bac-s-maths

Papers (167)
2025
bac-spe-maths__amerique-nord_j1 4 bac-spe-maths__amerique-nord_j2 5 bac-spe-maths__amerique-sud_j1 4 bac-spe-maths__amerique-sud_j2 7 bac-spe-maths__asie-sept_j1 4 bac-spe-maths__asie_j1 4 bac-spe-maths__asie_j2 4 bac-spe-maths__caledonie_j1 4 bac-spe-maths__caledonie_j2 4 bac-spe-maths__centres-etrangers_j1 6 bac-spe-maths__centres-etrangers_j2 4 bac-spe-maths__metropole-sept_j1 4 bac-spe-maths__metropole-sept_j2 5 bac-spe-maths__metropole_j1 4 bac-spe-maths__metropole_j2 5 bac-spe-maths__polynesie-sept_j1 4 bac-spe-maths__polynesie_j1 4 bac-spe-maths__polynesie_j2 4
2024
bac-spe-maths__amerique-nord_j1 5 bac-spe-maths__amerique-nord_j2 4 bac-spe-maths__amerique-sud_j1 4 bac-spe-maths__amerique-sud_j2 4 bac-spe-maths__asie_j1 7 bac-spe-maths__asie_j2 4 bac-spe-maths__centres-etrangers_j1 5 bac-spe-maths__centres-etrangers_j2 4 bac-spe-maths__metropole-sept_j1 4 bac-spe-maths__metropole-sept_j2 4 bac-spe-maths__metropole_j1 4 bac-spe-maths__metropole_j2 4 bac-spe-maths__polynesie-sept 4 bac-spe-maths__polynesie_j1 4 bac-spe-maths__polynesie_j2 4 bac-spe-maths__suede 4
2023
bac-spe-maths__amerique-nord_j1 4 bac-spe-maths__amerique-nord_j2 5 bac-spe-maths__amerique-sud_j1 6 bac-spe-maths__amerique-sud_j2 4 bac-spe-maths__asie_j1 4 bac-spe-maths__asie_j2 4 bac-spe-maths__caledonie_j1 4 bac-spe-maths__caledonie_j2 4 bac-spe-maths__centres-etrangers_j1 9 bac-spe-maths__centres-etrangers_j2 8 bac-spe-maths__europe_j1 4 bac-spe-maths__europe_j2 4 bac-spe-maths__metropole-sept_j1 7 bac-spe-maths__metropole-sept_j2 4 bac-spe-maths__metropole_j1 8 bac-spe-maths__metropole_j2 4 bac-spe-maths__polynesie-sept 4 bac-spe-maths__polynesie_j1 4 bac-spe-maths__polynesie_j2 4 bac-spe-maths__reunion_j1 4 bac-spe-maths__reunion_j2 4
2022
bac-spe-maths__amerique-nord_j1 4 bac-spe-maths__amerique-nord_j2 4 bac-spe-maths__amerique-sud_j1 4 bac-spe-maths__amerique-sud_j2 4 bac-spe-maths__asie_j1 4 bac-spe-maths__asie_j2 4 bac-spe-maths__caledonie_j1 4 bac-spe-maths__caledonie_j2 4 bac-spe-maths__centres-etrangers_j1 4 bac-spe-maths__centres-etrangers_j2 4 bac-spe-maths__madagascar_j1 4 bac-spe-maths__madagascar_j2 4 bac-spe-maths__metropole-sept_j1 9 bac-spe-maths__metropole-sept_j2 4 bac-spe-maths__metropole_j1 4 bac-spe-maths__metropole_j2 4 bac-spe-maths__polynesie-sept 4 bac-spe-maths__polynesie_j1 4 bac-spe-maths__polynesie_j2 4
2021
bac-spe-maths__amerique-nord 5 bac-spe-maths__asie_j1 5 bac-spe-maths__asie_j2 5 bac-spe-maths__centres-etrangers_j1 9 bac-spe-maths__centres-etrangers_j2 7 bac-spe-maths__metropole-juin_j1 5 bac-spe-maths__metropole-juin_j2 5 bac-spe-maths__metropole-sept_j1 8 bac-spe-maths__metropole-sept_j2 5 bac-spe-maths__metropole_j1 5 bac-spe-maths__metropole_j2 5 bac-spe-maths__polynesie 5
2020
antilles-guyane 9 caledonie 5 metropole 9 polynesie 9
2019
amerique-nord 5 amerique-sud 6 antilles-guyane 5 asie 6 caledonie 3 centres-etrangers 6 integrale-annuelle 4 liban 9 metropole 5 metropole-sept 5 polynesie 5
2018
amerique-nord 5 amerique-sud 5 antilles-guyane 6 asie 4 caledonie 5 centres-etrangers 17 liban 6 metropole 3 metropole-sept 5 polynesie 7 pondichery 7
2017
amerique-nord 6 amerique-sud 5 antilles-guyane 6 asie 8 caledonie 6 centres-etrangers 8 liban 5 metropole 5 metropole-sept 4 polynesie 7
2016
amerique-nord 5 amerique-sud 6 antilles-guyane 10 asie 5 caledonie 6 centres-etrangers 8 liban 6 metropole 7 metropole-sept 4 polynesie 5 pondichery 6
2015
amerique-nord 4 amerique-sud 8 antilles-guyane 4 asie 7 caledonie 7 centres-etrangers 9 liban 5 metropole 7 metropole-sept 9 polynesie 6 pondichery 5
2014
amerique-nord 4 amerique-sud 7 antilles-guyane 5 asie 4 caledonie 7 centres-etrangers 7 liban 7 metropole 5 metropole-sept 5 polynesie 5 pondichery 4
2013
amerique-nord 5 amerique-sud 4 antilles-guyane 9 asie 5 caledonie 5 centres-etrangers 8 liban 4 metropole 5 metropole-sept 5 polynesie 4 pondichery 4
2007
integrale-annuelle2 6
2025 bac-spe-maths__metropole_j1

