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
2015 liban

5 maths questions

Q1 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
ABCDEFGH is a cube.
I is the midpoint of segment $[\mathrm{AB}]$, J is the midpoint of segment $[\mathrm{EH}]$, K is the midpoint of segment [BC] and L is the midpoint of segment [CG]. We equip space with the orthonormal coordinate system (A ; $\overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}}$).
  1. a) Prove that the line (FD) is orthogonal to the plane (IJK). b) Deduce a Cartesian equation of the plane (IJK).
  2. Determine a parametric representation of the line (FD).
  3. Let $M$ be the point of intersection of the line (FD) and the plane (IJK). Determine the coordinates of point $M$.
  4. Determine the nature of triangle IJK and calculate its area.
  5. Calculate the volume of the tetrahedron FIJK.
  6. Are the lines (IJ) and (KL) intersecting?
Q2 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
We define the sequence $\left(u_{n}\right)$ as follows: for every natural integer $n$, $u_{n} = \int_{0}^{1} \frac{x^{n}}{1+x} \mathrm{~d}x$.
  1. Calculate $u_{0} = \int_{0}^{1} \frac{1}{1+x} \mathrm{~d}x$.
  2. a) Prove that, for every natural integer $n$, $u_{n+1} + u_{n} = \frac{1}{n+1}$. b) Deduce the exact value of $u_{1}$.
  3. a) Copy and complete the algorithm below so that it displays as output the term of rank $n$ of the sequence $(u_{n})$ where $n$ is a natural integer entered as input by the user.
    Variables :$i$ and $n$ are natural integers, $u$ is a real number
    Input :Enter $n$
    Initialization :Assign to $u$ the value ...
    Processing :\begin{tabular}{l} For $i$ varying from 1 to... | Assign to $u$ the value . . .
    End For
    \hline & \hline Output : & Display $u$ \hline \end{tabular}
    b) Using this algorithm, the following table of values was obtained:
    $n$0123451050100
    $u_{n}$0,69310,30690,19310,14020,10980,09020,04750,00990,0050

    What conjectures concerning the behavior of the sequence $(u_{n})$ can be made?
  4. a) Prove that the sequence $(u_{n})$ is decreasing. b) Prove that the sequence $(u_{n})$ is convergent.
  5. We call $\ell$ the limit of the sequence $(u_{n})$. Prove that $\ell = 0$.
Q3 3 marks Tangents, normals and gradients Prove a given line is tangent to a curve View
We consider the curve $\mathscr{C}$ with equation $y = \mathrm{e}^{x}$.
For every strictly positive real $m$, we denote by $\mathscr{D}_{m}$ the line with equation $y = mx$.
  1. In this question, we choose $m = \mathrm{e}$.
    Prove that the line $\mathscr{D}_{\mathrm{e}}$, with equation $y = \mathrm{e}x$, is tangent to the curve $\mathscr{C}$ at its point with abscissa 1.
  2. Conjecture, according to the values taken by the strictly positive real $m$, the number of intersection points of the curve $\mathscr{C}$ and the line $\mathscr{D}_{m}$.
  3. Prove this conjecture.
Q4a Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
Candidates who have not followed the specialization course
In preparation for an election between two candidates A and B, a polling institute collects the voting intentions of future voters. Among the 1200 people who responded to the survey, $47\%$ state they want to vote for candidate A and the others for candidate B.
Given the profile of the candidates, the polling institute estimates that $10\%$ of people declaring they want to vote for candidate A are not telling the truth and actually vote for candidate B, while $20\%$ of people declaring they want to vote for candidate B are not telling the truth and actually vote for candidate A.
We randomly choose a person who responded to the survey and we denote:
  • A the event ``The person interviewed states they want to vote for candidate A'';
  • $B$ the event ``The person interviewed states they want to vote for candidate B'';
  • $V$ the event ``The person interviewed is telling the truth''.

  1. Construct a probability tree representing the situation.
  2. a) Calculate the probability that the person interviewed is telling the truth. b) Given that the person interviewed is telling the truth, calculate the probability that they state they want to vote for candidate A.
  3. Prove that the probability that the chosen person actually votes for candidate A is 0.529.
  4. The polling institute then publishes the following results:
    \begin{displayquote} $52.9\%$ of voters* would vote for candidate A. *estimate after adjustment, based on a survey of a representative sample of 1200 people. \end{displayquote}
    At the 95\% confidence level, can candidate A believe in their victory?
  5. To conduct this survey, the institute conducted a telephone survey at a rate of 10 calls per half-hour. The probability that a person contacted agrees to respond to this survey is 0.4. The polling institute wishes to obtain a sample of 1200 responses. What average time, expressed in hours, should the institute plan to achieve this objective?
Q4b Matrices Matrix Power Computation and Application View
Candidates who have followed the specialization course
A smoker decides to quit smoking. We choose to use the following model:
  • if they do not smoke on a given day, they do not smoke the next day with a probability of 0.9;
  • if they smoke on a given day, they smoke the next day with a probability of 0.6.

We call $p_{n}$ the probability of not smoking on the $n$-th day after their decision to quit smoking and $q_{n}$, the probability of smoking on the $n$-th day after their decision to quit smoking. We assume that $p_{0} = 0$ and $q_{0} = 1$.
  1. Calculate $p_{1}$ and $q_{1}$.
  2. We use a spreadsheet to automate the calculation of successive terms of the sequences $(p_{n})$ and $(q_{n})$. A screenshot of this spreadsheet is provided below:
    \cline{2-5} \multicolumn{1}{c|}{}ABCD
    1$n$$p_{n}$$q_{n}$
    2001
    31
    42
    53

    Column A contains the values of the natural integer $n$. What formulas can be written in cells B3 and C3 so that by copying them downward, we obtain respectively in columns B and C the successive terms of the sequences $(p_{n})$ and $(q_{n})$?
  3. We define the matrices $M$ and, for every natural integer $n$, $X_{n}$ by
    $$M = \left(\begin{array}{ll} 0,9 & 0,4 \\ 0,1 & 0,6 \end{array}\right) \quad \text{and} \quad X_{n} = \binom{p_{n}}{q_{n}}.$$
    We admit that $X_{n+1} = M \times X_{n}$ and that, for every natural integer $n$, $X_n = M^{n} \times X_{0}$. We define the matrices $A$ and $B$ by $A = \left(\begin{array}{ll} 0,8 & 0,8 \\ 0,2 & 0,2 \end{array}\right)$ and $B = \left(\begin{array}{cc} 0,2 & -0,8 \\ -0,2 & 0,8 \end{array}\right)$. a) Prove that $M = A + 0,5 B$. b) Verify that $A^{2} = A$, and that $A \times B = B \times A = \left(\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right)$.
    We admit in the following that, for every strictly positive natural integer $n$, $A^{n} = A$ and $B^{n} = B$. c) Prove that, for every natural integer $n$, $M^{n} = A + 0,5^{n} B$. d) Deduce that, for every natural integer $n$, $p_{n} = 0,8 - 0,8 \times 0,5^{n}$. e) In the long term, can we assert with certainty that the smoker will quit smoking?