $\sqrt { 15 }$

On the basis of the model, calculate the total length of the borehole rounded to the nearest metre.

A fountain mounted on a pole consists of a marble sphere resting in a bronze bowl. The marble sphere touches the four inner walls of the bronze bowl at exactly one point each. The bronze bowl is described in the model by the lateral faces of the pyramid ABCDS, the marble sphere by a sphere with center $M ( 0 | 0 | 4 )$ and radius $r$. The $x _ { 1 } x _ { 2 }$-plane of the coordinate system represents the horizontally running ground in the model; one unit of length corresponds to one decimeter in reality.
Show that the highest point of the fountain is approximately 64 cm above the ground.

Given are the points $P ( 4 | 5 | - 19 ) , Q ( 5 | 9 | - 18 )$ and $R ( 3 | 7 | - 17 )$, which lie in the plane $E$, as well as the line $g : \vec { X } = \left( \begin{array} { c } - 12 \\ 11 \\ 0 \end{array} \right) + \lambda \cdot \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) , \lambda \in \mathbb { R }$.
Determine the length of the line segment $[ \mathrm { PQ } ]$. Show that the triangle PQR is right-angled at $R$, and use this to justify that the line segment $[ \mathrm { PQ } ]$ is the diameter of the circumcircle of triangle PQR.
(for verification: $\overline { \mathrm { PQ } } = 3 \sqrt { 2 }$ )

A photographer is to take underwater photos for a travel magazine.
The photographer swims along the shortest possible path from the shoreline to the buoy. Determine the length of this path.

From the buoy, the photographer dives vertically with respect to the water surface downward to a location whose distance to the sea floor is exactly three meters and is represented in the model by the point $K$.
Determine by calculation what depth below the water surface the photographer reaches during this dive.

Calculate the length of the line segment [AB] and state the special position of this line segment in the coordinate system. (for verification: $\overline { \mathrm { AB } } = \sqrt { 2 }$ )

134- What is the initial distance from the line passing through point $(1, 2, -3)$ parallel to the vector with components $(4, -3, -5)$?
(1) $\dfrac{\sqrt{5}}{2}$ (2) $\sqrt{3}$ (3) $\sqrt{5}$ (4) $2\sqrt{3}$

134. The distance from point $(1,3,2)$ to the line of intersection of the plane $2x - y - z = 4$ with the plane $xOy$ is:
(1) $2$ (2) $\sqrt{6}$ (3) $3$ (4) $\sqrt{10}$

Consider the line $r$ passing through the two points $A(1, -2, 0)$ and $B(2, 3, -1)$, determine the Cartesian equation of the spherical surface with center $C(1, -6, 7)$ and tangent to $r$.

From a point $P(\lambda, \lambda, \lambda)$, perpendiculars $PQ$ and $PR$ are drawn respectively on the lines $y = x, z = 1$ and $y = -x, z = -1$. If $P$ is such that $\angle QPR$ is a right angle, then the possible value(s) of $\lambda$ is(are)
(A) $\sqrt{2}$
(B) $1$
(C) $-1$
(D) $-\sqrt{2}$

The distance of the point $(1, 3, -7)$ from the plane passing through the point $(1, -1, -1)$, having normal perpendicular to both the lines $\dfrac{x-1}{1} = \dfrac{y+2}{-2} = \dfrac{z-4}{3}$ and $\dfrac{x-2}{2} = \dfrac{y+1}{-1} = \dfrac{z+7}{-1}$, is:
(1) $\dfrac{20}{\sqrt{74}}$
(2) $\dfrac{10}{\sqrt{83}}$
(3) $\dfrac{5}{\sqrt{83}}$
(4) $\dfrac{10}{\sqrt{74}}$

The length of the projection of the line segment joining the points $( 5 , - 1,4 )$ and $( 4 , - 1,3 )$ on the plane, $x + y + z = 7$ is
(1) $\sqrt { \frac { 2 } { 3 } }$
(2) $\frac { 2 } { \sqrt { 3 } }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 1 } { 3 }$

The vertices $B$ and $C$ of a $\triangle A B C$ lie on the line, $\frac { x + 2 } { 3 } = \frac { y - 1 } { 0 } = \frac { z } { 4 }$ such that $B C = 5$ units. Then the area (in sq. units) of this triangle, given the point $A ( 1 , - 1,2 )$, is
(1) 6
(2) $2 \sqrt { 34 }$
(3) $\sqrt { 34 }$
(4) $5 \sqrt { 17 }$

If for $a > 0$, the feet of perpendiculars from the points $A ( a , - 2 a , 3 )$ and $B ( 0,4,5 )$ on the plane $l x + m y + n z = 0$ are points $C ( 0 , - a , - 1 )$ and $D$ respectively, then the length of line segment $C D$ is equal to :
(1) $\sqrt { 31 }$
(2) $\sqrt { 41 }$
(3) $\sqrt { 55 }$
(4) $\sqrt { 66 }$

Let $S$ be the mirror image of the point $Q ( 1,3,4 )$ with respect to the plane $2 x - y + z + 3 = 0$ and let $R ( 3,5 , \gamma )$ be a point of this plane. Then the square of the length of the line segment $S R$ is

Let $Q$ be the foot of perpendicular drawn from the point $P(1, 2, 3)$ to the plane $x + 2y + z = 14$. If $R$ is a point on the plane such that $\angle PRQ = 60^\circ$, then the area of $\triangle PQR$ is equal to

The length of the perpendicular from the point $( 1 , - 2,5 )$ on the line passing through $( 1,2,4 )$ and parallel to the line $x + y - z = 0 = x - 2 y + 3 z - 5$ is:
(1) $\sqrt { \frac { 21 } { 2 } }$
(2) $\sqrt { \frac { 9 } { 2 } }$
(3) $\sqrt { \frac { 73 } { 2 } }$
(4) 1

Let $Q$ and $R$ be two points on the line $\frac { x + 1 } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 1 } { 2 }$ at a distance $\sqrt { 26 }$ from the point $P ( 4,2,7 )$. Then the square of the area of the triangle $PQR$ is $\_\_\_\_$.

If the length of the perpendicular drawn from the point $P ( a , 4,2 ) , a > 0$ on the line $\frac { x + 1 } { 2 } = \frac { y - 3 } { 3 } = \frac { z - 1 } { - 1 }$ is $2 \sqrt { 6 }$ units and $Q \left( \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } \right)$ is the image of the point $P$ in this line, then $a + \sum _ { i = 1 } ^ { 3 } \alpha _ { i }$ is equal to
(1) 7
(2) 8
(3) 12
(4) 14

The distance of the point $P ( 4,6 , - 2 )$ from the line passing through the point $( - 3,2,3 )$ and parallel to a line with direction ratios $3,3 , - 1$ is equal to:
(1) 3
(2) $\sqrt { 6 }$
(3) $2 \sqrt { 3 }$
(4) $\sqrt { 14 }$

Let the co-ordinates of one vertex of $\triangle A B C$ be $A ( 0,2 , \alpha )$ and the other two vertices lie on the line $\frac { \mathrm { x } + \alpha } { 5 } = \frac { \mathrm { y } - 1 } { 2 } = \frac { \mathrm { z } + 4 } { 3 }$. For $\alpha \in \mathbb { Z }$, if the area of $\triangle A B C$ is 21 sq. units and the line segment $B C$ has length $2 \sqrt { 21 }$ units, then $\alpha ^ { 2 }$ is equal to $\_\_\_\_$ .