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
2019 polynesie

5 maths questions

Q1 5 marks Exponential Distribution View
5 points
The probabilities requested should be rounded to 0.01.
A shopkeeper has just equipped himself with an Italian ice cream dispenser.
  1. The duration, in months, of operation without breakdown of his Italian ice cream dispenser is modelled by a random variable $X$ which follows an exponential distribution with parameter $\lambda$ where $\lambda$ is a strictly positive real number (recall that the density function $f$ of the exponential distribution is given on $\left[ 0 ; + \infty \left[ \text{ by } f ( x ) = \lambda \mathrm { e } ^ { - \lambda x } \right. \right.$).

The seller of the device assures that the average duration of operation without breakdown of this type of dispenser, that is to say the mathematical expectation of $X$, is 10 months. a. Justify that $\lambda = 0.1$. b. Calculate the probability that the Italian ice cream dispenser has experienced no breakdown during the first six months. c. Given that the dispenser has experienced no breakdown during the first six months, what is the probability that it experiences none until the end of the first year? Justify. d. The shopkeeper will replace his Italian ice cream dispenser after a time $t$, expressed in months, which verifies that the probability of the event ( $X > t$ ) is equal to 0.05. Determine the value of $t$ rounded to the nearest integer.
2. The dispenser manual specifies that the dispenser provides Italian ice creams whose mass is between 55 g and 65 g. Consider the random variable $M$ representing the mass, in grams, of a distributed ice cream. It is admitted that $M$ follows the normal distribution with expectation 60 and standard deviation 2.5. a. Calculate the probability that the mass of an Italian ice cream chosen at random from those distributed is between 55 g and 65 g. b. Determine the largest value of $m$, rounded to the nearest gram, such that the probability $P ( M \geqslant m )$ is greater than or equal to 0.99.
3. The Italian ice cream dispenser allows you to choose only one of two flavours: vanilla or strawberry. To better manage his purchases of raw materials, the shopkeeper makes the hypothesis that there will be in proportion two vanilla ice cream buyers for one strawberry ice cream buyer. On the first day of use of his dispenser, he observes that out of 120 consumers, 65 chose vanilla ice cream. For what mathematical reason could he doubt his hypothesis? Justify.
Q2 Integration by Parts Integration by Parts within Function Analysis View
The flow of water from a tap has a constant and moderate flow rate.
We are particularly interested in a part of the flow profile represented in appendix 1 by the curve $C$ in an orthonormal coordinate system.
Part A
We consider that the curve $C$ given in appendix 1 is the graphical representation of a function $f$ differentiable on the interval $] 0 ; 1 ]$ which respects the following three conditions:
$$( H ) : f ( 1 ) = 0 \quad f ^ { \prime } ( 1 ) = 0.25 \quad \text { and } \lim _ { \substack { x \rightarrow 0 \\ x > 0 } } f ( x ) = - \infty .$$
  1. Can the function $f$ be a polynomial function of degree two? Why?
  2. Let $g$ be the function defined on the interval $]0 ; 1]$ by $g ( x ) = k \ln x$. a. Determine the real number $k$ so that the function $g$ respects the three conditions $( H )$. b. Does the representative curve of the function $g$ coincide with the curve $C$ ? Why?
  3. Let $h$ be the function defined on the interval $]0; 1]$ by $h ( x ) = \frac { a } { x ^ { 4 } } + b x$ where $a$ and $b$ are real numbers. Determine $a$ and $b$ so that the function $h$ respects the three conditions ( $H$ ).

Part B
We admit in this part that the curve $C$ is the graphical representation of a function $f$ continuous, strictly increasing, defined and differentiable on the interval $] 0 ; 1 ]$ with expression:
$$f ( x ) = \frac { 1 } { 20 } \left( x - \frac { 1 } { x ^ { 4 } } \right)$$
  1. Justify that the equation $f ( x ) = - 5$ admits on the interval $] 0 ; 1 ]$ a unique solution which will be denoted $\alpha$. Determine an approximate value of $\alpha$ to $10 ^ { - 2 }$ near.
  2. It is admitted that the volume of water in $\mathrm { cm } ^ { 3 }$, contained in the first 5 centimetres of the flow, is given by the formula: $V = \int _ { \alpha } ^ { 1 } \pi x ^ { 2 } f ^ { \prime } ( x ) \mathrm { d } x$. a. Let $u$ be the function differentiable on $] 0; 1]$ defined by $u ( x ) = \frac { 1 } { 2 x ^ { 2 } }$. Determine its derivative function. b. Determine the exact value of $V$. Using the approximate value of $\alpha$ obtained in question 1, give an approximate value of $V$.
Q3 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
We consider the sequence $\left( I _ { n } \right)$ defined by $I _ { 0 } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { 1 - x } \mathrm {~d} x$ and for every non-zero natural number $n$
$$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { x ^ { n } } { 1 - x } \mathrm {~d} x$$
  1. Show that $I _ { 0 } = \ln ( 2 )$.
  2. a. Calculate $I _ { 0 } - I _ { 1 }$. b. Deduce $I _ { 1 }$.
  3. a. Show that, for every natural number $n , I _ { n } - I _ { n + 1 } = \frac { \left( \frac { 1 } { 2 } \right) ^ { n + 1 } } { n + 1 }$. b. Propose an algorithm to determine, for a given natural number $n$, the value of $I _ { n }$.
  4. Let $n$ be a non-zero natural number.

