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
2025 bac-spe-maths__amerique-nord_j2

5 maths questions

Q1A Binomial Distribution Compute Exact Binomial Probability View
In basketball, it is possible to score baskets worth one point, two points or three points.
The coach of a basketball team decides to study the success statistics of his players' shots. He observes that during training, when Victor attempts a three-point shot, he succeeds with a probability of 0.32. During a training session, Victor makes a series of 15 three-point shots. We assume that these shots are independent.
Let $N$ be the random variable giving the number of baskets scored. The results of the requested probabilities should be, if necessary, rounded to the nearest thousandth.
  1. We admit that the random variable $N$ follows a binomial distribution. Specify its parameters.
  2. Calculate the probability that Victor succeeds in exactly 4 baskets during this series.
  3. Determine the probability that Victor succeeds in at most 6 baskets during this series.
  4. Determine the expected value of the random variable $N$.
  5. Let $T$ be the random variable giving the number of points scored after this series of shots. a. Express $T$ as a function of $N$. b. Deduce the expected value of the random variable $T$. Give an interpretation of this value in the context of the exercise. c. Calculate $P ( 12 \leqslant T \leqslant 18 )$.
Q1B Central limit theorem View
Let $X$ be the random variable giving the number of points scored by Victor during a match.
We admit that the expected value $E ( X ) = 22$ and the variance $V ( X ) = 65$. Victor plays $n$ matches, where $n$ is a strictly positive integer. Let $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ be the random variables giving the number of points scored during the $1 ^ { \text {st} } , 2 ^ { \mathrm { nd } } , \ldots , n$-th matches. We admit that the random variables $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are independent and follow the same distribution as $X$. We set $\quad M _ { n } = \frac { X _ { 1 } + X _ { 2 } + \ldots + X _ { n } } { n }$.
  1. In this question, we take $n = 50$. a. What does the random variable $M _ { 50 }$ represent? b. Determine the expected value and variance of $M _ { 50 }$. c. Prove that $P \left( \left| M _ { 50 } - 22 \right| \geqslant 3 \right) \leqslant \frac { 13 } { 90 }$. d. Deduce that the probability of the event ``$19 < M _ { 50 } < 25$'' is strictly greater than 0.85.
  2. Indicate, by justifying, whether the following statement is true or false: ``There is no natural number $n$ such that $P \left( \left| M _ { n } - 22 \right| \geqslant 3 \right) < 0.01$ ''.
Q2 5 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
One of the objectives of this exercise is to determine an approximation of the real number $\ln ( 2 )$, using one of the methods of the English mathematician Henry Briggs in the XVI${}^{\text{th}}$ century.
We denote by ( $u _ { n }$ ) the sequence defined by:
$$u _ { 0 } = 2 \quad \text { and, for every natural number } n , \quad u _ { n + 1 } = \sqrt { u _ { n } }$$
Part A
  1. a. Give the exact value of $u _ { 1 }$ and $u _ { 2 }$. b. Make a conjecture, using a calculator, about the direction of variation and the possible limit of the sequence.
  2. a. Show by induction that for every natural number $n , \quad 1 \leqslant u _ { n + 1 } \leqslant u _ { n }$. b. Deduce that the sequence ( $u _ { n }$ ) is convergent. c. Solve in the interval [ $0 ; + \infty$ [ the equation $\sqrt { x } = x$. d. Determine, by justifying, the limit of the sequence $\left( u _ { n } \right)$.

