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__polynesie_j2

4 maths questions

Q1 7 marks Laws of Logarithms Analyze a Logarithmic Function (Limits, Monotonicity, Zeros, Extrema) View
This exercise is a multiple choice questionnaire. For each of the six following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers, or no answer to a question earns no points and loses no points.
  1. Consider the function $f$ defined and differentiable on $] 0 ; + \infty [$ by: $$f ( x ) = x \ln ( x ) - x + 1 .$$ Among the four expressions below, which one is the derivative of $f$?
    a. $\ln ( x )$b. $\frac { 1 } { x } - 1$c. $\ln ( x ) - 2$d. $\ln ( x ) - 1$

  2. Consider the function $g$ defined on $] 0$; $+ \infty \left[ \text{ by } g ( x ) = x ^ { 2 } [ 1 - \ln ( x ) ] \right.$. Among the four statements below, which one is correct?
    a. $\lim _ { x \rightarrow 0 } g ( x ) = + \infty$b. $\lim _ { x \rightarrow 0 } g ( x ) = - \infty$c. $\lim _ { x \rightarrow 0 } g ( x ) = 0$\begin{tabular}{ l } d. The function $g$
    does not have a li-
    mit at 0.
    \hline \end{tabular}
  3. Consider the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x ^ { 3 } - 0,9 x ^ { 2 } - 0,1 x$. The number of solutions to the equation $f ( x ) = 0$ on $\mathbb { R }$ is:
    a. 0b. 1c. 2d. 3

  4. If $H$ is an antiderivative of a function $h$ defined and continuous on $\mathbb { R }$, and if $k$ is the function defined on $\mathbb { R }$ by $k ( x ) = h ( 2 x )$, then an antiderivative $K$ of $k$ is defined on $\mathbb { R }$ by:
    a. $K ( x ) = H ( 2 x )$b. $K ( x ) = 2 H ( 2 x )$c. $K ( x ) = \frac { 1 } { 2 } H ( 2 x )$d. $K ( x ) = 2 H ( x )$

  5. The equation of the tangent line at the point with abscissa 1 to the curve of the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x \mathrm { e } ^ { x }$ is:
    a. $y = \mathrm { e } x + \mathrm { e }$b. $y = 2 \mathrm { e } x - \mathrm { e }$c. $y = 2 \mathrm { e } x + \mathrm { e }$d. $y = \mathrm { e } x$

  6. The integers $n$ that are solutions to the inequality $( 0,2 ) ^ { n } < 0,001$ are all integers $n$ such that:
    a. $n \leqslant 4$b. $n \leqslant 5$c. $n \geqslant 4$d. $n \geqslant 5$

Q2 7 marks Conditional Probability Bayes' Theorem with Production/Source Identification View
Customs authorities are interested in imports of headphones bearing the logo of a certain brand. Customs seizures allow them to estimate that:
  • $20 \%$ of headphones bearing this brand's logo are counterfeits;
  • $2 \%$ of non-counterfeit headphones have a design defect;
  • $10 \%$ of counterfeit headphones have a design defect.
The fraud agency randomly orders a headphone displaying the brand's logo from an internet site. Consider the following events:
  • C: ``the headphone is counterfeit'';
  • $D$: ``the headphone has a design defect'';
  • $\bar { C }$ and $\bar { D }$ denote respectively the complementary events of $C$ and $D$.
Throughout the exercise, probabilities will be rounded to $10 ^ { - 3 }$ if necessary.
Part 1
  1. Calculate $P ( C \cap D )$. You may use a probability tree.
  2. Prove that $P ( D ) = 0,036$.
  3. The headphone has a defect. What is the probability that it is counterfeit?

