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 amerique-sud

7 maths questions

Q1A Normal Distribution Finding Unknown Standard Deviation from a Given Probability Condition View
A football is compliant with regulations if it meets, depending on its size, two conditions simultaneously (on its mass and on its circumference). In particular, a standard-sized football is compliant with regulations when its mass, expressed in grams, belongs to the interval [410;450] and its circumference, expressed in centimetres, belongs to the interval [68;70].
  1. Let $X$ denote the random variable which, for each standard-sized football chosen at random in the company, associates its mass in grams. It is admitted that $X$ follows a normal distribution with mean 430 and standard deviation 10. Determine an approximate value to $10 ^ { - 3 }$ of the probability $P ( 410 \leqslant X \leqslant 450 )$.
  2. Let $Y$ denote the random variable which, for each standard-sized football chosen at random in the company, associates its circumference in centimetres. It is admitted that $Y$ follows a normal distribution with mean 69 and standard deviation $\sigma$. Determine the value of $\sigma$, to the nearest hundredth, knowing that $97 \%$ of standard-sized footballs have a circumference compliant with regulations. You may use the following result: when $Z$ is a random variable that follows the standard normal distribution, then $P ( - \beta \leqslant Z \leqslant \beta ) = 0,97$ for $\beta \approx 2,17$.
Q1B Modelling and Hypothesis Testing View
The company claims that $98 \%$ of its standard-sized footballs are compliant with regulations. A check is then carried out on a sample of 250 standard-sized footballs. It is found that 233 of them are compliant with regulations. Does the result of this check call into question the company's claim? Justify your answer. (You may use the confidence interval)
Q1C 6 marks Conditional Probability Bayes' Theorem with Production/Source Identification View
The company produces $40 \%$ of small-sized footballs and $60 \%$ of standard-sized footballs. It is admitted that $2 \%$ of small-sized footballs and $5 \%$ of standard-sized footballs do not comply with regulations. A football is chosen at random in the company.
Consider the events: $A$ : ``the football is small-sized'', $B$ : ``the football is standard-sized'', $C$ : ``the football complies with regulations'' and $\bar { C }$, the opposite event of C.
  1. Represent this random experiment using a probability tree.
  2. Calculate the probability that the football is small-sized and complies with regulations.
  3. Show that the probability of event $C$ is equal to 0.962.
  4. The football chosen does not comply with regulations. What is the probability that this football is small-sized? Round the result to $10 ^ { - 3 }$.
Q2 Vectors: Lines & Planes MCQ: Identify Correct Equation or Representation View
This exercise is a multiple choice questionnaire. No justification is required. For each question, only one of the four propositions is correct. Each correct answer earns one point. An incorrect answer or no answer does not deduct any points.
  1. In an orthonormal coordinate system in space, consider the points $\mathrm { A } ( 2 ; 5 ; - 1 ) , \mathrm { B } ( 3 ; 2 ; 1 )$ and $\mathrm { C } ( 1 ; 3 ; - 2 )$. Triangle ABC is: a. right-angled and not isosceles b. isosceles and not right-angled c. right-angled and isosceles d. equilateral
  2. In an orthonormal coordinate system in space, consider the plane $P$ with equation $2 x - y + 3 z - 1 = 0$ and the point $\mathrm { A } ( 2 ; 5 ; - 1 )$. A parametric representation of the line $d$, perpendicular to plane $P$ and passing through A is: a. $\left\{ \begin{aligned} x & = 2 + 2 t \\ y & = 5 + t \\ z & = - 1 + 3 t \end{aligned} \right.$ b. $\left\{ \begin{aligned} x & = 2 + 2 t \\ y & = - 1 + 5 t \\ z & = 3 - t \end{aligned} \right.$ c. $\left\{ \begin{aligned} x & = 6 - 2 t \\ y & = 3 + t \\ z & = 5 - 3 t \end{aligned} \right.$ d. $\left\{ \begin{aligned} x & = 1 + 2 t \\ y & = 4 - t \\ z & = - 2 + 3 t \end{aligned} \right.$
  3. Let A and B be two distinct points in the plane. The set of points $M$ in the plane such that $\overrightarrow { M A } \cdot \overrightarrow { M B } = 0$ is: a. the empty set b. the perpendicular bisector of segment [AB] c. the circle with diameter $[ \mathrm { AB } ]$ d. the line (AB)
  4. The figure below represents a cube ABCDEFGH. Points I and J are the midpoints of edges $[ \mathrm { GH } ]$ and $[ \mathrm { FG } ]$ respectively. Points M and N are the centres of faces ABFE and BCGF respectively. Lines (IJ) and (MN) are: a. perpendicular b. intersecting, non-perpendicular c. orthogonal d. parallel
Q3 (non-specialization) 5 marks Sequences and series, recurrence and convergence Conjecture from numerical data or computation View
Consider the numerical sequence $(u_n)$ defined on $\mathbb{N}$ by:
$$u _ { 0 } = 2 \quad \text { and for every natural number } n , \quad u _ { n + 1 } = - \frac { 1 } { 2 } u _ { n } ^ { 2 } + 3 u _ { n } - \frac { 3 } { 2 } .$$
Part A: Conjecture
  1. Calculate the exact values, given as irreducible fractions, of $u _ { 1 }$ and $u _ { 2 }$.
  2. Give an approximate value to $10 ^ { - 5 }$ of the terms $u _ { 3 }$ and $u _ { 4 }$.
  3. Conjecture the direction of variation and the convergence of the sequence $(u_n)$.

