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 polynesie

5 maths questions

Q1 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
In an orthonormal coordinate system in space, we consider the points $$\mathrm { A } ( 5 ; - 5 ; 2 ) , \mathrm { B } ( - 1 ; 1 ; 0 ) , \mathrm { C } ( 0 ; 1 ; 2 ) \text { and } \mathrm { D } ( 6 ; 6 ; - 1 ) .$$
  1. Determine the nature of triangle BCD and calculate its area.
  2. a. Show that the vector $\vec { n } \left( \begin{array} { c } - 2 \\ 3 \\ 1 \end{array} \right)$ is a normal vector to the plane (BCD). b. Determine a Cartesian equation of the plane (BCD).
  3. Determine a parametric representation of the line $\mathfrak { D }$ perpendicular to the plane (BCD) and passing through point A.
  4. Determine the coordinates of point H, the intersection of line $\mathcal { D }$ and plane (BCD).
  5. Determine the volume of tetrahedron ABCD.

Recall that the volume of a tetrahedron is given by the formula $\mathcal { V } = \frac { 1 } { 3 } \mathcal { B } \times h$, where $\mathcal { B }$ is the area of a base of the tetrahedron and h is the corresponding height. 6. We admit that $\mathrm { AB } = \sqrt { 76 }$ and $\mathrm { AC } = \sqrt { 61 }$.
Determine an approximate value to the nearest tenth of a degree of the angle $\widehat { \mathrm { BAC } }$.
Q2a Sequences and series, recurrence and convergence Direct term computation from recurrence View
(For candidates who have NOT followed the specialization course)
We consider the sequence $(u_n)$ defined by $$u_0 = 0 \quad \text{and, for every natural integer } n, u_{n+1} = u_n + 2n + 2.$$
  1. Calculate $u_1$ and $u_2$.
  2. We consider the following two algorithms:
    \multicolumn{2}{|l|}{Algorithm 1}\multicolumn{2}{|l|}{Algorithm 2}
    Variables :$n$ is a natural integer $u$ is a real numberVariables :$n$ is a natural integer $u$ is a real number
    Input : Processing:Input : Processing:
    \begin{tabular}{l} Enter the value of $n$ $u$ takes the value 0
    For $i$ ranging from 1 to $n$ : $u$ takes the value $u + 2i + 2$
    & &
    Enter the value of $n$ $u$ takes the value 0
    For $i$ ranging from 0 to $n - 1$ : $u$ takes the value $u + 2i + 2$
    \hline & End For & & End For \hline Output : & Display $u$ & Output : & Display $u$ \hline \end{tabular}
    Of these two algorithms, which one allows the output to display the value of $u_n$, with the value of the natural integer $n$ being entered by the user?
  3. Using the algorithm, we obtained the table and the scatter plot below where $n$ is on the horizontal axis and $u_n$ is on the vertical axis.
    $n$$u_n$
    00
    12
    26
    312
    420
    530
    642
    756
    872
    990
    10110
    11132
    12156

    a. What conjecture can be made about the direction of variation of the sequence $\left( u_n \right)$? Prove this conjecture. b. The parabolic shape of the scatter plot leads to conjecturing the existence of three real numbers $a, b$ and $c$ such that, for every natural integer $n, u_n = an^2 + bn + c$. Within the framework of this conjecture, find the values of $a, b$ and $c$ using the information provided.
  4. We define, for every natural integer $n$, the sequence $\left( v_n \right)$ by: $v_n = u_{n+1} - u_n$. a. Express $v_n$ as a function of the natural integer $n$. What is the nature of the sequence $\left( v_n \right)$? b. We define, for every natural integer $n, S_n = \sum_{k=0}^{n} v_k = v_0 + v_1 + \cdots + v_n$. Prove that, for every natural integer $n, S_n = (n+1)(n+2)$. c. Prove that, for every natural integer $n, S_n = u_{n+1} - u_0$, then express $u_n$ as a function of $n$.
Q2b Number Theory Modular Arithmetic Computation View
(For candidates who have followed the specialization course)
In this exercise, we call the number of the day of birth the rank of this day in the month and the number of the month of birth, the rank of the month in the year. For example, for a person born on May 14, the number of the day of birth is 14 and the number of the month of birth is 5.
Part A
During a performance, a magician asks spectators to perform the following calculation program (A): ``Take the number of your day of birth and multiply it by 12. Take the number of your month of birth and multiply it by 37. Add the two numbers obtained. I will then be able to give you the date of your birthday''.
A spectator announces 308 and in a few seconds, the magician declares: ``Your birthday falls on August $1^{\text{st}}$!''.
  1. Verify that for a person born on August $1^{\text{st}}$, calculation program (A) indeed gives the number 308.
  2. a. For a given spectator, we denote $j$ the number of their day of birth, $m$ that of their month of birth and $z$ the result obtained by applying calculation program (A). Express $z$ as a function of $j$ and $m$ and prove that $z$ and $m$ are congruent modulo 12. b. Then find the birthday of a spectator who obtained the number 474 by applying calculation program (A).

