Exercise 1 — 7 points

Topics: Probability

• If the customer rents a road bike, the probability that it is an electric bike is 0.4;
• If the customer rents a mountain bike, the probability that it is an electric bike is 0.7;
• The probability that the customer rents an electric bike is 0.58.

• R: ``the customer rents a road bike'';
• $E$ : ``the customer rents an electric bike'';
• $\bar { R }$ and $\bar { E }$, complementary events of $R$ and $E$.

1. Copy this tree onto your answer sheet and complete it.
2. a. Show that $p ( E ) = 0.7 - 0.3 \alpha$. b. Deduce that: $\alpha = 0.4$.
3. We know that the customer rented an electric bike. Determine the probability that they rented a mountain bike. Give the result rounded to the nearest hundredth.
4. What is the probability that the customer rents an electric mountain bike?
5. The daily rental price of a non-electric road bike is 25 euros, that of a non-electric mountain bike is 35 euros. For each type of bike, choosing the electric version increases the daily rental price by 15 euros. We denote by $X$ the random variable modeling the daily rental price of a bike. a. Give the probability distribution of $X$. Present the results in the form of a table. b. Calculate the expected value of $X$ and interpret this result.
6. When 30 of Hugo's customers are chosen at random, we treat this choice as sampling with replacement. We denote by $Y$ the random variable associating to a sample of 30 randomly chosen customers the number of customers who rent an electric bike. We recall that the probability of event $E$ is: $p ( E ) = 0.58$. a. Justify that $Y$ follows a binomial distribution and specify its parameters. b. Determine the probability that a sample contains exactly 20 customers who rent an electric bike. Give the result rounded to the nearest thousandth. c. Determine the probability that a sample contains at least 15 customers who rent an electric bike. Give the result rounded to the nearest thousandth.

Exercise 2 — 7 points

1. We consider the sequences $\left( a _ { n } \right)$ and $\left( b _ { n } \right)$ defined by $a _ { 0 } = 1$ and, for every natural number $n$, $a _ { n + 1 } = 0.5 a _ { n } + 1$ and $b _ { n } = a _ { n } - 2$. We can affirm that: a. $\left( a _ { n } \right)$ is arithmetic; b. $\left( b _ { n } \right)$ is geometric; c. $\left( a _ { n } \right)$ is geometric; d. $\left( b _ { n } \right)$ is arithmetic.
2. In questions 2. and 3., we consider the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ defined by: $$u _ { 0 } = 2 , v _ { 0 } = 1 \mathrm { and } , \text { for every natural number } n : \left\{ \begin{array} { l } u _ { n + 1 } = u _ { n } + 3 v _ { n } \\ v _ { n + 1 } = u _ { n } + v _ { n } . \end{array} \right.$$ We can affirm that: a. $\left\{ \begin{array} { l } u _ { 2 } = 5 \\ v _ { 2 } = 3 \end{array} \right.$ b. $u _ { 2 } ^ { 2 } - 3 v _ { 2 } ^ { 2 } = - 2 ^ { 2 }$ c. $\frac { u _ { 2 } } { v _ { 2 } } = 1.75$ d. $5 u _ { 1 } = 3 v _ { 1 }$.
3. We consider the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ defined by: $$u _ { 0 } = 2 , v _ { 0 } = 1 \mathrm { and } , \text { for every natural number } n : \left\{ \begin{array} { l } u _ { n + 1 } = u _ { n } + 3 v _ { n } \\ v _ { n + 1 } = u _ { n } + v _ { n } . \end{array} \right.$$ We consider the program below written in Python language: \begin{verbatim} def valeurs() : u = 2 v = 1 for k in range(1,11) c = u u = u + 3*v v = c + v return (u, v) \end{verbatim} This program returns: a. $u _ { 11 }$ and $v _ { 11 }$; b. $u _ { 10 }$ and $v _ { 11 }$; c. the values of $u _ { n }$ and $v _ { n }$ for $n$ ranging from 1 to 10; d. $u _ { 10 }$ and $v _ { 10 }$.
4. For questions 4. and 5., we consider a function $f$ twice differentiable on the interval $[-4; 2]$. We denote by $f ^ { \prime }$ the derivative function of $f$ and $f ^ { \prime \prime }$ the second derivative of $f$. We are given below the representative curve $\mathscr { C } ^ { \prime }$ of the derivative function $f ^ { \prime }$ in a coordinate system of the plane. We are also given the points $\mathrm { A } ( - 2 ; 0 ) , \mathrm { B } ( 1 ; 0 )$ and $\mathrm { C } ( 0 ; 5 )$. The function $f$ is: a. concave on $[-2; 1]$; b. convex on $[-4; 0]$; c. convex on $[ - 2 ; 1 ]$; d. convex on $[ 0 ; 2 ]$.
5. For questions 4. and 5., we consider a function $f$ twice differentiable on the interval $[-4; 2]$. We denote by $f ^ { \prime }$ the derivative function of $f$ and $f ^ { \prime \prime }$ the second derivative of $f$. We are given below the representative curve $\mathscr { C } ^ { \prime }$ of the derivative function $f ^ { \prime }$ in a coordinate system of the plane. We are also given the points $\mathrm { A } ( - 2 ; 0 ) , \mathrm { B } ( 1 ; 0 )$ and $\mathrm { C } ( 0 ; 5 )$. We admit that the line (BC) is tangent to the curve $\mathscr { C } ^ { \prime }$ at point B. We have: a. $f ^ { \prime } ( 1 ) < 0$; b. $f ^ { \prime } ( 1 ) = 5$; c. $f ^ { \prime \prime } ( 1 ) > 0$; d. $f ^ { \prime \prime } ( 1 ) = - 5$.
6. Let $f$ be the function defined on $\mathbb { R }$ by $f ( x ) = \left( x ^ { 2 } + 1 \right) \mathrm { e } ^ { x }$. The antiderivative $F$ of $f$ on $\mathbb { R }$ such that $F ( 0 ) = 1$ is defined by: a. $F ( x ) = \left( x ^ { 2 } - 2 x + 3 \right) \mathrm { e } ^ { x }$; b. $F ( x ) = \left( x ^ { 2 } - 2 x + 3 \right) \mathrm { e } ^ { x } - 2$; c. $F ( x ) = \left( \frac { 1 } { 3 } x ^ { 3 } + x \right) \mathrm { e } ^ { x } + 1$; d. $F ( x ) = \left( \frac { 1 } { 3 } x ^ { 3 } + x \right) \mathrm { e } ^ { x }$.

