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
2022 bac-spe-maths__metropole-sept_j1

9 maths questions

QExercise 2 7 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
A hotel located near a prehistoric tourism site offers two visits in the surrounding area, one to a museum and one to a cave.
A study showed that $70\%$ of the hotel's clients visit the museum. Furthermore, among clients visiting the museum, $60\%$ visit the cave. The study also shows that $6\%$ of the hotel's clients make no visits. We randomly question a hotel client and note:
  • $M$ the event: ``the client visits the museum'';
  • $G$ the event: ``the client visits the cave''.
We denote by $\bar { M }$ the complementary event of $M$, $\bar { G }$ the complementary event of $G$, and for any event $E$, we denote by $p ( E )$ the probability of $E$. Thus, according to the problem statement, we have: $p ( \bar { M } \cap \bar { G } ) = 0.06$.
  1. a. Verify that $p _ { \bar { M } } ( \bar { G } ) = 0.2$, where $p _ { \bar { M } } ( \bar { G } )$ denotes the probability that the questioned client does not visit the cave given that he does not visit the museum. b. The weighted tree opposite models the situation. Copy and complete this tree by indicating on each branch the associated probability. c. What is the probability of the event ``the client visits the cave and does not visit the museum''? d. Show that $p ( G ) = 0.66$.
  2. The hotel manager claims that among clients who visit the cave, more than half also visit the museum. Is this claim correct?
  3. The prices for visits are as follows:
    • museum visit: 12 euros;
    • cave visit: 5 euros.
    We consider the random variable $T$ which models the amount spent by a hotel client for these visits. a. Give the probability distribution of $T$. Present the results in the form of a table. b. Calculate the mathematical expectation of $T$. c. For profitability reasons, the hotel manager estimates that the average amount of visit revenues must be greater than 700 euros per day. Determine the average number of clients per day needed to achieve this objective.
  4. To increase revenues, the manager wishes the expectation of the random variable modeling the amount spent by a hotel client for these visits to increase to 15 euros, without changing the museum visit price which remains at 12 euros. What price should be set for the cave visit to achieve this objective? (We will assume that the increase in the cave entrance price does not change the frequency of visits to the two sites).
  5. We randomly choose 100 hotel clients, treating this choice as a draw with replacement. What is the probability that at least three-quarters of these clients visited the cave during their stay at the hotel? Give a value of the result to $10 ^ { - 3 }$ near.
QExercise 3 7 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
Part A
Consider the function $f$ defined on the interval $[ 1 ; + \infty [$ by $$f ( x ) = \frac { \ln x } { x }$$ where ln denotes the natural logarithm function.
  1. Give the limit of the function $f$ at $+ \infty$.
  2. We admit that the function $f$ is differentiable on the interval $[ 1 ; + \infty [$ and we denote by $f ^ { \prime }$ its derivative function. a. Show that, for every real number $x \geqslant 1$, $f ^ { \prime } ( x ) = \frac { 1 - \ln x } { x ^ { 2 } }$. b. Justify the following sign table, giving the sign of $f ^ { \prime } ( x )$ according to the values of $x$.
    $x$1e$+ \infty$
    $f ^ { \prime } ( x )$+0-

    c. Draw up the complete variation table of the function $f$.
  3. Let $k$ be a non-negative real number. a. Show that, if $0 \leqslant k \leqslant \frac { 1 } { \mathrm { e } }$, the equation $f ( x ) = k$ admits a unique solution on the interval $[1; e]$. b. If $k > \frac { 1 } { \mathrm { e } }$, does the equation $f ( x ) = k$ admit solutions on the interval $[ 1 ; + \infty [$? Justify.

