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
2023 bac-spe-maths__amerique-sud_j1

6 maths questions

Q1A Sign Change & Interval Methods View
We consider the function $f$ defined on the set $] 0 ; + \infty [$ by
$$f ( x ) = 1 + x ^ { 2 } - 2 x ^ { 2 } \ln ( x )$$
We admit that $f$ is differentiable on the interval and we denote $f ^ { \prime }$ its derivative function.
  1. Justify that $\lim _ { x \rightarrow 0 } f ( x ) = 1$ and, by noting that $f ( x ) = 1 + x ^ { 2 } [ 1 - 2 \ln ( x ) ]$, justify that $\lim _ { x \rightarrow + \infty } f ( x ) = - \infty$.
  2. Show that for all real $x$ in the interval $] 0 ; + \infty \left[ , f ^ { \prime } ( x ) = - 4 x \ln ( x ) \right.$.
  3. Study the sign of $f ^ { \prime } ( x )$ on the interval $] 0$; $+ \infty [$, then draw up the table of variations of the function on the interval $] 0 ; + \infty [$.
  4. Prove that the equation $f ( x ) = 0$ admits a unique solution $\alpha$ in the interval $[ 1 ; + \infty [$ and that $\alpha \in [ 1 ; \mathrm { e } ]$.

We admit in the rest of the exercise that the equation $f ( x ) = 0$ has no solution on the interval $] 0 ; 1]$.
5. We are given the function below written in Python. The instruction from lycee import* allows access to the function $\ln$.
\begin{verbatim} from lycee import * def f(x) : return 1 + x**2 - 2*x**2*ln(x) def dichotomie(p) a=1 b=2.7 while b - a > 10**(-p) : if f(a)*f((a+b)/2) < 0 : | b = (a+b)/2 else : |a =(a+b)/2 return (a,b) \end{verbatim}
It writes in the execution console:
\begin{verbatim} >>> dichotomie(1) \end{verbatim}
Among the four propositions below, copy the one displayed by the previous instruction. Justify your answer (you may proceed by elimination).
Proposition A: $\quad ( 1.75,1.9031250000000002 )$ Proposition B : ( $1.85,1.9031250000000002 )$ Proposition C : $\quad ( 2.75,2.9031250000000002 )$ Proposition D : (2.85, 2.9031250000000002)
Q1B Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View
We consider the function $g$ defined on the interval $] 0 ; + \infty [$, by
$$g ( x ) = \frac { \ln ( x ) } { 1 + x ^ { 2 } }$$
We admit that $g$ is differentiable on the interval $] 0 ; + \infty \left[ \right.$ and we denote $g ^ { \prime }$ its derivative function. We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in the plane with respect to a coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } )$.
We also consider the function $f$ defined on $]0;+\infty[$ by $f(x) = 1 + x^2 - 2x^2\ln(x)$, and $\alpha$ denotes the unique solution of $f(x)=0$ in $[1;+\infty[$. We admit that $g(\alpha) = \frac{1}{2\alpha^2}$.
  1. Prove that for all real $x$ in the interval $] 0 ; + \infty \left[ , \quad g ^ { \prime } ( x ) = \frac { f ( x ) } { x \left( 1 + x ^ { 2 } \right) ^ { 2 } } \right.$.
  2. Prove that the function $g$ admits a maximum at $x = \alpha$.
  3. We denote $T _ { 1 }$ the tangent line to $\mathscr { C } _ { g }$ at the point with abscissa 1 and we denote $T _ { \alpha }$ the tangent line to $\mathscr { C } _ { g }$ at the point with abscissa $\alpha$. Determine, as a function of $\alpha$, the coordinates of the intersection point of the lines $T _ { 1 }$ and $T _ { \alpha }$.
Q2 Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
  1. Between 1998 and 2020, in France 18221965 deliveries were recorded, of which 293898 resulted in the birth of twins and 4921 resulted in the birth of at least three children. a. With a precision of $0.1\%$ calculate, among all recorded deliveries, the percentage of deliveries resulting in the birth of twins over the period 1998-2020. b. Verify that the percentage of deliveries that resulted in the birth of at least three children is less than $0.1\%$.

