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
2024 bac-spe-maths__asie_j2

4 maths questions

Q1 Differentiating Transcendental Functions Full function study with transcendental functions View
Consider the function $f$ defined on $] 0 ; + \infty [$ by
$$f ( x ) = x ^ { 2 } - x \ln ( x ) .$$
We admit that $f$ is twice differentiable on $] 0 ; + \infty [$. We denote $f ^ { \prime }$ the derivative function of $f$ and $f ^ { \prime \prime }$ the derivative function of $f ^ { \prime }$.
Part A: Study of the function $f$
  1. Determine the limits of the function $f$ at 0 and at $+ \infty$.
  2. For all strictly positive real $x$, calculate $f ^ { \prime } ( x )$.
  3. Show that for all strictly positive real $x$: $$f ^ { \prime \prime } ( x ) = \frac { 2 x - 1 } { x }$$
  4. Study the variations of the function $f ^ { \prime }$ on $] 0 ; + \infty [$, then draw up the table of variations of the function $f ^ { \prime }$ on $] 0 ; + \infty [$. Care should be taken to show the exact value of the extremum of the function $f ^ { \prime }$ on $] 0 ; + \infty [$. The limits of the function $f ^ { \prime }$ at the boundaries of the domain of definition are not expected.
  5. Show that the function $f$ is strictly increasing on $] 0 ; + \infty [$.

Part B: Study of an auxiliary function for solving the equation $f ( x ) = x$
We consider in this part the function $g$ defined on $] 0 ; + \infty [$ by
$$g ( x ) = x - \ln ( x )$$
We admit that the function $g$ is differentiable on $] 0 ; + \infty [$, we denote $g ^ { \prime }$ its derivative.
  1. For all strictly positive real, calculate $g ^ { \prime } ( x )$, then draw up the table of variations of the function $g$. The limits of the function $g$ at the boundaries of the domain of definition are not expected.
  2. We admit that 1 is the unique solution of the equation $g ( x ) = 1$. Solve, on the interval $] 0 ; + \infty [$, the equation $f ( x ) = x$.

Part C: Study of a recursive sequence
We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = \frac { 1 } { 2 }$ and for all natural integer $n$,
$$u _ { n + 1 } = f \left( u _ { n } \right) = u _ { n } ^ { 2 } - u _ { n } \ln \left( u _ { n } \right) .$$
  1. Show by induction that for all natural integer $n$: $$\frac { 1 } { 2 } \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 1 .$$
  2. Justify that the sequence $( u _ { n } )$ converges. We call $\ell$ the limit of the sequence $( u _ { n } )$ and we admit that $\ell$ satisfies the equality $f ( \ell ) = \ell$.
  3. Determine the value of $\ell$.
Q2 Geometric Sequences and Series Prove a Transformed Sequence is Geometric View
Léa spends a good part of her days playing a video game and is interested in the chances of winning her next games.
She estimates that if she has just won a game, she wins the next one in $70\%$ of cases. But if she has just suffered a defeat, according to her, the probability that she wins the next one is 0.2. Furthermore, she thinks she has an equal chance of winning the first game as of losing it.
For all non-zero natural integer $n$, we define the following events:
  • $G _ { n }$: ``Léa wins the $n$-th game of the day'';
  • $D _ { n }$: ``Léa loses the $n$-th game of the day''.

