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
2021 bac-spe-maths__metropole-sept_j1

8 maths questions

QExercise 2 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Let $f$ be the function defined on the interval $] - \frac { 1 } { 3 } ; + \infty [$ by: $$f ( x ) = \frac { 4 x } { 1 + 3 x }$$ We consider the sequence $(u _ { n })$ defined by: $u _ { 0 } = \frac { 1 } { 2 }$ and, for every natural number $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$.
  1. Calculate $u _ { 1 }$.
  2. We admit that the function $f$ is increasing on the interval $] - \frac { 1 } { 3 } ; + \infty [$. a. Show by induction that, for every natural number $n$, we have: $\frac { 1 } { 2 } \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 2$. b. Deduce that the sequence $(u _ { n })$ is convergent. c. We call $\ell$ the limit of the sequence $(u _ { n })$. Determine the value of $\ell$.
  3. a. Copy and complete the Python function below which, for every positive real number $E$, determines the smallest value $P$ such that: $1 - u _ { P } < E$. \begin{verbatim} def seuil(E) : u=0.5 n = 0 while u = n = n + 1 return n \end{verbatim} b. Give the value returned by this program in the case where $E = 10 ^ { - 4 }$.
  4. We consider the sequence $(v _ { n })$ defined, for every natural number $n$, by: $$v _ { n } = \frac { u _ { n } } { 1 - u _ { n } }$$ a. Show that the sequence $(v _ { n })$ is geometric with common ratio 4. Deduce, for every natural number $n$, the expression of $v _ { n }$ as a function of $n$. b. Prove that, for every natural number $n$, we have: $u _ { n } = \frac { v _ { n } } { v _ { n } + 1 }$. c. Show then that, for every natural number $n$, we have: $$u _ { n } = \frac { 1 } { 1 + 0.25 ^ { n } }$$ Find by calculation the limit of the sequence $(u _ { n })$.
QExercise 3 Applied differentiation Applied modeling with differentiation View
In the Pyrenees National Park, a researcher is working on the decline of a protected species in high-mountain lakes: the ``midwife toad''. Parts I and II can be approached independently.
Part I: Effect of the introduction of a new species
In certain lakes in the Pyrenees, trout have been introduced by humans to enable fishing activities in the mountains. The researcher studied the impact of this introduction on the midwife toad population in a lake. His previous studies lead him to model the evolution of this population as a function of time by the following function $f$: $$f ( t ) = \left( 0.04 t ^ { 2 } - 8 t + 400 \right) \mathrm { e } ^ { \frac { t } { 50 } } + 40 \text { for } t \in [ 0 ; 120 ]$$ The variable $t$ represents the elapsed time, in days, from the introduction at time $t = 0$ of trout into the lake, and $f ( t )$ models the number of toads at time $t$.
  1. Determine the number of toads present in the lake when the trout are introduced.
  2. We admit that the function $f$ is differentiable on the interval $[0 ; 120]$ and we denote $f ^ { \prime }$ its derivative function. Show, by displaying the calculation steps, that for every real number $t$ belonging to the interval $[0 ; 120]$ we have: $$f ^ { \prime } ( t ) = t ( t - 100 ) \mathrm { e } ^ { \frac { t } { 50 } } \times 8 \times 10 ^ { - 4 }$$
  3. Study the variations of the function $f$ on the interval $[0 ; 120]$, then draw up the variation table of $f$ on this interval (approximate values to the nearest hundredth will be given).
  4. According to this model: a. Determine the number of days $J$ necessary for the number of toads to reach its minimum. What is this minimum number? b. Justify that, after reaching its minimum, the number of toads will one day exceed 140 individuals. c. Using a calculator, determine the duration in days from which the number of toads will exceed 140 individuals.

