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__metropole-sept_j2

4 maths questions

Q1 Vectors: Lines & Planes Volume of Pyramid/Tetrahedron Using Planes and Lines View
We consider the cube ABCDEFGH represented below. The points I and J are the midpoints of segments $[\mathrm{AB}]$ and $[\mathrm{CG}]$ respectively. The point N is the midpoint of segment [IJ]. The objective of this exercise is to calculate the volume of the tetrahedron HFIJ. We place ourselves in the orthonormal coordinate system ($A$; $\overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE}$).
  1. a. Give the coordinates of points I and J.
    Deduce the coordinates of N. b. Justify that the vectors $\overrightarrow{\mathrm{IJ}}$ and $\overrightarrow{\mathrm{NF}}$ have the respective coordinates: $$\overrightarrow{\mathrm{IJ}} \left(\begin{array}{c} 0.5 \\ 1 \\ 0.5 \end{array}\right) \text{ and } \overrightarrow{\mathrm{NF}} \left(\begin{array}{c} 0.25 \\ -0.5 \\ 0.75 \end{array}\right)$$ c. Prove that the vectors $\overrightarrow{\mathrm{IJ}}$ and $\overrightarrow{\mathrm{NF}}$ are orthogonal.
    We admit that $\mathrm{NF} = \frac{\sqrt{14}}{4}$. d. Deduce that the area of triangle FIJ is equal to $\frac{\sqrt{21}}{8}$.
  2. We consider the vector $\vec{u}\left(\begin{array}{c} 4 \\ -1 \\ -2 \end{array}\right)$. a. Prove that the vector $\vec{u}$ is normal to the plane (FIJ). b. Deduce that a Cartesian equation of the plane (FIJ) is: $4x - y - 2z - 2 = 0$. c. We denote by $d$ the line perpendicular to the plane (FIJ) passing through point H. Determine a parametric representation of the line $d$. d. Show that the distance from point H to the plane (FIJ) is equal to $\frac{5\sqrt{21}}{21}$. e. We recall that the volume of a pyramid is given by the formula $V = \frac{1}{3} \times \mathscr{B} \times h$ where $\mathscr{B}$ is the area of a base and $h$ is the length of the height relative to this base. Calculate the volume of the tetrahedron HFIJ. Give the answer in the form of an irreducible fraction.
Q2 5 marks Geometric Sequences and Series Prove a Transformed Sequence is Geometric View
A robot is positioned on a horizontal axis and moves several times by one meter on this axis, randomly to the right or to the left. During the first movement, the probability that the robot moves to the right is equal to $\frac{1}{3}$. If it moves to the right, the probability that the robot moves to the right again during the next movement is equal to $\frac{3}{4}$. If it moves to the left, the probability that the robot moves to the left again during the next movement is equal to $\frac{1}{2}$. For every natural integer $n \geqslant 1$, we denote:
  • $D_n$ the event: ``the robot moves to the right during the $n$-th movement'';
  • $\overline{D_n}$ the complementary event of $D_n$;
  • $p_n$ the probability of event $D_n$.
We therefore have $p_1 = \frac{1}{3}$.
Part A: study of the special case where $n = 2$ In this part, the robot performs two successive movements.
  1. Reproduce and complete the following weighted tree.
  2. Determine the probability that the robot moves to the right twice.
  3. Show that $p_2 = \frac{7}{12}$.
  4. The robot moved to the left during the second movement. What is the probability that it moved to the right during the first movement?

Part B: study of the sequence $(p_n)$. We wish to estimate the movement of the robot after a large number of steps.
  1. Prove that for every natural integer $n \geqslant 1$, we have: $$p_{n+1} = \frac{1}{4} p_n + \frac{1}{2}.$$ You may use a tree to help.
  2. a. Show by induction that for every natural integer $n \geqslant 1$, we have: $$p_n \leqslant p_{n+1} < \frac{2}{3}.$$ b. Is the sequence $(p_n)$ convergent? Justify.
  3. We consider the sequence $(u_n)$ defined for every natural integer $n \geqslant 1$, by $u_n = p_n - \frac{2}{3}$. a. Show that the sequence $(u_n)$ is geometric and specify its first term and its common ratio. b. Determine the limit of the sequence $(p_n)$ and interpret the result in the context of the exercise.

