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
2022 bac-spe-maths__madagascar_j1

4 maths questions

Q1 6 marks Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
Exercise 1 — 6 points
Main topics covered: Probability
At a ski resort, there are two types of passes depending on the skier's age:
  • a JUNIOR pass for people under 25 years old;
  • a SENIOR pass for others.
Furthermore, a user can choose, in addition to the pass corresponding to their age, the skip-the-line option which allows them to reduce waiting time at the ski lifts. We assume that:
  • $20 \%$ of skiers have a JUNIOR pass;
  • $80 \%$ of skiers have a SENIOR pass;
  • among skiers with a JUNIOR pass, $6 \%$ choose the skip-the-line option;
  • among skiers with a SENIOR pass, $12.5 \%$ choose the skip-the-line option.
We interview a skier at random and consider the events:
  • $J$ : ``the skier has a JUNIOR pass'';
  • $C$ : ``the skier chooses the skip-the-line option''.
The two parts can be worked on independently
Part A
  1. Represent the situation with a probability tree.
  2. Calculate the probability $P ( J \cap C )$.
  3. Prove that the probability that the skier chooses the skip-the-line option is equal to 0.112.
  4. The skier has chosen the skip-the-line option. What is the probability that this is a skier with a SENIOR pass? Round the result to $10 ^ { - 3 }$.
  5. Is it true that people under twenty-five years old represent less than $15 \%$ of skiers who chose the skip-the-line option? Explain.

Part B
We recall that the probability that a skier chooses the skip-the-line option is equal to 0.112. We consider a sample of 30 skiers chosen at random. Let $X$ be the random variable that counts the number of skiers in the sample who chose the skip-the-line option.
  1. We assume that the random variable $X$ follows a binomial distribution. Give the parameters of this distribution.
  2. Calculate the probability that at least one of the 30 skiers chose the skip-the-line option. Round the result to $10 ^ { - 3 }$.
  3. Calculate the probability that at most one of the 30 skiers chose the skip-the-line option. Round the result to $10 ^ { - 3 }$.
  4. Calculate the expected value of the random variable $X$.
Q2 6 marks Laws of Logarithms Analyze a Logarithmic Function (Limits, Monotonicity, Zeros, Extrema) View
Exercise 2 — 6 points
Theme: Exponential function Main topics covered: Sequences; Functions, Logarithm function. This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.
A correct answer earns one point. An incorrect answer, a multiple answer, or no answer to a question earns or loses no points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
  1. A container initially containing 1 litre of water is left in the sun. Every hour, the volume of water decreases by $15 \%$. After how many whole hours does the volume of water become less than a quarter of a litre? a. 2 hours b. 8 hours. c. 9 hours d. 13 hours
  2. We consider the function $f$ defined on the interval $] 0$; $+ \infty [ \operatorname { by } f ( x ) = 4 \ln ( 3 x )$. For every real $x$ in the interval $] 0$; $+ \infty [$, we have: a. $f ( 2 x ) = f ( x ) + \ln ( 24 )$ b. $f ( 2 x ) = f ( x ) + \ln ( 16 )$ c. $f ( 2 x ) = \ln ( 2 ) + f ( x )$ d. $f ( 2 x ) = 2 f ( x )$
  3. We consider the function $g$ defined on the interval $] 1 ; + \infty [$ by: $$g ( x ) = \frac { \ln ( x ) } { x - 1 } .$$ We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in an orthogonal coordinate system. The curve $\mathscr { C } _ { g }$ has: a. a vertical asymptote and a horizontal asymptote. b. a vertical asymptote and no horizontal asymptote. c. no vertical asymptote and a horizontal asymptote. d. no vertical asymptote and no horizontal asymptote.
    In the rest of the exercise, we consider the function $h$ defined on the interval ]0;2] by: $$h ( x ) = x ^ { 2 } [ 1 + 2 \ln ( x ) ] .$$ We denote $\mathscr { C } _ { h }$ the representative curve of $h$ in a coordinate system of the plane. We assume that $h$ is twice differentiable on the interval ]0; 2]. We denote $h ^ { \prime }$ its derivative and $h ^ { \prime \prime }$ its second derivative. We assume that, for every real $x$ in the interval ] 0 ; 2], we have: $$h ^ { \prime } ( x ) = 4 x ( 1 + \ln ( x ) ) .$$
  4. On the interval $\left. ] \frac { 1 } { \mathrm { e } } ; 2 \right]$, the function $h$ equals zero: a. exactly 0 times. b. exactly 1 time. c. exactly 2 times. d. exactly 3 times.
  5. An equation of the tangent line to $\mathscr { C } _ { h }$ at the point with abscissa $\sqrt { \mathrm { e } }$ is: a. $y = \left( 6 \mathrm { e } ^ { \frac { 1 } { 2 } } \right) \cdot x$ b. $y = ( 6 \sqrt { \mathrm { e } } ) \cdot x + 2 \mathrm { e }$ c. $y = 6 \mathrm { e } ^ { \frac { x } { 2 } }$ d. $y = \left( 6 \mathrm { e } ^ { \frac { 1 } { 2 } } \right) \cdot x - 4 \mathrm { e }$.
  6. On the interval $] 0 ; 2 ]$, the number of inflection points of the curve $\mathscr { C } _ { h }$ is equal to: a. 0 b. 1 c. 2 d. 3
  7. We consider the sequence $\left( u _ { n } \right)$ defined for every natural number $n$ by $$u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + 3 \quad \text { and } \quad u _ { 0 } = 6 .$$ We can affirm that: a. the sequence $\left( u _ { n } \right)$ is strictly increasing. b. the sequence $( u _ { n } )$ is strictly decreasing. c. the sequence $( u _ { n } )$ is not monotonic. d. the sequence $( u _ { n } )$ is constant.
Q3 6 marks Exponential Functions Variation and Monotonicity Analysis View
Exercise 3 — 6 points
Theme: Exponential function Main topics covered: Sequences; Functions, Exponential function.
Part A
We consider the function $f$ defined for every real $x$ by: $$f ( x ) = 1 + x - \mathrm { e } ^ { 0,5 x - 2 } .$$ We assume that the function $f$ is differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ its derivative.
  1. a. Determine the limit of the function $f$ at $- \infty$. b. Prove that, for every non-zero real $x$, $f ( x ) = 1 + 0,5 x \left( 2 - \frac { \mathrm { e } ^ { 0,5 x } } { 0,5 x } \times \mathrm { e } ^ { - 2 } \right)$. Deduce the limit of the function $f$ at $+ \infty$.
  2. a. Determine $f ^ { \prime } ( x )$ for every real $x$. b. Prove that the set of solutions of the inequality $f ^ { \prime } ( x ) < 0$ is the interval $] 4 + 2 \ln ( 2 ) ; + \infty [$.
  3. Deduce from the previous questions the variation table of the function $f$ on $\mathbb { R }$. The exact value of the image of $4 + 2 \ln ( 2 )$ by $f$ should be shown.
  4. Show that the equation $f ( x ) = 0$ has a unique solution on the interval $[ - 1 ; 0 ]$.

