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
2013 metropole

5 maths questions

Q1 4 marks Conditional Probability Bayes' Theorem with Production/Source Identification View
A garden centre sells young tree saplings that come from three horticulturists: $35 \%$ of the plants come from horticulturist $\mathrm { H } _ { 1 } , 25 \%$ from horticulturist $\mathrm { H } _ { 2 }$ and the rest from horticulturist $\mathrm { H } _ { 3 }$. Each horticulturist supplies two categories of trees: conifers and deciduous trees. The delivery from horticulturist $\mathrm { H } _ { 1 }$ contains $80 \%$ conifers while that from horticulturist $\mathrm { H } _ { 2 }$ contains only $50 \%$ and that from horticulturist $\mathrm { H } _ { 3 }$ only $30 \%$.
We consider the following events: $H _ { 1 }$ : ``the tree chosen was purchased from horticulturist $\mathrm { H } _ { 1 }$'', $H _ { 2 }$ : ``the tree chosen was purchased from horticulturist $\mathrm { H } _ { 2 }$'', $H _ { 3 }$ : ``the tree chosen was purchased from horticulturist $\mathrm { H } _ { 3 }$'', $C$ : ``the tree chosen is a conifer'', $F$ : ``the tree chosen is a deciduous tree''.
  1. The garden centre manager chooses a tree at random from his stock.
    1. [a.] Construct a probability tree representing the situation.
    2. [b.] Calculate the probability that the tree chosen is a conifer purchased from horticulturist $\mathrm { H } _ { 3 }$.
    3. [c.] Justify that the probability of event $C$ is equal to 0.525.
    4. [d.] The tree chosen is a conifer. What is the probability that it was purchased from horticulturist $\mathrm { H } _ { 1 }$? Round to $10 ^ { - 3 }$.
  2. A random sample of 10 trees is chosen from the stock of this garden centre. We assume that this stock is large enough that this choice can be treated as sampling with replacement of 10 trees from the stock. Let $X$ be the random variable that gives the number of conifers in the chosen sample.
    1. [a.] Justify that $X$ follows a binomial distribution and specify its parameters.
    2. [b.] What is the probability that the sample contains exactly 5 conifers? Round to $10 ^ { - 3 }$.
    3. [c.] What is the probability that this sample contains at least two deciduous trees? Round to $10 ^ { - 3 }$.
Q2 Stationary points and optimisation Determine parameters from given extremum conditions View
In the plane with an orthonormal coordinate system ($\mathrm { O } ; \vec { \imath } , \vec { \jmath }$), the representative curve $\mathscr { C }$ of a function $f$ defined and differentiable on the interval $] 0 ; + \infty [$ is given.
We have the following information:
  • the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ have coordinates $(1,0)$, $(1,2)$, $(0,2)$ respectively;
  • the curve $\mathscr { C }$ passes through point B and the line (BC) is tangent to $\mathscr { C }$ at B;
  • there exist two positive real numbers $a$ and $b$ such that for every strictly positive real number $x$, $$f ( x ) = \frac { a + b \ln x } { x } .$$

    1. [a.] Using the graph, give the values of $f ( 1 )$ and $f ^ { \prime } ( 1 )$.
    2. [b.] Verify that for every strictly positive real number $x , f ^ { \prime } ( x ) = \frac { ( b - a ) - b \ln x } { x ^ { 2 } }$.
    3. [c.] Deduce the real numbers $a$ and $b$.
    1. [a.] Justify that for every real number $x$ in the interval $] 0 , + \infty [$, $f ^ { \prime } ( x )$ has the same sign as $- \ln x$.
    2. [b.] Determine the limits of $f$ at 0 and at $+ \infty$. We may note that for every strictly positive real number $x$, $f ( x ) = \frac { 2 } { x } + 2 \frac { \ln x } { x }$.
    3. [c.] Deduce the table of variations of the function $f$.
    1. [a.] Prove that the equation $f ( x ) = 1$ has a unique solution $\alpha$ on the interval $] 0,1 ]$.
    2. [b.] By similar reasoning, we prove that there exists a unique real number $\beta$ in the interval $] 1 , + \infty [$ such that $f ( \beta ) = 1$. Determine the integer $n$ such that $n < \beta < n + 1$.
  4. The following algorithm is given.
    \begin{tabular}{l} Variables:
    $a , b$ and $m$ are real numbers.
    Initialization:
    Assign to $a$ the value 0.
    Assign to $b$ the value 1.
    Processing:
    While $b - a > 0.1$
    Assign to $m$ the value $\frac { 1 } { 2 } ( a + b )$.
    If $f ( m ) < 1$ then Assign to $a$ the value $m$. Otherwise Assign to $b$ the value $m$.
    End If.
    End While.
    Output:
    Display $a$.
    Display $b$.
    \hline \end{tabular}
    1. [a.] Run this algorithm by completing the table below, which you will copy onto your answer sheet.
      step 1step 2step 3step 4step 5
      $a$0
      $b$1
      $b - a$
      $m$