We then consider that this percentage is negligible. We call an ordinary delivery a delivery resulting in the birth of a single child. We call a double delivery a delivery resulting in the birth of exactly two children. We consider in the rest of the exercise that a delivery is either ordinary or double. The probability of an ordinary delivery is equal to 0.984 and that of a double delivery is then equal to 0.016. The probabilities calculated in the rest will be rounded to the nearest thousandth.
2. We admit that on a given day in a maternity ward, $n$ deliveries are performed. We consider that these $n$ deliveries are independent of each other. We denote $X$ the random variable that gives the number of double deliveries performed that day. a. In the case where $n = 20$, specify the probability distribution followed by the random variable $X$ and calculate the probability that exactly one double delivery is performed. b. By the method of your choice that you will explain, determine the smallest value of $n$ such that $P ( X \geqslant 1 ) \geqslant 0.99$. Interpret the result in the context of the exercise.
3. In this maternity ward, among double births, it is estimated that there are $30\%$ monozygotic twins (called ``identical twins'' which are necessarily of the same sex: two boys or two girls) and therefore $70\%$ dizygotic twins (called ``fraternal twins'', which can be of different sexes: two boys, two girls or one boy and one girl). In the case of double births, we admit that, as for ordinary births, the probability of being a girl at birth is equal to 0.49 and that of being a boy at birth is equal to 0.51. In the case of a double birth of dizygotic twins, we also admit that the sex of the second newborn of the twins is independent of the sex of the first newborn. We randomly choose a double delivery performed in this maternity ward and we consider the following events:
  • $M$ : ``the twins are monozygotic'';
  • $F _ { 1 }$ : ``the first newborn is a girl'';
  • $F _ { 2 }$ : ``the second newborn is a girl''.
We will denote $P ( A )$ the probability of event $A$ and $\bar { A }$ the opposite event of $A$. a. Copy and complete the probability tree. b. Show that the probability that the two newborns are girls is 0.315 07. c. The two newborns are twin girls. Calculate the probability that they are monozygotic.
Q3 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
In space with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ), we consider the points
$$\mathrm { A } ( 0 ; 4 ; 16 ) , \quad \mathrm { B } ( 0 ; 4 ; - 10 ) , \quad \mathrm { C } ( 4 ; - 8 ; 0 ) \quad \text { and } \quad \mathrm { K } ( 0 ; 4 ; 3 ) .$$
We define the sphere $S$ with center K and radius 13 as the set of points M such that $\mathrm { KM } = 13$.
  1. a. Verify that point C belongs to sphere $S$. b. Show that triangle ABC is right-angled at C.
  2. a. Show that the vector $\vec { n } \left( \begin{array} { l } 3 \\ 1 \\ 0 \end{array} \right)$ is a normal vector to plane (ABC). b. Determine a Cartesian equation of plane (ABC).
  3. We admit that sphere $S$ intersects the x-axis at two points, one having a positive abscissa and the other a negative abscissa. We denote D the one with positive abscissa. a. Show that point D has coordinates $( 12 ; 0 ; 0 )$. b. Give a parametric representation of the line $\Delta$ passing through D and perpendicular to plane (ABC). c. Determine the distance from point D to plane (ABC).
  4. Calculate an approximate value, to the nearest unit of volume, of the volume of tetrahedron ABCD. We recall the formula for the volume V of a tetrahedron $$V = \frac { 1 } { 3 } \times \mathscr { B } \times h$$ where $\mathscr { B }$ is the area of a base and h the associated height.
Q4A Sequences and series, recurrence and convergence Monotonicity and boundedness analysis View
The purpose of Part A is to study the behavior of the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 0.3$ and by the recurrence relation, for all natural integer $n$ :
$$u _ { n + 1 } = 2 u _ { n } \left( 1 - u _ { n } \right)$$
This recurrence relation is written $u _ { n + 1 } = f \left( u _ { n } \right)$, where $f$ is the function defined on $\mathbb { R }$ by :
$$f ( x ) = 2 x ( 1 - x )$$
  1. Prove that the function $f$ is strictly increasing on the interval $\left[ 0 ; \frac { 1 } { 2 } \right]$.
  2. We admit that for all natural integer $n , 0 \leqslant u _ { n } \leqslant \frac { 1 } { 2 }$. Calculate $u _ { 1 }$ then perform a proof by induction to demonstrate that for all natural integer $n , u _ { n } \leqslant u _ { n + 1 }$.
  3. Deduce that the sequence $( u _ { n } )$ is convergent.
  4. Justify that the limit of the sequence $( u _ { n } )$ is equal to $\frac { 1 } { 2 }$.
Q4B Sequences and series, recurrence and convergence Applied/contextual sequence problem View
The purpose of this part is to study a model of population evolution. In 2022, this population has 3000 individuals. We denote $P _ { n }$ the population size in thousands in the year $2022 + n$. Thus $P _ { 0 } = 3$. According to a model inspired by the Verhulst model, a Belgian mathematician of the XIX${}^{\mathrm{th}}$ century, we consider that, for all natural integer $n$ :
$$P _ { n + 1 } - P _ { n } = P _ { n } \left( 1 - b \times P _ { n } \right) , \text { where } b \text { is a strictly positive real number. }$$
The real number $b$ is a damping factor that allows us to account for the limited nature of the resources in the environment in which these individuals evolve.
  1. In this question $b = 0$. a. Justify that the sequence $\left( P _ { n } \right)$ is a geometric sequence and specify its common ratio. b. Determine the limit of $P _ { n }$.
  2. In this question $b = 0.2$. a. For all natural integer $n$, we set $v _ { n } = 0.1 \times P _ { n }$. Calculate $v _ { 0 }$ then show that, for all natural integer $n , v _ { n + 1 } = 2 v _ { n } \left( 1 - v _ { n } \right)$. b. In this model, justify that the population will stabilize around a value that you will specify.