4 maths questions

Q1 5 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
There are four blood groups in the human species: $\mathrm { A } , \mathrm { B } , \mathrm { AB }$ and O. Each blood group can present a rhesus factor. When it is present, we say that the rhesus is positive, otherwise we say that it is negative.
Within the French population, we know that:
  • $45 \%$ of individuals belong to group A, and among them $85 \%$ are rhesus positive;
  • $10 \%$ of individuals belong to group B, and among them $84 \%$ are rhesus positive;
  • $3 \%$ of individuals belong to group AB, and among them $82 \%$ are rhesus positive.

We randomly choose a person from the French population. We denote by:
  • A the event ``The chosen person is of blood group A'';
  • B the event ``The chosen person is of blood group B'';
  • $AB$ the event ``The chosen person is of blood group AB'';
  • O the event ``The chosen person is of blood group O'';
  • $R$ the event ``The chosen person has a positive rhesus factor''.
For any event $E$, we denote by $\bar { E }$ the complementary event of $E$ and $p ( E )$ the probability of $E$.
  1. Copy the tree opposite and complete the ten blanks.
  2. Show that $p ( B \cap R ) = 0{,}084$. Interpret this result in the context of the exercise.
  3. We specify that $p ( R ) = 0{,}8397$. Show that $p _ { O } ( R ) = 0{,}83$.
  4. We say that an individual is a ``universal donor'' when their blood can be transfused to any person without risk of incompatibility. Blood group O with negative rhesus is the only one satisfying this characteristic. Show that the probability that an individual randomly chosen from the French population is a universal donor is 0,0714.
  5. During a blood collection, a sample of 100 people is chosen from the population of a French city. This population is large enough to assimilate this choice to sampling with replacement. We denote by $X$ the random variable that associates to each sample of 100 people the number of universal donors in that sample. a. Justify that $X$ follows a binomial distribution and specify its parameters. b. Determine to $10 ^ { - 3 }$ near the probability that there are at most 7 universal donors in this sample. c. Show that the expectation $E ( X )$ of the random variable $X$ is equal to 7,14 and that its variance $V ( X )$ is equal to 6,63 to $10 ^ { - 2 }$ near.
  6. During the national blood donation week, a blood collection is organized in $N$ randomly chosen French cities numbered $1,2,3 , \ldots , N$ where $N$ is a non-zero natural integer. We consider the random variable $X _ { 1 }$ which associates to each sample of 100 people from city 1 the number of universal donors in that sample. We define in the same way the random variables $X _ { 2 }$ for city $2 , \ldots , X _ { N }$ for city $N$. We assume that these random variables are independent and that they have the same expectation equal to 7,14 and the same variance equal to 6,63. We consider the random variable $M _ { N } = \frac { X _ { 1 } + X _ { 2 } + \ldots + X _ { N } } { N }$. a. What does the random variable $M _ { N }$ represent in the context of the exercise? b. Calculate the expectation $E \left( M _ { N } \right)$. c. We denote by $V \left( M _ { N } \right)$ the variance of the random variable $M _ { N }$. Show that $V \left( M _ { N } \right) = \frac { 6{,}63 } { N }$. d. Determine the smallest value of $N$ for which the Bienaymé-Chebyshev inequality allows us to assert that: $$P \left( 7 < M _ { N } < 7{,}28 \right) \geqslant 0{,}95 .$$
Q2 6 marks Applied differentiation Tangent line computation and geometric consequences View
We consider a function $f$ defined on the interval $] 0 ; + \infty [$. We admit that it is twice differentiable on the interval $] 0 ; + \infty[$. We denote by $f ^ { \prime }$ its derivative function and $f ^ { \prime \prime }$ its second derivative function.
In an orthogonal coordinate system, we have drawn:
  • the representative curve of $f$, denoted $\mathscr { C } _ { f }$ on the interval $] 0$; 3 ];
  • the line $T _ { \mathrm { A } }$, tangent to $\mathscr { C } _ { f }$ at point $\mathrm { A } ( 1 ; 2 )$;
  • the line $T _ { \mathrm { B } }$ tangent to $\mathscr { C } _ { f }$ at point $\mathrm { B } ( \mathrm { e } ; \mathrm { e } )$.
We further specify that the tangent $T _ { \mathrm { A } }$ passes through point $\mathrm { C } ( 3 ; 0 )$.
Part A: Graphical readings
Answer the following questions by justifying them using the graph.
  1. Determine the derivative number $f ^ { \prime } ( 1 )$.
  2. How many solutions does the equation $f ^ { \prime } ( x ) = 0$ have in the interval $] 0$; 3 ]?
  3. What is the sign of $f ^ { \prime \prime } ( 0{,}2 )$ ?