Exercise 3 — 7 points

Parts $\mathbf { B }$ and $\mathbf { C }$ are independent

Part A

1. Determine the limit of $f ( x )$ as $x$ tends to 0.
2. Determine the limit of $f ( x )$ as $x$ tends to $+ \infty$.
3. We admit that the function $f$ is differentiable on $] 0 ; + \infty \left[ \right.$ and we denote by $f ^ { \prime }$ its derivative function. a. Prove that, for every real number $x > 0$, we have: $f ^ { \prime } ( x ) = - \ln x$. b. Deduce the variations of the function $f$ on $] 0 ; + \infty [$ and draw its variation table.
4. Solve the equation $f ( x ) = x$ on $] 0$; $+ \infty [$.

Part B

1. We recall that the function $f$ is increasing on the interval $[ 0.5 ; 1 ]$. Prove by induction that, for every natural number $n$, we have: $0.5 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 1$.
2. a. Show that the sequence $( u _ { n } )$ is convergent. b. We denote by $\ell$ the limit of the sequence $( u _ { n } )$. Determine the value of $\ell$.

Part C

1. For every real number $k$, show that $f _ { k }$ admits a maximum $y _ { k }$ attained at $x _ { k } = \mathrm { e } ^ { k - 1 }$.
2. Verify that, for every real number $k$, we have: $x _ { k } = y _ { k }$.

Exercise 4 — 7 points

• the line $\mathscr { D }$ passing through the point $\mathrm { A } ( 2 ; 4 ; 0 )$ and whose direction vector is $\vec { u } \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)$;
• the line $\mathscr { D } ^ { \prime }$ whose parametric representation is: $\left\{ \begin{array} { r l } x & = 3 \\ y & = 3 + t \\ z & = 3 + t \end{array} , t \in \mathbb { R } \right.$.

1. a. Give the coordinates of a direction vector $\overrightarrow { u ^ { \prime } }$ of the line $\mathscr { D } ^ { \prime }$. b. Show that the lines $\mathscr { D }$ and $\mathscr { D } ^ { \prime }$ are not parallel. c. Determine a parametric representation of the line $\mathscr { D }$.

\setcounter{enumi}{1}
1. Show that the vector $\vec { v } \left( \begin{array} { c } 2 \\ - 1 \\ 1 \end{array} \right)$ is a direction vector of the line $\Delta$.
2. We denote by $\mathscr { P }$ the plane containing the lines $\mathscr { D }$ and $\Delta$, that is, the plane passing through point A and with direction vectors $\vec { u }$ and $\vec { v }$. a. Show that the vector $\vec { n } \left( \begin{array} { c } 2 \\ - 1 \\ - 5 \end{array} \right)$ is a normal vector to the plane $\mathscr { P }$. b. Deduce that an equation of the plane $\mathscr { P }$ is: $2 x - y - 5 z = 0$. c. We recall that $\mathrm { M } ^ { \prime }$ is the intersection point of the lines $\Delta$ and $\mathscr { D } ^ { \prime }$. Justify that $\mathrm { M } ^ { \prime }$ is also the intersection point of $\mathscr { D } ^ { \prime }$ and the plane $\mathscr { P }$. Deduce that the coordinates of point $\mathrm { M } ^ { \prime }$ are $( 3 ; 1 ; 1 )$.
3. a. Determine a parametric representation of the line $\Delta$. b. Justify that point M has coordinates $( 1 ; 2 ; 0 )$. c. Calculate the distance $\mathrm { MM } ^ { \prime }$.
4. We consider the line $d$ with parametric representation $\left\{ \begin{aligned} x & = 5 t \\ y & = 2 + 5 t \\ z & = 1 + t \end{aligned} \right.$ with $t \in \mathbb { R }$. a. Show that the line $d$ is parallel to the plane $\mathscr { P }$. b. We denote by $\ell$ the distance from a point N of the line $d$ to the plane $\mathscr { P }$. Express the volume of the tetrahedron $\mathrm { ANMM } ^ { \prime }$ as a function of $\ell$. We recall that the volume of a tetrahedron is given by: $V = \frac { 1 } { 3 } \times B \times h$ where $B$ denotes the area of a base and $h$ the height relative to this base. c. Justify that, if $\mathrm { N } _ { 1 }$ and $\mathrm { N } _ { 2 }$ are any two points of the line $d$, the tetrahedra $A N _ { 1 } M M ^ { \prime }$ and $A N _ { 2 } M M ^ { \prime }$ have the same volume.