It is admitted that if $x$ belongs to the interval $\left[ 0 ; \frac { 1 } { 2 } \right]$ then $0 \leqslant \frac { x ^ { n } } { 1 - x } \leqslant \frac { 1 } { 2 ^ { n - 1 } }$. a. Show that for every non-zero natural number $n$, $0 \leqslant I _ { n } \leqslant \frac { 1 } { 2 ^ { n } }$. b. Deduce the limit of the sequence ( $I _ { n }$ ) as $n$ tends to $+ \infty$.
5. For every non-zero natural number $n$, we set
$$S _ { n } = \frac { 1 } { 2 } + \frac { \left( \frac { 1 } { 2 } \right) ^ { 2 } } { 2 } + \frac { \left( \frac { 1 } { 2 } \right) ^ { 3 } } { 3 } + \ldots + \frac { \left( \frac { 1 } { 2 } \right) ^ { n } } { n }$$
a. Show that for every non-zero natural number $n$, $S _ { n } = I _ { 0 } - I _ { n }$. b. Determine the limit of $S _ { n }$ as $n$ tends to $+ \infty$.
Q4a 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 4 — For candidates who have not followed the speciality
5 points
On the figure given in appendix 2 to be returned with the copy:
  • ABCDEFGH is a rectangular parallelepiped such that $\mathrm { AB } = 12 , \mathrm { AD } = 18$ and $\mathrm { AE } = 6$
  • EBDG is a tetrahedron.

Space is referred to an orthonormal coordinate system with origin A in which the points $\mathrm { B } , \mathrm { D }$ and E have respective coordinates $\mathrm { B } ( 12 ; 0 ; 0 ) , \mathrm { D } ( 0 ; 18 ; 0 )$ and $\mathrm { E } ( 0 ; 0 ; 6 )$.
  1. Prove that the plane (EBD) has the Cartesian equation $3 x + 2 y + 6 z - 36 = 0$.
  2. a. Determine a parametric representation of the line (AG). b. Deduce that the line (AG) intersects the plane (EBD) at a point K with coordinates (4;6;2).
  3. Is the line (AG) orthogonal to the plane (EBD)? Justify.
  4. a. Let M be the midpoint of the segment $[ \mathrm { ED } ]$. Prove that the points B, K and M are collinear. b. Then construct the point K on the figure given in appendix 2 to be returned with the copy.
  5. We denote by P the plane parallel to the plane (ADE) passing through the point K. a. Prove that the plane P intersects the plane (EBD) along a line parallel to the line (ED). b. Then construct on appendix 2 to be returned with the copy the intersection of the plane P and the face EBD of the tetrahedron EBDG.
Q4b Number Theory Properties of Integer Sequences and Digit Analysis View
Exercise 4 — For candidates who have followed the speciality course
We consider the matrix $M = \left( \begin{array} { l l } 2 & 3 \\ 1 & 2 \end{array} \right)$ and the sequences of natural numbers $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ defined by: $u _ { 0 } = 1 , v _ { 0 } = 0$, and for every natural number $n , \binom { u _ { n + 1 } } { v _ { n + 1 } } = M \binom { u _ { n } } { v _ { n } }$. The two parts can be treated independently.
Part A
The first terms of the sequence $\left( v _ { n } \right)$ have been calculated:
$n$0123456789101112
$v _ { n }$0141556209780291110864405451513165647192107560

  1. Conjecture the possible values of the units digit of the terms of the sequence $\left( v _ { n } \right)$.
  2. It is admitted that for every natural number $n , \binom { u _ { n + 3 } } { v _ { n + 3 } } = M ^ { 3 } \binom { u _ { n } } { v _ { n } }$. a. Justify that for every natural number $n , \left\{ \begin{array} { l } u _ { n + 3 } = 26 u _ { n } + 45 v _ { n } \\ v _ { n + 3 } = 15 u _ { n } + 26 v _ { n } \end{array} \right.$. b. Deduce that for every natural number $n : v _ { n + 3 } \equiv v _ { n } [ 5 ]$.
  3. Let $r$ be a fixed natural number. Prove, using a proof by induction, that, for every natural number $q , v _ { 3 q + r } \equiv v _ { r }$ [5].
  4. Deduce that for every natural number $n$ the term $v _ { n }$ is congruent to 0, to 1 or to 4 modulo 5.
  5. Conclude regarding the set of values taken by the units digit of the terms of the sequence $\left( v _ { n } \right)$.

Part B
The objective of this part is to prove that $\sqrt { 3 }$ is not a rational number using the matrix $M$.
To do this, we perform a proof by contradiction and assume that $\sqrt { 3 }$ is a rational number. In this case, $\sqrt { 3 }$ can be written in the form of an irreducible fraction $\frac { p } { q }$ where $p$ and $q$ are non-zero natural numbers, with $q$ the smallest possible natural number.
  1. Show that $q < p < 2 q$.
  2. It is admitted that the matrix $M$ is invertible. Give its inverse $M ^ { - 1 }$ (no justification is expected). Let the pair $\left( p ^ { \prime } ; q ^ { \prime } \right)$ be defined by $\binom { p ^ { \prime } } { q ^ { \prime } } = M ^ { - 1 } \binom { p } { q }$.
  3. a. Verify that $p ^ { \prime } = 2 p - 3 q$ and that $q ^ { \prime } = - p + 2 q$. b. Justify that ( $p ^ { \prime } ; q ^ { \prime }$ ) is a pair of relative integers. c. Recall that $p = q \sqrt { 3 }$. Show that $p ^ { \prime } = q ^ { \prime } \sqrt { 3 }$. d. Show that $0 < q ^ { \prime } < q$. e. Deduce that $\sqrt { 3 }$ is not a rational number.