Part B: Study of the function $f$
We admit in this part that the function $f$ is defined on the interval $] 0 ; + \infty [$ by $$f ( x ) = x \left[ 2 ( \ln x ) ^ { 2 } - 3 \ln x + 2 \right]$$ where ln denotes the natural logarithm function.
  1. Solve in $\mathbb { R }$ the equation $2 X ^ { 2 } - 3 X + 2 = 0$. Deduce that $\mathscr { C } _ { f }$ does not intersect the x-axis.
  2. Determine, by justifying, the limit of $f$ as $+ \infty$. We will admit that the limit of $f$ at 0 is equal to 0.
  3. We admit that for all $x$ belonging to $] 0 ; + \infty [$, $f ^ { \prime } ( x ) = 2 ( \ln x ) ^ { 2 } + \ln x - 1$. a. Show that for all $x$ belonging to $] 0 ; + \infty [$, $f ^ { \prime \prime } ( x ) = \frac { 1 } { x } ( 4 \ln x + 1 )$. b. Study the convexity of the function $f$ on the interval $] 0 ; + \infty [$ and specify the exact value of the abscissa of the inflection point. c. Show that the curve $\mathscr { C } _ { f }$ is above the tangent $T _ { \mathrm { B } }$ on the interval $] 1 ; + \infty [$.

Part C: Area calculation
  1. Justify that the tangent $T _ { \mathrm { B } }$ has the reduced equation $y = 2 x - \mathrm { e }$.
  2. Using integration by parts, show that $$\int _ { 1 } ^ { \mathrm { e } } x \ln x \mathrm {~d} x = \frac { \mathrm { e } ^ { 2 } + 1 } { 4 }$$
  3. We denote by $\mathscr { A }$ the area of the shaded region in the figure, bounded by the curve $\mathscr { C } _ { f }$, the tangent $T _ { \mathrm { B } }$, and the lines with equations $x = 1$ and $x = \mathrm { e }$. We admit that $\int _ { 1 } ^ { \mathrm { e } } x ( \ln x ) ^ { 2 } \mathrm {~d} x = \frac { \mathrm { e } ^ { 2 } - 1 } { 4 }$. Deduce the exact value of $\mathscr { A }$ in square units.
Q3 Vectors: Lines & Planes True/False or Verify a Given Statement View
For each of the following statements, indicate whether it is true or false. Justify each answer. An unjustified answer earns no points.
We equip space with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ).
  1. We consider the points $\mathrm { A } ( - 1 ; 0 ; 5 )$ and $\mathrm { B } ( 3 ; 2 ; - 1 )$.
    Statement 1: A parametric representation of the line (AB) is $$\left\{ \begin{aligned} x & = 3 - 2 t \\ y & = 2 - t \text { with } t \in \mathbb { R } \\ z & = - 1 + 3 t \end{aligned} \right.$$
    Statement 2: The vector $\vec { n } \left( \begin{array} { c } 5 \\ - 2 \\ 1 \end{array} \right)$ is normal to the plane (OAB).
  2. We consider:
    • the line $d$ with parametric representation $\left\{ \begin{aligned} x & = 15 + k \\ y & = 8 - k \\ z & = - 6 + 2 k \end{aligned} \right.$ with $k \in \mathbb { R }$;
    • the line $d ^ { \prime }$ with parametric representation $\left\{ \begin{array} { l } x = 1 + 4 s \\ y = 2 + 4 s \\ z = 1 - 6 s \end{array} \right.$ with $s \in \mathbb { R }$.