Part II: Effect of Chytridiomycosis on a tadpole population
One of the main causes of the decline of this toad species in high mountains is a disease, ``Chytridiomycosis'', caused by a fungus. The researcher considers that:
  • Three quarters of the mountain lakes in the Pyrenees are not infected by the fungus, that is, they contain no contaminated tadpoles (toad larvae).
  • In the remaining lakes, the probability that a tadpole is contaminated is 0.74.
The researcher randomly chooses a lake in the Pyrenees and takes samples from it. For the rest of the exercise, results will be rounded to the nearest thousandth when necessary. The researcher randomly takes a tadpole from the chosen lake to perform a test before releasing it. We denote $T$ the event ``The tadpole is contaminated by the disease'' and $L$ the event ``The lake is infected by the fungus''. We denote $\bar { L }$ the opposite event of $L$ and $\bar { T }$ the opposite event of $T$.
  1. Copy and complete the following probability tree using the data from the problem statement.
  2. Show that the probability $P ( T )$ that the sampled tadpole is contaminated is 0.185.
  3. The tadpole is not contaminated. What is the probability that the lake is infected?
QExercise A Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
We consider the cube ABCDEFGH. We are given three points I, J and K satisfying: $$\overrightarrow { \mathrm { EI } } = \frac { 1 } { 4 } \overrightarrow { \mathrm { EH } } , \quad \overrightarrow { \mathrm { EJ } } = \frac { 1 } { 4 } \overrightarrow { \mathrm { EF } } , \quad \overrightarrow { \mathrm { BK } } = \frac { 1 } { 4 } \overrightarrow { \mathrm { BF } }$$ We use the orthonormal coordinate system $(A ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } })$.
  1. Give without justification the coordinates of points I, J and K.
  2. Prove that the vector $\overrightarrow { \mathrm { AG } }$ is normal to the plane (IJK).
  3. Show that a Cartesian equation of the plane (IJK) is $4 x + 4 y + 4 z - 5 = 0$.
  4. Determine a parametric representation of the line (BC).
  5. Deduce the coordinates of point L, the point of intersection of the line (BC) with the plane (IJK).
  6. On the figure in the appendix, place point L and construct the intersection of the plane (IJK) with the face (BCGF).
  7. Let $\mathrm { M } \left( \frac { 1 } { 4 } ; 1 ; 0 \right)$. Show that the points I, J, L and M are coplanar.
QExercise B Differentiating Transcendental Functions Full function study with transcendental functions View
Part I
We consider the function $h$ defined on the interval $] 0 ; + \infty [$ by: $$h ( x ) = 1 + \frac { \ln ( x ) } { x }$$
  1. Determine the limit of the function $h$ at 0.
  2. Determine the limit of the function $h$ at $+ \infty$.
  3. We denote $h ^ { \prime }$ the derivative function of $h$. Prove that, for every real number $x$ in $] 0 ; + \infty [$, we have: $$h ^ { \prime } ( x ) = \frac { 1 - \ln ( x ) } { x ^ { 2 } }$$
  4. Draw up the variation table of the function $h$ on the interval $] 0 ; + \infty [$.
  5. Prove that the equation $h ( x ) = 0$ has a unique solution $\alpha$ in $] 0 ; + \infty [$. Justify that we have: $0.5 < \alpha < 0.6$.

Part II
In this part, we consider the functions $f$ and $g$ defined on $] 0 ; + \infty [$ by: $$f ( x ) = x \ln ( x ) - x ; \quad g ( x ) = \ln ( x )$$ We denote $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ the curves representing respectively the functions $f$ and $g$ in an orthonormal coordinate system $(O ; \vec { \imath } , \vec { \jmath })$. For every strictly positive real number $a$, we call:
  • $T _ { a }$ the tangent to $\mathscr { C } _ { f }$ at its point with abscissa $a$;
  • $D _ { a }$ the tangent to $\mathscr { C } _ { g }$ at its point with abscissa $a$.
We are looking for possible values of $a$ for which the lines $T _ { a }$ and $D _ { a }$ are perpendicular. Let $a$ be a real number belonging to the interval $] 0 ; + \infty [$.
  1. Justify that the line $D _ { a }$ has slope $\frac { 1 } { a }$.
  2. Justify that the line $T _ { a }$ has slope $\ln ( a )$.
  3. We recall that in an orthonormal coordinate system, two lines with slopes $m$ and $m ^ { \prime }$ respectively are perpendicular if and only if $m m ^ { \prime } = - 1$. Prove that there exists a unique value of $a$, which you will identify, for which the lines $T _ { a }$ and $D _ { a }$ are perpendicular.
Q1 Tangents, normals and gradients Find tangent line equation at a given point View
The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$.
We can assert that: a. $f ^ { \prime } ( - 0.5 ) = 0$ b. if $x \in ] - \infty ; - 0.5 \left[ \right.$, then $f ^ { \prime } ( x ) < 0$ c. $f ^ { \prime } ( 0 ) = 15$ d. the derivative function $f ^ { \prime }$ does not change sign on $\mathbb { R }$.
Q2 Differentiating Transcendental Functions Determine parameters from function or curve conditions View
The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$.
We admit that the function $f$ represented above is defined on $\mathbb { R }$ by $f ( x ) = ( a x + b ) \mathrm { e } ^ { x }$, where $a$ and $b$ are two real numbers and that its curve intersects the x-axis at the point with coordinates ($-0.5$; 0). We can assert that: a. $a = 10$ and $b = 5$ b. $a = 2.5$ and $b = -0.5$ c. $a = -1.5$ and $b = 5$ d. $a = 0$ and $b = 5$
Q3 Applied differentiation Convexity and inflection point analysis View
The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$.
We admit that the second derivative of the function $f$ is defined on $\mathbb { R }$ by: $$f ^ { \prime \prime } ( x ) = ( 10 x + 25 ) \mathrm { e } ^ { x }$$ We can assert that: a. The function $f$ is convex on $\mathbb { R }$ b. The function $f$ is concave on $\mathbb { R }$ c. Point C is the unique inflection point of $\mathscr { C } _ { f }$ d. $\mathscr { C } _ { f }$ has no inflection point
Q4 Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
We consider two sequences $(U _ { n })$ and $(V _ { n })$ defined on $\mathbb { N }$ such that:
  • for every natural number $n$, $U _ { n } \leqslant V _ { n }$;
  • $\lim _ { n \rightarrow + \infty } V _ { n } = 2$.
We can assert that: a. the sequence $(U _ { n })$ converges b. for every natural number $n$, $V _ { n } \leqslant 2$ c. the sequence $(U _ { n })$ diverges d. the sequence $(U _ { n })$ is bounded above