Part B
Let $g$ be the function defined on $\mathbb { R }$ by: $$g ( x ) = \mathrm { e } ^ { \frac { x } { 4 } } .$$ We consider the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 1$ and, for every natural integer $n$: $u _ { n + 1 } = e ^ { \frac { u _ { n } } { 4 } }$, that is: $u _ { n + 1 } = g \left( u _ { n } \right)$.
  1. Justify that the function $g$ is increasing on $\mathbb { R }$.
  2. Show by induction that, for every natural integer $n$, we have: $u _ { n } \leqslant u _ { n + 1 } \leqslant \mathrm { e }$.
  3. Deduce that the sequence $( u _ { n } )$ is convergent.
We denote by $\ell$ the limit of the sequence $( u _ { n } )$ and we admit that $\ell$ is a solution of the equation: $$\mathrm { e } ^ { \frac { x } { 4 } } = x .$$
    \setcounter{enumi}{3}
  1. Deduce that $\ell$ is a solution of the equation $f ( x ) = \frac { 1 } { 4 }$, where $f$ is the function studied in Part A.
  2. Give an approximate value to $10 ^ { - 2 }$ near of the limit $\ell$ of the sequence $( u _ { n } )$.
QExercise 4 7 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
In space with respect to an orthonormal coordinate system $( \mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k } )$, we consider the points $$\mathrm { A } ( - 1 ; - 1 ; 3 ) , \quad \mathrm { B } ( 1 ; 1 ; 2 ) , \quad \mathrm { C } ( 1 ; - 1 ; 7 )$$ We also consider the line $\Delta$ passing through the points $\mathrm { D } ( - 1 ; 6 ; 8 )$ and $\mathrm { E } ( 11 ; - 9 ; 2 )$.
  1. a. Verify that the line $\Delta$ admits the following parametric representation: $$\left\{ \begin{aligned} x & = - 1 + 4 t \\ y & = 6 - 5 t \quad \text { with } t \in \mathbb { R } \\ z & = 8 - 2 t \end{aligned} \right.$$ b. Specify a parametric representation of the line $\Delta ^ { \prime }$ parallel to $\Delta$ and passing through the origin O of the coordinate system. c. Does the point $\mathrm { F } ( 1.36 ; - 1.7 ; - 0.7 )$ belong to the line $\Delta ^ { \prime }$?
  2. a. Show that the points $\mathrm { A }$, $\mathrm { B }$ and $\mathrm { C }$ define a plane. b. Show that the line $\Delta$ is perpendicular to the plane (ABC). c. Show that a Cartesian equation of the plane (ABC) is: $4 x - 5 y - 2 z + 5 = 0$.
  3. a. Show that the point $\mathrm { G } ( 7 ; - 4 ; 4 )$ belongs to the line $\Delta$. b. Determine the coordinates of the point H, the orthogonal projection of point G onto the plane (ABC). c. Deduce that the distance from point G to the plane (ABC) is equal to $3 \sqrt { 5 }$.
  4. a. Show that the triangle ABC is right-angled at A. b. Calculate the volume $V$ of the tetrahedron ABCG. We recall that the volume $V$ of a tetrahedron is given by the formula $V = \frac { 1 } { 3 } \times B \times h$ where B is the area of a base and h the height corresponding to this base.
Q1 Exponential Functions MCQ on Function Properties View
Consider the function $g$ defined on $\mathbb { R }$ by: $$g ( x ) = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 } .$$ The representative curve of the function $g$ admits as an asymptote at $+ \infty$ the line with equation: a. $x = 2$; b. $y = 2$; c. $y = 0$; d. $x = - 1$
Q2 Applied differentiation Convexity and inflection point analysis View
Consider a function $f$ defined and twice differentiable on $\mathbb { R }$. We call $\mathscr { C }$ its graphical representation. We denote by $f ^ { \prime \prime }$ the second derivative of $f$. The curve of $f ^ { \prime \prime }$, denoted $\mathscr { C } ^ { \prime \prime }$, is represented in the graph opposite. a. $\mathscr { C }$ admits a unique inflection point; b. $f$ is convex on the interval $[ - 1 ; 2 ]$; c. $f$ is convex on $] - \infty ; - 1 ]$ and on $[2; + \infty [$; d. $f$ is convex on $\mathbb { R }$.
Q3 Geometric Sequences and Series True/False or Multiple-Statement Verification View
We are given the sequence $( u _ { n } )$ defined by: $u _ { 0 } = 0$ and for every natural integer $n$, $u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + 1$. The sequence $\left( v _ { n } \right)$, defined for every natural integer $n$ by $v _ { n } = u _ { n } - 2$, is: a. arithmetic with common difference $- 2$; b. geometric with common ratio $- 2$; c. arithmetic with common difference $1$; d. geometric with common ratio $\frac { 1 } { 2 }$.
Q4 Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
Consider a sequence $( u _ { n } )$ such that, for every natural integer, we have: $$1 + \left( \frac { 1 } { 4 } \right) ^ { n } \leqslant u _ { n } \leqslant 2 - \frac { n } { n + 1 }$$ We can affirm that the sequence $\left( u _ { n } \right)$: a. converges to $2$; b. converges to $1$; c. diverges to $+ \infty$; d. has no limit.
Q5 Integration by Parts Multiple-Choice Primitive Identification View
Let $f$ be the function defined on $] 0 ; + \infty \left[ \text{ by } f ( x ) = x ^ { 2 } \ln x \right.$.
A primitive $F$ of $f$ on $] 0$; $+ \infty [$ is defined by: a. $F ( x ) = \frac { 1 } { 3 } x ^ { 3 } \left( \ln x - \frac { 1 } { 3 } \right)$; b. $F ( x ) = \frac { 1 } { 3 } x ^ { 3 } ( \ln x - 1 )$; c. $F ( x ) = \frac { 1 } { 3 } x ^ { 2 }$; d. $F ( x ) = \frac { 1 } { 3 } x ^ { 2 } ( \ln x - 1 )$.
Q6 Exponential Functions Algebraic Simplification and Expression Manipulation View
For every real $x$, the expression $2 + \frac { 3 \mathrm { e } ^ { - x } - 5 } { \mathrm { e } ^ { - x } + 1 }$ is equal to: a. $\frac { 5 - 3 \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$; b. $\frac { 5 + 3 \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }$; c. $\frac { 5 + 3 \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$; d. $\frac { 5 - 3 \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }$.