For all non-zero natural integer $n$, we denote $g _ { n }$ the probability of event $G _ { n }$. We have therefore $g _ { 1 } = 0.5$.
  1. What is the value of the conditional probability $p _ { G _ { 1 } } \left( D _ { 2 } \right)$?
  2. Copy and complete the probability tree below which models the situation for the first two games of the day.
  3. Calculate $g _ { 2 }$.
  4. Let $n$ be a non-zero natural integer. a. Copy and complete the probability tree below which models the situation for the $n$-th and $(n+1)$-th games of the day. b. Justify that for all non-zero natural integer $n$, $$g _ { n + 1 } = 0.5 g _ { n } + 0.2 .$$
  5. For all non-zero natural integer $n$, we set $v _ { n } = g _ { n } - 0.4$. a. Show that the sequence $( v _ { n } )$ is geometric. We will specify its first term and its common ratio. b. Show that, for all non-zero natural integer $n$: $$g _ { n } = 0.1 \times 0.5 ^ { n - 1 } + 0.4 .$$
  6. Study the variations of the sequence $( g _ { n } )$.
  7. Give, by justifying, the limit of the sequence $( g _ { n } )$. Interpret the result in the context of the problem.
  8. Determine, by calculation, the smallest integer $n$ such that $g _ { n } - 0.4 \leqslant 0.001$.
  9. Copy and complete lines 4, 5 and 6 of the following function, written in Python language, so that it returns the smallest rank from which the terms of the sequence $\left( g _ { n } \right)$ are all less than or equal to $0.4 + e$, where $e$ is a strictly positive real number. \begin{verbatim} def seuil(e) : g = 0.5 n = 1 while...: g = 0.5 * g + 0.2 n = ... return (n) \end{verbatim}
Q3 Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
  1. Let $( u _ { n } )$ be a sequence defined for all natural integer $n$ and satisfying the following relation: $$\text{for all natural integer } n , \frac { 1 } { 2 } < u _ { n } \leqslant \frac { 3 n ^ { 2 } + 4 n + 7 } { 6 n ^ { 2 } + 1 } .$$ Statement 1: $\lim _ { n \rightarrow + \infty } u _ { n } = \frac { 1 } { 2 }$.
  2. Let $h$ be a function defined and differentiable on the interval $[-4;4]$. The graphical representation $\mathscr { C } _ { h ^ { \prime } }$ of its derivative function $h ^ { \prime }$ is given below. Statement 2: The function $h$ is convex on $[ - 1 ; 3]$.
  3. The code of a building is composed of 4 digits (which may be identical) followed by two distinct letters among A, B and C (example: 1232BA). Statement 3: There exist 20634 codes that contain at least one 0.
  4. We consider the function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = x \ln x$. Statement 4: The function $f$ is a solution on $] 0 ; + \infty [$ of the differential equation $$x y ^ { \prime } - y = x .$$
Q4 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
In an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ of space, we consider the plane $(P)$ with equation:
$$(P) : \quad 2x + 2y - 3z + 1 = 0 .$$
We consider the three points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ with coordinates:
$$\mathrm{A}(1;0;1), \quad \mathrm{B}(2;-1;1) \quad \text{and} \quad \mathrm{C}(-4;-6;5).$$
The purpose of this exercise is to study the ratio of areas between a triangle and its orthogonal projection onto a plane.
Part A
  1. For each of the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$, verify whether it belongs to the plane $(P)$.
  2. Show that the point $\mathrm{C}^{\prime}(0;-2;-1)$ is the orthogonal projection of point $\mathrm{C}$ onto the plane $(P)$.
  3. Determine a parametric representation of the line (AB).
  4. We admit the existence of a unique point H satisfying the two conditions $$\left\{ \begin{array}{l} \mathrm{H} \in (\mathrm{AB}) \\ (\mathrm{AB}) \text{ and } (\mathrm{HC}) \text{ are orthogonal.} \end{array} \right.$$ Determine the coordinates of point H.

Part B
We admit that the coordinates of the vector $\overrightarrow{\mathrm{HC}}$ are: $\overrightarrow{\mathrm{HC}} \left( \begin{array}{c} -\frac{11}{2} \\ -\frac{11}{2} \\ 4 \end{array} \right)$.
  1. Calculate the exact value of $\| \overrightarrow{\mathrm{HC}} \|$.
  2. Let $S$ be the area of triangle ABC. Determine the exact value of $S$.

Part C
We admit that $\mathrm{HC}^{\prime} = \sqrt{\frac{17}{2}}$.
  1. Let $\alpha = \widehat{\mathrm{CHC}^{\prime}}$. Determine the value of $\cos(\alpha)$.
  2. a. Show that the lines $(\mathrm{C}^{\prime}\mathrm{H})$ and (AB) are perpendicular. b. Calculate $S^{\prime}$ the area of triangle $\mathrm{ABC}^{\prime}$, give the exact value. c. Give a relationship between $S$, $S^{\prime}$ and $\cos(\alpha)$.