Part B: Validation of conjectures
Consider the numerical sequence $\left( v _ { n } \right)$ defined for every natural number $n$, by: $v _ { n } = u _ { n } - 3$.
  1. Show that, for every natural number $n , v _ { n + 1 } = - \frac { 1 } { 2 } v _ { n } ^ { 2 }$.
  2. Prove by induction that, for every natural number $n , - 1 \leqslant v _ { n } \leqslant 0$.
  3. a. Prove that, for every natural number $n , v _ { n + 1 } - v _ { n } = - v _ { n } \left( \frac { 1 } { 2 } v _ { n } + 1 \right)$. b. Deduce the direction of variation of the sequence $\left( v _ { n } \right)$.
  4. Why can we then affirm that the sequence $\left( v _ { n } \right)$ converges?
  5. Let $\ell$ denote the limit of the sequence $(v_n)$. It is admitted that $\ell$ belongs to the interval $[ - 1 ; 0 ]$ and satisfies the equality: $\ell = - \frac { 1 } { 2 } \ell ^ { 2 }$. Determine the value of $\ell$.
  6. Are the conjectures made in Part A validated?
Q3 (specialization) 5 marks Matrices Matrix Power Computation and Application View
A city has a bike-sharing network where two stations A and B are located at the top of a hill. It is admitted that no bikes from other stations arrive at stations A and B.
It is observed that for each hour $n$ on average:
  • $20 \%$ of the bikes present at hour $n - 1$ at station A are still at this station. $60 \%$ of the bikes present at hour $n - 1$ at station A are at station B and the others are in other stations of the network or in circulation.
  • $10 \%$ of the bikes present at hour $n - 1$ at station B are at station $\mathrm { A } , 30 \%$ are still at station B and the others are in other stations of the network or in circulation.
  • At the beginning of the day, station A has 50 bikes, station B has 60 bikes.

Part A
After $n$ hours, let $a _ { n }$ denote the average number of bikes present at station A and $b _ { n }$ the average number of bikes present at station B. Let $U _ { n }$ denote the column matrix $\binom { a _ { n } } { b _ { n } }$ and thus $U _ { 0 } = \binom { 50 } { 60 }$.
  1. Determine the matrix $M$ such that $U _ { n + 1 } = M \times U _ { n }$.
  2. Determine $U _ { 1 }$ and $U _ { 2 }$.
  3. After how many hours is there only one bike left in station A?

Part B
The service decides to study the effects of a supply of stations A and B consisting of bringing 30 bikes to station A and 10 bikes to station B after each hour of operation.
After $n$ hours, let $\alpha _ { n }$ denote the average number of bikes present at station A and $\beta _ { n }$ the average number of bikes present at station B. Let $V _ { n }$ denote the column matrix $\binom { \alpha _ { n } } { \beta _ { n } }$ and $V _ { 0 } = \binom { 50 } { 60 }$. Under these conditions $V _ { n + 1 } = M \times V _ { n } + R$ with $R = \binom { 30 } { 10 }$.
  1. Let $I$ denote the matrix $\left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$ and $N$ the matrix $I - M$. a. Let $V$ denote a column matrix with two rows. Show that $V = M \times V + R$ is equivalent to $N \times V = R$. b. It is admitted that $N$ is an invertible matrix and that $N ^ { - 1 } = \left( \begin{array} { l l } 1,4 & 0,2 \\ 1,2 & 1,6 \end{array} \right)$. Deduce that $V = \binom { 44 } { 52 }$.
  2. For every natural number $n$, let $W _ { n } = V _ { n } - V$. a. Show that $W _ { n + 1 } = M \times W _ { n }$. b. It is admitted that: for every natural number $n , W _ { n } = M ^ { n } \times W _ { 0 }$, and $$\text{for every natural number } n \geqslant 1 , M ^ { n } = \frac { 1 } { 2 ^ { n - 1 } } \left( \begin{array} { l l } 0,2 & 0,1 \\ 0,6 & 0,3 \end{array} \right) .$$ Calculate, for every natural number $n \geqslant 1 , V _ { n }$ as a function of $n$. c. Does the average number of bikes present in stations A and B tend to stabilize?
Q4 Applied differentiation Applied modeling with differentiation View
It is desired to create a gate. Each leaf measures 2 metres wide.
Part A: modelling the upper part of the gate
The upper edge of the right leaf of the gate is modelled with a function $f$ defined on the interval [0;2] by
$$f ( x ) = \left( x + \frac { 1 } { 4 } \right) \mathrm { e } ^ { - 4 x } + b$$
where $b$ is a real number. Let $f ^ { \prime }$ denote the derivative function of $f$ on the interval $[ 0 ; 2 ]$.
  1. a. Calculate $f ^ { \prime } ( x )$, for every real $x$ belonging to the interval $[ 0 ; 2 ]$. b. Deduce the direction of variation of the function $f$ on the interval $[ 0 ; 2 ]$.
  2. Determine the number $b$ so that the maximum height of the gate is equal to $1{,}5 \mathrm{~m}$.

In the following, the function $f$ is defined on the interval $[ 0 ; 2 ]$ by
$$f ( x ) = \left( x + \frac { 1 } { 4 } \right) \mathrm { e } ^ { - 4 x } + \frac { 5 } { 4 }$$
Part B: determination of an area
Each leaf is made using a metal plate. We want to calculate the area of each plate, knowing that the lower edge of the leaf is at $0{,}05 \mathrm{~m}$ height from the ground.
  1. Show that the function $F$ defined on the interval $[ 0 ; 2 ]$ by $$F ( x ) = \left( - \frac { 1 } { 4 } x - \frac { 5 } { 16 } \right) \mathrm { e } ^ { - 4 x } + \frac { 5 } { 4 } x$$ is an antiderivative of $f$ on the interval $[ 0 ; 2 ]$.
  2. Calculate the area, in square metres, of each metal plate.