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__caledonie_j2

4 maths questions

Q1 Vectors: Lines & Planes True/False or Verify a Given Statement View
For each of the four following statements, indicate whether it is true or false, by justifying the answer. An unjustified answer is not taken into account. An absence of answer is not penalised.
Consider a cube ABCDEFGH with edge length 1 and the point I defined by $\overrightarrow { \mathrm { FI } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { FB } }$. One may place oneself in the orthonormal coordinate system of space $( \mathrm { A } ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } } )$.
  1. Consider the triangle HAC.

Statement 1: The triangle HAC is a right-angled triangle.
2. Consider the lines (HF) and (DI).
Statement 2: The lines (HF) and (DI) are secant.
3. Consider a real number $\alpha$ belonging to the interval $] 0 ; \pi [$.
Consider the vector $\vec { u }$ with coordinates $\left( \begin{array} { c } \sin ( \alpha ) \\ \sin ( \pi - \alpha ) \\ \sin ( - \alpha ) \end{array} \right)$. Statement 3: The vector $\vec { u }$ is a normal vector to the plane (FAC).
4. The cube ABCDEFGH has 8 vertices. We are interested in the number $N$ of segments that can be constructed by connecting 2 distinct vertices of the cube. Statement 4: $N = \frac { 8 ^ { 2 } } { 2 }$.
Q2 6 marks Areas Between Curves Prove or Verify an Area Result View
In the orthonormal coordinate system (O; I, J), we have represented:
  • the line with equation $y = x$;
  • the line with equation $y = 1$;
  • the line with equation $x = 1$;
  • the parabola with equation $y = x ^ { 2 }$.
We can thus divide the square OIKJ into three zones.
Part A Prove the results shown in the table below.
ZONEZONE 1ZONE 2ZONE 3
AREA$\frac { 1 } { 2 }$$\frac { 1 } { 3 }$$\frac { 1 } { 6 }$

Part B: a first game
A player throws a dart at the square above. It is admitted that the probability that it lands on a zone is equal to the area of that zone. Thus, the probability that the dart lands on ZONE 3 is equal to $\frac { 1 } { 6 }$.
  • If the dart lands on ZONE 3, then the player tosses a fair coin. If the coin lands on HEADS, then the player wins, otherwise he loses.
  • If the dart lands on a zone other than ZONE 3, then the player rolls a fair six-sided die. If the die lands on FACE 6, then the player wins, otherwise he loses.
We note the following events: $T$: ``the dart lands on ZONE 3''; $G$: ``the player wins''.
  1. Represent the situation with a weighted tree.
  2. Prove that the probability of event $G$ is equal to $\frac { 2 } { 9 }$.
  3. Given that the player has won, what is the probability that the dart landed on ZONE 3?

Part C: a second game
A player, called player $n^{\mathrm{o}}1$, throws a dart at the previous square. As in Part B, it is admitted that the probability that the dart lands on each of the zones is equal to the area of that zone. The player wins a sum equal, in euros, to the number of the zone. For example, if the dart lands on ZONE 3, the player wins 3 euros. We denote $X _ { 1 }$ the random variable giving the winnings of player $n^{\circ}1$. We denote respectively $E \left( X _ { 1 } \right)$ and $V \left( X _ { 1 } \right)$ the expectation and variance of the random variable $X _ { 1 }$.
  1. a. Calculate $E \left( X _ { 1 } \right)$. b. Show that $V \left( X _ { 1 } \right) = \frac { 5 } { 9 }$.
  2. A player $n^{\circ}2$ and a player $n^{\circ}3$ play in turn, under the same conditions as player $n^{\circ}1$. It is admitted that the games of these three players are independent of each other. We denote $X _ { 2 }$ and $X _ { 3 }$ the random variables giving the winnings of players $n^{\circ}2$ and $n^{\circ}3$. We denote $Y$ the random variable defined by $Y = X _ { 1 } + X _ { 2 } + X _ { 3 }$. a. Determine the probability that $Y = 9$. b. Calculate $E ( Y )$. c. Justify that $V ( Y ) = \frac { 5 } { 3 }$.
Q3 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Consider the function $f$ defined for all real $x$ by: $$f ( x ) = \ln \left( \mathrm { e } ^ { \frac { x } { 2 } } + 2 \right)$$ It is admitted that the function $f$ is differentiable on $\mathbb { R }$. Consider the sequence $(u_n)$ defined by $u _ { 0 } = \ln ( 9 )$ and, for all natural integer $n$, $$u _ { n + 1 } = f \left( u _ { n } \right)$$
  1. Show that the function $f$ is strictly increasing on $\mathbb { R }$.
  2. Show that $f ( 2 \ln ( 2 ) ) = 2 \ln ( 2 )$.
  3. Show that $u _ { 1 } = \ln ( 5 )$.
  4. Show by induction that for all natural integer $n$, we have: $$2 \ln ( 2 ) \leqslant u _ { n + 1 } \leqslant u _ { n }$$
  5. Deduce that the sequence $(u_n)$ converges.
  6. a. Solve in $\mathbb { R }$ the equation $X ^ { 2 } - X - 2 = 0$. b. Deduce the set of solutions on $\mathbb { R }$ of the equation: $$\mathrm { e } ^ { x } - \mathrm { e } ^ { \frac { x } { 2 } } - 2 = 0$$ c. Deduce the set of solutions on $\mathbb { R }$ of the equation $f ( x ) = x$. d. Determine the limit of the sequence $\left( u _ { n } \right)$.
Q4 Applied differentiation Full function study (variation table, limits, asymptotes) View
Consider the function $f$ defined on the interval $] 0 ; + \infty [$ by: $$f ( x ) = \frac { \ln ( x ) } { x ^ { 2 } } + 1$$ We denote $\mathscr { C } _ { f }$ the representative curve of the function $f$ in an orthonormal coordinate system. It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$ and we denote $f ^ { \prime }$ its derivative function.
  1. Determine the limits of the function $f$ at 0 and at $+ \infty$. Deduce the possible asymptotes to the curve $\mathscr { C } _ { f }$.
  2. Show that, for all real $x$ in the interval $] 0 ; + \infty [$, we have: $$f ^ { \prime } ( x ) = \frac { 1 - 2 \ln ( x ) } { x ^ { 3 } }$$
  3. Deduce the variation table of the function $f$ on the interval $] 0 ; + \infty [$.
  4. a. Show that the equation $f ( x ) = 0$ has a unique solution, denoted $\alpha$, on the interval $] 0 ; + \infty [$. b. Give an interval for the real number $\alpha$ with amplitude 0.01. c. Deduce the sign of the function $f$ on the interval $] 0 ; + \infty [$.
  5. Consider the function $g$ defined on the interval $] 0 ; + \infty [$ by: $$g ( x ) = \ln ( x )$$ We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in an orthonormal coordinate system with origin O. We consider a strictly positive real $x$ and the point M of the curve $\mathscr{C} _ { g }$ with abscissa $x$. We denote OM the distance between points O and M. a. Express the quantity $\mathrm { OM } ^ { 2 }$ as a function of the real $x$. b. Show that, when the real $x$ ranges over the interval $] 0 ; + \infty [$, the quantity $\mathrm { OM } ^ { 2 }$ admits a minimum at $\alpha$. c. The minimum value of the distance OM, when the real $x$ ranges over the interval $] 0 ; + \infty [$, is called the distance from point O to the curve $\mathscr { C } _ { g }$. We denote $d$ this distance. Express $d$ in terms of $\alpha$.