    Statement 3: The lines $d$ and $d ^ { \prime }$ are not coplanar.
  3. We consider the plane $\mathscr { P }$ with equation $x - y + z + 1 = 0$.
    Statement 4: The distance from point $\mathrm { C } ( 2 ; - 1 ; 2 )$ to the plane $\mathscr { P }$ is equal to $2 \sqrt { 3 }$.
Q4 Differential equations Applied Modeling with Differential Equations View
A team of biologists is studying the evolution of the area covered by a marine algae called seagrass, on the bottom of Alycastre Bay, near the island of Porquerolles. The studied area has a total area of 20 hectares (ha), and on July 1, 2024, the seagrass covered 1 ha of this area.
Part A: Study of a discrete model
For any natural integer $n$, we denote by $u _ { n }$ the area of the zone, in hectares, covered by seagrass on July 1 of the year $2024 + n$. Thus, $u _ { 0 } = 1$. A study conducted on this area made it possible to establish that for any natural integer $n$: $$u _ { n + 1 } = - 0{,}02 u _ { n } ^ { 2 } + 1{,}3 u _ { n }$$
  1. Calculate the area that seagrass should cover on July 1, 2025 according to this model.
  2. We denote by $h$ the function defined on $[0;20]$ by $$h ( x ) = - 0{,}02 x ^ { 2 } + 1{,}3 x$$ We admit that $h$ is increasing on $[0;20]$. a. Prove that for any natural integer $n$, $1 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 20$. b. Deduce that the sequence $(u _ { n })$ converges. We denote by $L$ its limit. c. Justify that $L = 15$.
  3. The biologists wish to know after how long the area covered by seagrass will exceed 14 hectares. a. Without any calculation, justify that, according to this model, this will occur. b. Copy and complete the following algorithm so that at the end of execution, it displays the answer to the biologists' question. \begin{verbatim} def seuil(): n=0 u= 1 while ...... : n=...... u=...... return n \end{verbatim}

Part B: Study of a continuous model
We wish to describe the area of the studied zone covered by seagrass over time with a continuous model. In this model, for a duration $t$, in years, elapsed from July 1, 2024, the area of the studied zone covered by seagrass is given by $f ( t )$, where $f$ is a function defined on $[ 0 ; + \infty [$ satisfying:
  • $f ( 0 ) = 1$;
  • $f$ does not vanish on $[ 0 ; + \infty [$;
  • $f$ is differentiable on $[ 0 ; + \infty [$;
  • $f$ is a solution on $[ 0 ; + \infty [$ of the differential equation $$\left( E _ { 1 } \right) : \quad y ^ { \prime } = 0{,}02 y ( 15 - y ) .$$
We admit that such a function $f$ exists; the purpose of this part is to determine an expression for it. We denote by $f ^ { \prime }$ the derivative function of $f$.
  1. Let $g$ be the function defined on $\left[ 0 ; + \infty \left[ \text{ by } g ( t ) = \frac { 1 } { f ( t ) } \right. \right.$. Show that $g$ is a solution of the differential equation $$\left( E _ { 2 } \right) : \quad y ^ { \prime } = - 0{,}3 y + 0{,}02 .$$
  2. Give the solutions of the differential equation $( E _ { 2 } )$.
  3. Deduce that for all $t \in [ 0 ; + \infty [$: $$f ( t ) = \frac { 15 } { 14 \mathrm { e } ^ { - 0{,}3 t } + 1 }$$
  4. Determine the limit of $f$ as $+ \infty$.
  5. Solve in the interval $[ 0 ; + \infty [$ the inequality $f ( t ) > 14$. Interpret the result in the context of the exercise.