Part B
We denote by ( $\nu _ { n }$ ) the sequence defined for every natural number $n$ by $\nu _ { n } = \ln \left( u _ { n } \right)$.
  1. a. Prove that the sequence ( $v _ { n }$ ) is geometric with common ratio $\frac { 1 } { 2 }$. b. Express $v _ { n }$ as a function of $n$, for every natural number $n$. c. Deduce that, for every natural number $n , \quad \ln ( 2 ) = 2 ^ { n } \ln \left( u _ { n } \right)$.
  2. We have traced below in an orthonormal coordinate system the curve $\mathscr { C }$ of the function ln and the tangent T to the curve $\mathscr { C }$ at the point with abscissa 1. An equation of the line T is $y = x - 1$. The points $\mathrm { A } _ { 0 } , \mathrm {~A} _ { 1 } , \mathrm {~A} _ { 2 }$ have abscissas $u _ { 0 } , u _ { 1 }$ and $u _ { 2 }$ respectively and ordinate 0. We decide to take $x - 1$ as an approximation of $\ln ( x )$ when $x$ belongs to the interval $] 0,99 ; 1,01 [$. a. Using a calculator, determine the smallest natural number $k$ such that $u _ { k }$ belongs to the interval $] 0,99 ; 1,01 [$ and give an approximate value of $u _ { k }$ to $10 ^ { - 5 }$ near. b. Deduce an approximation of $\ln \left( u _ { k } \right)$. c. Deduce from questions 1.c. and 2.b. of Part B an approximation of $\ln ( 2 )$.
  3. We generalize the previous method to any real number $a$ strictly greater than 1.
    Copy and complete the algorithm below so that the call Briggs(a) returns an approximation of $\ln ( a )$.
    We recall that the instruction in Python language sqrt (a) corresponds to $\sqrt { a }$.
    \begin{verbatim} from math import* def Briggs(a): n = 0 while a >= 1.01: a = sqrt(a) n = n+1 L =... return L \end{verbatim}
Q3 Vectors: Lines & Planes True/False or Verify a Given Statement View
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
PART A
ABCDEFGH is a cube with edge length 1. The points I, J, K, L and M are the midpoints respectively of the edges [AB], [BF], [AE], [CD] and [DH].
Statement 1: $\ll \overrightarrow { \mathrm { JH } } = 2 \overrightarrow { \mathrm { BI } } + \overrightarrow { \mathrm { DM } } - \overrightarrow { \mathrm { CB } } \gg$ Statement 2: ``The triplet of vectors ( $\overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AH } } , \overrightarrow { \mathrm { AG } }$ ) is a basis of space.'' Statement 3: ``$\overrightarrow { \mathrm { IB } } \cdot \overrightarrow { \mathrm { LM } } = - \frac { 1 } { 4 }$.''
PART B
In space equipped with an orthonormal coordinate system, we consider:
  • the plane $\mathscr { P }$ with Cartesian equation $2 x - y + 3 z + 6 = 0$
  • the points $\mathrm { A } ( 2 ; 0 ; - 1 )$ and $\mathrm { B } ( 5 ; - 3 ; 7 )$

Statement 4: ``The plane $\mathscr { P }$ and the line ( AB ) are parallel.'' Statement 5: ``The plane $\mathscr { P } ^ { \prime }$ parallel to $\mathscr { P }$ passing through B has Cartesian equation $- 2 x + y - 3 z + 34 = 0$'' Statement 6: ``The distance from point A to plane $\mathscr { P }$ is equal to $\frac { \sqrt { 14 } } { 2 }$.'' We denote by (d) the line with parametric representation
$$\left\{ \begin{array} { r l } x & = - 12 + 2 k \\ y & = 6 \\ z & = 3 - 5 k \end{array} , \text { where } k \in \mathbb { R } \right.$$
Statement 7: ``The lines (AB) and (d) are not coplanar.''
Q4 5 marks Integration by Parts Integration by Parts within Function Analysis View
We denote by $f$ the function defined on the interval $[ 0 ; \pi ]$ by
$$f ( x ) = \mathrm { e } ^ { x } \sin ( x )$$
We denote by $\mathscr { C } _ { f }$ the representative curve of $f$ in a coordinate system.
PART A
  1. a. Prove that for every real number $x$ in the interval $[ 0 ; \pi ]$, $$f ^ { \prime } ( x ) = \mathrm { e } ^ { x } [ \sin ( x ) + \cos ( x ) ]$$ b. Justify that the function $f$ is strictly increasing on the interval $\left[ 0 ; \frac { \pi } { 2 } \right]$
  2. a. Determine an equation of the tangent $T$ to the curve $\mathscr { C } _ { f }$ at the point with abscissa 0. b. Prove that the function $f$ is convex on the interval $\left[ 0 ; \frac { \pi } { 2 } \right]$. c. Deduce that for every real number $x$ in the interval $\left[ 0 ; \frac { \pi } { 2 } \right] , \mathrm { e } ^ { x } \sin ( x ) \geqslant x$.
  3. Justify that the point with abscissa $\frac { \pi } { 2 }$ of the representative curve of the function $f$ is an inflection point.

PART B
We denote
$$I = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { x } \sin ( x ) \mathrm { d } x \text { and } \quad J = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { x } \cos ( x ) \mathrm { d } x$$
  1. By integrating by parts the integral $I$ in two different ways, establish the following two relations: $$I = 1 + J \quad \text { and } \quad I = \mathrm { e } ^ { \frac { \pi } { 2 } } - J$$
  2. Deduce that $I = \frac { 1 + \mathrm { e } ^ { \frac { \pi } { 2 } } } { 2 }$.
  3. We denote by $g$ the function defined on $\mathbb { R }$ by $g ( x ) = x$. Calculate the exact value of the area of the shaded region situated between the curves $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ and the lines with equations $x = 0$ and $x = \frac { \pi } { 2 }$.