Part C In this part, we consider another robot that performs ten movements of one meter independent of each other, each movement to the right having a fixed probability equal to $\frac{3}{4}$. What is the probability that it returns to its starting point after the ten movements? Round the result to $10^{-3}$ near.
Q3 Applied differentiation Full function study (variation table, limits, asymptotes) View
Part A We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = \frac{6}{1 + 5e^{-x}}$$ We have represented on the diagram below the representative curve $\mathscr{C}_f$ of the function $f$.
  1. Show that point A with coordinates $(\ln 5 ; 3)$ belongs to the curve $\mathscr{C}_f$.
  2. Show that the line with equation $y = 6$ is an asymptote to the curve $\mathscr{C}_f$.
  3. a. We admit that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function. Show that for every real $x$, we have: $$f'(x) = \frac{30\mathrm{e}^{-x}}{\left(1 + 5\mathrm{e}^{-x}\right)^2}.$$ b. Deduce the complete table of variations of $f$ on $\mathbb{R}$.
  4. We admit that:
    • $f$ is twice differentiable on $\mathbb{R}$, we denote $f''$ its second derivative;
    • for every real $x$,
    $$f''(x) = \frac{30\mathrm{e}^{-x}\left(5\mathrm{e}^{-x} - 1\right)}{\left(1 + 5\mathrm{e}^{-x}\right)^3}.$$ a. Study the convexity of $f$ on $\mathbb{R}$. In particular, we will show that the curve $\mathscr{C}_f$ admits an inflection point. b. Justify that for every real $x$ belonging to $]-\infty ; \ln 5]$, we have: $f(x) \geqslant \frac{5}{6}x + 1$.
  5. We consider a function $F_k$ defined on $\mathbb{R}$ by $F_k(x) = k\ln\left(\mathrm{e}^x + 5\right)$, where $k$ is a real constant. a. Determine the value of the real $k$ so that $F_k$ is a primitive of $f$ on $\mathbb{R}$. b. Deduce that the area, in square units, of the domain bounded by the curve $\mathscr{C}_f$, the $x$-axis, the $y$-axis and the line with equation $x = \ln 5$ is equal to $6\ln\left(\frac{5}{3}\right)$.

Part B The objective of this part is to study the following differential equation: $$(E) \quad y' = y - \frac{1}{6}y^2.$$ We recall that a solution of equation $(E)$ is a function $u$ defined and differentiable on $\mathbb{R}$ such that for every real $x$, we have: $$u'(x) = u(x) - \frac{1}{6}[u(x)]^2.$$
  1. Show that the function $f$ defined in part A is a solution of the differential equation $(E)$.
  2. Solve the differential equation $y' = -y + \frac{1}{6}$.
  3. We denote by $g$ a function differentiable on $\mathbb{R}$ that does not vanish. We denote by $h$ the function defined on $\mathbb{R}$ by $h(x) = \frac{1}{g(x)}$. We admit that $h$ is differentiable on $\mathbb{R}$. We denote $g'$ and $h'$ the derivative functions of $g$ and $h$. a. Show that if $h$ is a solution of the differential equation $y' = -y + \frac{1}{6}$, then $g$ is a solution of the differential equation $y' = y - \frac{1}{6}y^2$. b. For every positive real $m$, we consider the functions $g_m$ defined on $\mathbb{R}$ by: $$g_m(x) = \frac{6}{1 + 6m\mathrm{e}^{-x}}.$$ Show that for every positive real $m$, the function $g_m$ is a solution of the differential equation $(E): \quad y' = y - \frac{1}{6}y^2$.
Q4 5 marks Sequences and Series Evaluation of a Finite or Infinite Sum 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. The five questions of this exercise are independent.
  1. We consider the script written in Python language below. \begin{verbatim} def seuil(S) : n=0 u=7 while u < S : n=n+1 u=1.05*u+3 return(n) \end{verbatim} Statement 1: the instruction seuil(100) returns the value 18.
  2. Let $(S_n)$ be the sequence defined for every natural integer $n$ by $$S_n = 1 + \frac{1}{5} + \frac{1}{5^2} + \ldots + \frac{1}{5^n}.$$ Statement 2: the sequence $(S_n)$ converges to $\frac{5}{4}$.
  3. Statement 3: in a class composed of 30 students, we can form 870 different pairs of delegates.
  4. We consider the function $f$ defined on $[1 ; +\infty[$ by $f(x) = x(\ln x)^2$. Statement 4: the equation $f(x) = 1$ admits a unique solution in the interval $[1 ; +\infty[$.
  5. Statement 5: $$\int_0^1 x\mathrm{e}^{-x}\,\mathrm{d}x = \frac{\mathrm{e} - 2}{\mathrm{e}}.$$