Part B
During another performance, the magician decides to change their calculation program. For a spectator whose day of birth number is $j$ and month of birth number is $m$, the magician asks to calculate the number $z$ defined by $z = 12j + 31m$. In the following questions, we study different methods to find the spectator's birthday.
  1. First method:
    We consider the following algorithm: \begin{verbatim} Variables: j and m are natural integers Processing : For m ranging from 1 to 12 do: For j ranging from 1 to 31 do: z takes the value 12j+31m Display z End For End For \end{verbatim} Modify this algorithm so that it displays all values of $j$ and $m$ such that $12j + 31m = 503$.
  2. Second method: a. Prove that $7m$ and $z$ have the same remainder in the Euclidean division by 12. b. For $m$ varying from 1 to 12, give the remainder of the Euclidean division of $7m$ by 12. c. Deduce the birthday of a spectator who obtained the number 503 with calculation program (B).
  3. Third method: a. Prove that the pair $(-2; 17)$ is a solution of the equation $12x + 31y = 503$. b. Deduce that if a pair of relative integers $(x; y)$ is a solution of the equation $12x + 31y = 503$, then $12(x + 2) = 31(17 - y)$. c. Determine the set of all pairs of relative integers $(x; y)$, solutions of the equation $12x + 31y = 503$. d. Prove that there exists a unique pair of relative integers $(x; y)$ such that $1 \leqslant y \leqslant 12$. Deduce the birthday of a spectator who obtained the number 503 with calculation program (B).
Q3 5 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
For each of the following five statements, indicate whether it is true or false and justify the answer. An unjustified answer is not taken into account. An absence of answer is not penalized.
  1. Zoé goes to work on foot or by car. Where she lives, it rains one day out of four. When it rains, Zoé goes to work by car in $80\%$ of cases. When it does not rain, she goes to work on foot with a probability equal to 0.6.
    Statement $\mathbf{n^o 1}$: ``Zoé uses the car one day out of two.''
  2. In the set $E$ of outcomes of a random experiment, we consider two events $A$ and $B$.
    Statement $\mathbf{n^o 2}$: ``If $A$ and $B$ are independent, then $A$ and $\bar{B}$ are also independent.''
  3. We model the waiting time, expressed in minutes, at a counter, by a random variable $T$ that follows the exponential distribution with parameter 0.7.
    Statement $\mathbf{n^o 3}$: ``The probability that a customer waits at least five minutes at this counter is approximately 0.7.''
    Statement $\mathbf{n^o 4}$: ``The average waiting time at this counter is seven minutes.''
  4. We know that $39\%$ of the French population has blood group A+. We want to know if this proportion is the same among blood donors. We survey 183 blood donors and among them, $34\%$ have blood group A+.
    Statement $\mathbf{n^o 5}$: ``We cannot reject, at the $5\%$ significance level, the hypothesis that the proportion of people with blood group A+ among blood donors is $39\%$ as in the general population.''
Q4 Applied differentiation Tangent line computation and geometric consequences View
Let $f$ and $g$ be the functions defined on $\mathbb{R}$ by $$f(x) = \mathrm{e}^{x} \quad \text{and} \quad g(x) = 2\mathrm{e}^{\frac{x}{2}} - 1.$$ We denote $\mathcal{C}_f$ and $\mathcal{C}_g$ the representative curves of functions $f$ and $g$ in an orthogonal coordinate system.
  1. Prove that the curves $\mathcal{C}_f$ and $\mathcal{C}_g$ have a common point with abscissa 0 and that at this point, they have the same tangent line $\Delta$ whose equation we will determine.
  2. Study of the relative position of curve $\mathcal{C}_g$ and line $\Delta$
    Let $h$ be the function defined on $\mathbb{R}$ by $h(x) = 2\mathrm{e}^{\frac{x}{2}} - x - 2$. a. Determine the limit of function $h$ at $-\infty$. b. Justify that, for every real $x \neq 0, h(x) = x\left(\frac{\mathrm{e}^{\frac{x}{2}}}{\frac{x}{2}} - 1 - \frac{2}{x}\right)$. Deduce the limit of function $h$ at $+\infty$. c. We denote $h'$ the derivative function of function $h$ on $\mathbb{R}$. For every real $x$, calculate $h'(x)$ and study the sign of $h'(x)$ according to the values of $x$. d. Draw the variation table of function $h$ on $\mathbb{R}$. e. Deduce that, for every real $x, 2\mathrm{e}^{\frac{x}{2}} - 1 \geqslant x + 1$. f. What can we deduce about the relative position of curve $\mathcal{C}_g$ and line $\Delta$?
  3. Study of the relative position of curves $\mathcal{C}_f$ and $\mathcal{C}_g$. a. For every real $x$, expand the expression $\left(\mathrm{e}^{\frac{x}{2}} - 1\right)^2$. b. Determine the relative position of curves $\mathcal{C}_f$ and $\mathcal{C}_g$.