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
2014 antilles-guyane

5 maths questions

Q1 5 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
Exercise 1 (5 points)
Part A
An oyster farmer raises two species of oysters: ``the flat'' and ``the Japanese''. Each year, flat oysters represent $15 \%$ of his production. Oysters are said to be of size $\mathrm { n } ^ { \circ } 3$ when their mass is between 66 g and 85 g. Only $10 \%$ of flat oysters are of size $\mathrm { n } ^ { \circ } 3$, whereas $80 \%$ of Japanese oysters are.
  1. The health service randomly selects an oyster from the oyster farmer's production. We assume that all oysters have an equal chance of being selected. We consider the following events:
    • J: ``the selected oyster is a Japanese oyster'',
    • C: ``the selected oyster is of size $\mathrm { n } ^ { \circ } 3$''.
    a. Construct a complete weighted tree representing the situation. b. Calculate the probability that the selected oyster is a flat oyster of size $\mathrm { n } ^ { \mathrm { o } } 3$. c. Justify that the probability of obtaining an oyster of size $\mathrm { n } ^ { \circ } 3$ is 0.695. d. The health service selected an oyster of size $\mathrm { n } ^ { \circ } 3$. What is the probability that it is a flat oyster?
  2. The mass of an oyster can be modeled by a random variable $X$ following the normal distribution with mean $\mu = 90$ and standard deviation $\sigma = 2$. a. Give the probability that the oyster selected from the oyster farmer's production has a mass between 87 g and 89 g. b. Give $\mathrm { P } ( \mathrm { X } \geqslant 91 )$.

Part B
This oyster farmer claims that $60 \%$ of his oysters have a mass greater than 91 g.
A restaurant owner would like to buy a large quantity of oysters but would first like to verify the oyster farmer's claim.
The restaurant owner purchases 10 dozen oysters from this oyster farmer, which we will consider as a sample of 120 oysters drawn at random. His production is large enough that we can treat it as sampling with replacement. He observes that 65 of these oysters have a mass greater than 91 g.
  1. Let F be the random variable that associates to any sample of 120 oysters the frequency of those with a mass greater than 91 g. After verifying the conditions of application, give an asymptotic confidence interval at the $95 \%$ level for the random variable F.
  2. What can the restaurant owner think of the oyster farmer's claim?
Q2 Applied differentiation Full function study (variation table, limits, asymptotes) View
Exercise 2
We consider the function $f$ defined and differentiable on the set $\mathbb { R }$ of real numbers by
$$f ( x ) = x + 1 + \frac { x } { \mathrm { e } ^ { x } }$$
We denote by $\mathscr { C }$ its representative curve in an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ).
Part A
  1. Let $g$ be the function defined and differentiable on the set $\mathbb { R }$ by $$g ( x ) = 1 - x + \mathrm { e } ^ { x }$$ Draw up, by justifying, the table giving the variations of the function $g$ on $\mathbb { R }$ (the limits of $g$ at the boundaries of its domain are not required). Deduce the sign of $g ( x )$.
  2. Determine the limit of $f$ at $- \infty$ then the limit of $f$ at $+ \infty$.
  3. We call $f ^ { \prime }$ the derivative of the function $f$ on $\mathbb { R }$. Prove that, for all real $x$, $$f ^ { \prime } ( x ) = \mathrm { e } ^ { - x } g ( x )$$
  4. Deduce the variation table of the function $f$ on $\mathbb { R }$.
  5. Prove that the equation $f ( x ) = 0$ admits a unique real solution $\alpha$ on $\mathbb { R }$. Prove that $- 1 < \alpha < 0$.
  6. a. Prove that the line T with equation $y = 2 x + 1$ is tangent to the curve $\mathscr { C }$ at the point with abscissa 0. b. Study the relative position of the curve $\mathscr { C }$ and the line T.