Part 2
We order $n$ headphones bearing this brand's logo. We treat this experiment as a random draw with replacement. Let $X$ be the random variable giving the number of headphones with a design defect in this batch.
  1. In this question, $n = 35$. a. Justify that $X$ follows a binomial distribution $\mathscr { B } ( n , p )$ where $n = 35$ and $p = 0,036$. b. Calculate the probability that among the ordered headphones, exactly one has a design defect. c. Calculate $P ( X \leqslant 1 )$.
  2. In this question, $n$ is not fixed. What is the minimum number of headphones to order so that the probability that at least one headphone has a defect is greater than 0.99?
Q3 7 marks Sequences and series, recurrence and convergence Applied/contextual sequence problem View
At the beginning of 2021, a bird colony had 40 individuals. Observation leads to modeling the population evolution by the sequence $(u_n)$ defined for all natural integers $n$ by: $$\begin{cases} u _ { 0 } & = 40 \\ u _ { n + 1 } & = 0,008 u _ { n } \left( 200 - u _ { n } \right) \end{cases}$$ where $u _ { n }$ denotes the number of individuals at the beginning of the year $( 2021 + n )$.
  1. Give an estimate, according to this model, of the number of birds in the colony at the beginning of 2022.

Consider the function $f$ defined on the interval $[ 0 ; 100 ]$ by $f ( x ) = 0,008 x ( 200 - x )$.
    \setcounter{enumi}{1}
  1. Solve in the interval $[ 0 ; 100 ]$ the equation $f ( x ) = x$.
  2. a. Prove that the function $f$ is increasing on the interval $[ 0 ; 100 ]$ and draw its variation table. b. By noting that, for all natural integers $n , u _ { n + 1 } = f \left( u _ { n } \right)$, prove by induction that, for all natural integers $n$: $$0 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 100$$ c. Deduce that the sequence $(u_n)$ is convergent. d. Determine the limit $\ell$ of the sequence $(u_n)$. Interpret the result in the context of the exercise.
  3. Consider the following algorithm: \begin{verbatim} def seuil(p) : n=0 u=40 while u < p: n =n+1 u=0.008*u*(200-u) return(n+2021) \end{verbatim} The execution of seuil(100) returns no value. Explain why using question 3.
Q4 7 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Consider the cube ABCDEFGH with edge length 1. The space is equipped with the orthonormal frame $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. Point I is the midpoint of segment $[\mathrm{EF}]$, K is the center of square ADHE, and O is the midpoint of segment [AG].
The goal of the exercise is to calculate in two different ways the distance from point B to the plane (AIG).
Part 1. First method
  1. Give, without justification, the coordinates of points $\mathrm{A}$, $\mathrm{B}$, and G.
We admit that points I and K have coordinates $\mathrm{I}\left(\frac{1}{2}; 0; 1\right)$ and $\mathrm{K}\left(0; \frac{1}{2}; \frac{1}{2}\right)$.
    \setcounter{enumi}{1}
  1. Prove that the line (BK) is orthogonal to the plane (AIG).
  2. Verify that a Cartesian equation of the plane (AIG) is: $2x - y - z = 0$.
  3. Give a parametric representation of the line (BK).
  4. Deduce that the orthogonal projection L of point B onto the plane (AIG) has coordinates $\mathrm{L}\left(\frac{1}{3}; \frac{1}{3}; \frac{1}{3}\right)$.
  5. Determine the distance from point B to the plane (AIG).

Part 2. Second method
Recall that the volume $V$ of a pyramid is given by the formula $V = \frac{1}{3} \times b \times h$, where $b$ is the area of a base and $h$ is the height associated with this base.
  1. a. Justify that in the tetrahedron $\mathrm{ABIG}$, $[\mathrm{GF}]$ is the height relative to the base AIB. b. Deduce the volume of the tetrahedron ABIG.
  2. We admit that $\mathrm{AI} = \mathrm{IG} = \frac{\sqrt{5}}{2}$ and that $\mathrm{AG} = \sqrt{3}$. Prove that the area of the isosceles triangle AIG is equal to $\frac{\sqrt{6}}{4}$ square units.
  3. Deduce the distance from point B to the plane (AIG).