Part B
We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = 0$ and, for every natural number $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$ where $f$ is the function defined in Part A.
  1. a. Prove by induction that, for every natural number $n$, we have: $$u _ { n } \leqslant u _ { n + 1 } \leqslant 4 .$$ b. Deduce that the sequence $( u _ { n } )$ converges. We denote its limit by $\ell$.
  2. a. We recall that $f$ satisfies the relation $\ell = f ( \ell )$. Prove that $\ell = 4$. b. We consider the function value written below in the Python language: \begin{verbatim} def valeur (a) : u = 0 n = 0 while u <= a: u=1 + u - exp(0.5*u - 2) n = n+1 return n \end{verbatim} The instruction valeur(3.99) returns the value 12. Interpret this result in the context of the exercise.
Q4 6 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 4 — 6 points
Theme: Exponential function Main topics covered: Geometry in space The space is equipped with an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$. We consider the points $\mathrm { A } ( 5 ; 0 ; - 1 ) , \mathrm { B } ( 1 ; 4 ; - 1 ) , \mathrm { C } ( 1 ; 0 ; 3 ) , \mathrm { D } ( 5 ; 4 ; 3 )$ and $\mathrm { E } ( 10 ; 9 ; 8 )$.
  1. a. Let R be the midpoint of the segment $[ \mathrm { AB } ]$. Calculate the coordinates of point R as well as the coordinates of the vector $\overrightarrow { \mathrm { AB } }$. b. Let $\mathscr { P } _ { 1 }$ be the plane passing through point R and for which $\overrightarrow { \mathrm { AB } }$ is a normal vector. Prove that a Cartesian equation of the plane $\mathscr { P } _ { 1 }$ is: $$x - y - 1 = 0 .$$ c. Prove that point E belongs to the plane $\mathscr { P } _ { 1 }$ and that $\mathrm { EA } = \mathrm { EB }$.
  2. We consider the plane $\mathscr { P } _ { 2 }$ with Cartesian equation $x - z - 2 = 0$. a. Justify that the planes $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$ are secant. b. We denote $\Delta$ the line of intersection of $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$. Prove that a parametric representation of the line $\Delta$ is: $$\left\{ \begin{aligned} x & = 2 + t \\ y & = 1 + t \quad ( t \in \mathbb { R } ) . \\ z & = t \end{aligned} \right.$$
  3. We consider the plane $\mathscr { P } _ { 3 }$ with Cartesian equation $y + z - 3 = 0$. Justify that the line $\Delta$ is secant to the plane $\mathscr { P } _ { 3 }$ at a point $\Omega$ whose coordinates you will determine.
    If S and T are two distinct points in space, we recall that the set of points M in space such that $\mathrm{MS} = \mathrm{MT}$ is a plane, called the perpendicular bisector plane of the segment $[ \mathrm { ST } ]$. We assume that the planes $\mathscr { P } _ { 1 }$, $\mathscr { P } _ { 2 }$ and $\mathscr { P } _ { 3 }$ are the perpendicular bisector planes of the segments [AB], [AC] and [AD] respectively.
  4. a. Justify that $\Omega A = \Omega B = \Omega C = \Omega D$. b. Deduce that the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D belong to the same sphere, whose centre and radius you will specify.