Part B
  1. Let H be the function defined and differentiable on $\mathbb { R }$ by $$\mathrm { H } ( x ) = ( - x - 1 ) \mathrm { e } ^ { - x }.$$ Prove that H is a primitive on $\mathbb { R }$ of the function $h$ defined by $h ( x ) = x \mathrm { e } ^ { - x }$.
  2. We denote by $\mathscr { D }$ the domain bounded by the curve $\mathscr { C }$, the line T and the lines with equations $x = 1$ and $x = 3$. Calculate, in square units, the area of the domain $\mathscr { D }$.
Q3 4 marks Vectors: Lines & Planes True/False or Verify a Given Statement View
Exercise 3 (4 points)
For each of the four following propositions, indicate whether it is true or false by justifying the answer. One point is awarded for each correct answer with proper justification. An unjustified answer is not taken into account. An absence of answer is not penalized. Space is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We consider the points $\mathrm { A } ( 1 ; 2 ; 5 ) , \mathrm { B } ( - 1 ; 6 ; 4 ) , \mathrm { C } ( 7 ; - 10 ; 8 )$ and $\mathrm { D } ( - 1 ; 3 ; 4 )$.
  1. Proposition 1: The points $\mathrm { A } , \mathrm { B }$ and C define a plane.
  2. We admit that the points $\mathrm { A } , \mathrm { B }$ and D define a plane. Proposition 2: A Cartesian equation of the plane (ABD) is $x - 2 z + 9 = 0$.
  3. Proposition 3: A parametric representation of the line (AC) is $$\left\{ \begin{aligned} x & = \frac { 3 } { 2 } t - 5 \\ y & = - 3 t + 14 \quad t \in \mathbb { R } \\ z & = - \frac { 3 } { 2 } t + 2 \end{aligned} \right.$$
  4. Let $\mathscr { P }$ be the plane with Cartesian equation $2 x - y + 5 z + 7 = 0$ and $\mathscr { P } ^ { \prime }$ the plane with Cartesian equation $- 3 x - y + z + 5 = 0$. Proposition 4: The planes $\mathscr { P }$ and $\mathscr { P } ^ { \prime }$ are parallel.
Q4A Sequences and series, recurrence and convergence Proof by induction on sequence properties View
Exercise 4 — Candidates who have not chosen the specialization option
Let the numerical sequence ( $u _ { n }$ ) defined on the set of natural integers $\mathbb { N }$ by
$$\left\{ \begin{aligned} u _ { 0 } & = 2 \\ \text { and for all natural integer } n , u _ { n + 1 } & = \frac { 1 } { 5 } u _ { n } + 3 \times 0{,}5 ^ { n } . \end{aligned} \right.$$
  1. a. Copy and, using a calculator, complete the table of values of the sequence $\left( u _ { n } \right)$ approximated to $10 ^ { - 2 }$ near:
    $n$012345678
    $u _ { n }$2

    b. Based on this table, state a conjecture about the direction of variation of the sequence $\left( u _ { n } \right)$.
  2. a. Prove, by induction, that for all non-zero natural integer $n$ we have $$u _ { n } \geqslant \frac { 15 } { 4 } \times 0{,}5 ^ { n }$$ b. Deduce that, for all natural integer $n$ non-zero, $u _ { n + 1 } - u _ { n } \leqslant 0$. c. Prove that the sequence ( $u _ { n }$ ) is convergent.
  3. We propose, in this question, to determine the limit of the sequence $\left( u _ { n } \right)$. Let $\left( v _ { n } \right)$ be the sequence defined on $\mathbb { N }$ by $v _ { n } = u _ { n } - 10 \times 0{,}5 ^ { n }$. a. Prove that the sequence $\left( v _ { n } \right)$ is a geometric sequence with ratio $\frac { 1 } { 5 }$. We will specify the first term of the sequence $\left( v _ { n } \right)$. b. Deduce that for all natural integer $n$, $$u _ { n } = - 8 \times \left( \frac { 1 } { 5 } \right) ^ { n } + 10 \times 0{,}5 ^ { n }.$$ c. Determine the limit of the sequence ( $u _ { n }$ ).
  4. Copy and complete lines (1), (2) and (3) of the following algorithm, so that it displays the smallest value of $n$ such that $u _ { n } \leqslant 0{,}01$.
    Input:$n$ and $u$ are numbers
    Initialization :$n$ takes the value 0
    $u$ takes the value 2
    Processing :While $\ldots$(1)
    $n$ takes the value $\ldots$(2)
    $u$ takes the value $\ldots$(3)
    End While
    Output:Display $n$

Q4B Number Theory Linear Diophantine Equations View
Exercise 4 — Candidates who have chosen the specialization option
In the mountains, a hiker made reservations in two types of accommodations: Accommodation A and Accommodation B. One night in accommodation A costs $24 €$ and one night in accommodation B costs $45 €$. He remembers that the total cost of his reservation is $438 €$. We wish to find the numbers $x$ and $y$ of nights spent respectively in accommodation $A$ and accommodation $B$.
  1. a. Show that the numbers $x$ and $y$ are respectively less than or equal to 18 and 9. b. Copy and complete lines (1), (2) and (3) of the following algorithm so that it displays the possible pairs ( $x ; y$ ).
    Input: Processing:\begin{tabular}{l} $x$ and $y$ are numbers
    For $x$ varying from $0$ to $\ldots$ (1)
    For $y$ varying from $0$ to $\ldots$ (2)
    If $\ldots$ (3) Display $x$ and $y$ End If End For End For
    \hline \end{tabular}
  2. Justify that the total cost of the reservation is a multiple of 3.
  3. a. Justify that the equation $8 x + 15 y = 1$ admits at least one solution in integers. b. Determine such a solution. c. Solve the equation (E): $8 x + 15 y = 146$ where $x$ and $y$ are integers.
  4. The hiker remembers having spent at most 13 nights in accommodation A. Show then that he can find the exact number of nights spent in accommodation A and that of nights spent in accommodation B. Calculate these numbers.