    2. [b.] What do the values displayed by this algorithm represent?
    3. [c.] Modify the algorithm above so that it displays the two bounds of an interval containing $\beta$ with amplitude $10 ^ { - 1 }$.
  5. The purpose of this question is to prove that the curve $\mathscr { C }$ divides the rectangle OABC into two regions of equal area.
    1. [a.] Justify that this amounts to proving that $\int _ { \frac { 1 } { \mathrm { e } } } ^ { 1 } f ( x ) \mathrm { d } x = 1$.
    2. [b.] By noting that the expression of $f ( x )$ can be written as $\frac { 2 } { x } + 2 \times \frac { 1 } { x } \times \ln x$, complete the proof.
Q3 4 marks Complex Numbers Argand & Loci True/False or Multiple-Statement Verification View
For each of the four propositions below, indicate whether it is true or false and justify your chosen answer. One point is awarded for each correct answer with proper justification. An answer without justification is not taken into account. No answer is not penalized.
  1. Proposition 1: In the plane with an orthonormal coordinate system, the set of points $M$ whose affix $z$ satisfies the equality $| z - \mathrm { i } | = | z + 1 |$ is a line.
  2. Proposition 2: The complex number $( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 4 }$ is a real number.
  3. Let ABCDEFGH be a cube. Proposition 3: The lines (EC) and (BG) are orthogonal.
  4. Space is equipped with an orthonormal coordinate system ($\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$). Let the plane $\mathscr { P }$ with Cartesian equation $x + y + 3z + 4 = 0$. We denote S the point with coordinates $( 1 , - 2 , - 2 )$. Proposition 4: The line passing through S and perpendicular to the plane $\mathscr { P }$ has parametric representation $\left\{ \begin{array} { l } x = 2 + t \\ y = - 1 + t \\ z = 1 + 3 t \end{array} , t \in \mathbf { R } \right.$.
Q4a Sequences and series, recurrence and convergence Conjecture from numerical data or computation View
(For candidates who have not followed the specialization course) Let the numerical sequence $(u _ { n })$ defined on $\mathbf{N}$ by: $$u _ { 0 } = 2 \quad \text { and for every natural number } n , u _ { n + 1 } = \frac { 2 } { 3 } u _ { n } + \frac { 1 } { 3 } n + 1$$
    1. [a.] Calculate $u _ { 1 } , u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$. Approximate values to $10 ^ { - 2 }$ may be given.
    2. [b.] Form a conjecture about the monotonicity of this sequence.
    1. [a.] Prove that for every natural number $n$, $$u _ { n } \leqslant n + 3$$
    2. [b.] Prove that for every natural number $n$, $$u _ { n + 1 } - u _ { n } = \frac { 1 } { 3 } \left( n + 3 - u _ { n } \right)$$
    3. [c.] Deduce a validation of the previous conjecture.
  3. We denote by $\left( v _ { n } \right)$ the sequence defined on $\mathbf { N }$ by $v _ { n } = u _ { n } - n$.
    1. [a.] Prove that the sequence $\left( v _ { n } \right)$ is a geometric sequence with common ratio $\frac { 2 } { 3 }$.
    2. [b.] Deduce that for every natural number $n$, $$u _ { n } = 2 \left( \frac { 2 } { 3 } \right) ^ { n } + n$$
    3. [c.] Determine the limit of the sequence $(u _ { n })$.
  4. For every non-zero natural number $n$, we set: $$S _ { n } = \sum _ { k = 0 } ^ { n } u _ { k } = u _ { 0 } + u _ { 1 } + \ldots + u _ { n } \quad \text { and } \quad T _ { n } = \frac { S _ { n } } { n ^ { 2 } } .$$
    1. [a.] Express $S _ { n }$ as a function of $n$.
    2. [b.] Determine the limit of the sequence $(T _ { n })$.
Q4b 5 marks Matrices Matrix Power Computation and Application View
(For candidates who have followed the specialization course) We study the population of an imaginary region. On January 1, 2013, this region had 250,000 inhabitants, of which $70 \%$ lived in the countryside and $30 \%$ in the city. Examination of statistical data collected over several years leads to the choice of modelling the population evolution for the coming years as follows:
  • the total population is globally constant,
  • each year, $5 \%$ of those living in the city decide to move to the countryside and $1 \%$ of those living in the countryside choose to move to the city.
For every natural number $n$, we denote $v _ { n }$ the number of inhabitants of this region living in the city on January 1 of the year $(2013 + n)$ and $c _ { n }$ the number of those living in the countryside on the same date.
  1. For every natural number $n$, express $v _ { n + 1 }$ and $c _ { n + 1 }$ as functions of $v _ { n }$ and $c _ { n }$.
  2. Let the matrix $A = \left( \begin{array} { l l } 0{,}95 & 0{,}01 \\ 0{,}05 & 0{,}99 \end{array} \right)$.