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__centres-etrangers_j2

4 maths questions

Q1 Discrete Probability Distributions Combinatorial Counting in Probabilistic Context View
An opaque bag contains eight tokens numbered from 1 to 8, indistinguishable to the touch. Three times, a player draws a token from this bag, notes its number, then puts it back in the bag. In this context, we call a ``draw'' the ordered list of the three numbers obtained. For example, if the player draws token number 4, then token number 5, then token number 1, then the corresponding draw is $(4 ; 5 ; 1)$.
  1. Determine the number of possible draws.
    1. [a.] Determine the number of draws without repetition of numbers.
    2. [b.] Deduce from this the number of draws containing at least one repetition of numbers.

We denote $X_1$ the random variable equal to the number of the first token drawn, $X_2$ the one equal to the number of the second token drawn and $X_3$ the one equal to the number of the third token drawn. Since this is a draw with replacement, the random variables $X_1, X_2$, and $X_3$ are independent and follow the same probability distribution.
    \setcounter{enumi}{2}
  1. Establish the probability distribution of the random variable $X_1$.
  2. Determine the expectation of the random variable $X_1$.

We denote $S = X_1 + X_2 + X_3$ the random variable equal to the sum of the numbers of the three tokens drawn.
    \setcounter{enumi}{4}
  1. Determine the expectation of the random variable $S$.
  2. Determine $P(S = 24)$.
  3. If a player obtains a sum greater than or equal to 22, then they win a prize.
    1. [a.] Justify that there are exactly 10 draws allowing one to win a prize.
    2. [b.] Deduce from this the probability of winning a prize.
Q2 Differentiating Transcendental Functions Full function study with transcendental functions View
We consider the function $f$ defined on the interval $]-\infty; 1[$ by $$f(x) = \frac{\mathrm{e}^x}{x-1}$$ We admit that the function $f$ is differentiable on the interval $]-\infty; 1[$. We call $\mathscr{C}$ its representative curve in a coordinate system.
    1. [a.] Determine the limit of the function $f$ at 1.
    2. [b.] Deduce from this a graphical interpretation.
  2. Determine the limit of the function $f$ at $-\infty$.
    1. [a.] Show that for every real number $x$ in the interval $]-\infty; 1[$, we have $$f'(x) = \frac{(x-2)\mathrm{e}^x}{(x-1)^2}$$
    2. [b.] Draw up, by justifying, the table of variations of the function $f$ on the interval $]-\infty; 1[$.
  4. We admit that for every real number $x$ in the interval $]-\infty; 1[$, we have $$f''(x) = \frac{\left(x^2 - 4x + 5\right)\mathrm{e}^x}{(x-1)^3}.$$
    1. [a.] Study the convexity of the function $f$ on the interval $]-\infty; 1[$.
    2. [b.] Determine the reduced equation of the tangent line $T$ to the curve $\mathscr{C}$ at the point with abscissa 0.
    3. [c.] Deduce from this that, for every real number $x$ in the interval $]-\infty; 1[$, we have: $$\mathrm{e}^x \geqslant (-2x-1)(x-1).$$
    1. [a.] Justify that the equation $f(x) = -2$ admits a unique solution $\alpha$ on the interval $]-\infty; 1[$.
    2. [b.] Using a calculator, determine an interval containing $\alpha$ with amplitude $10^{-2}$.
Q3 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
The cube ABCDEFGH has edge length 1 cm. The point I is the midpoint of segment [AB] and the point J is the midpoint of segment [CG].
We place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.
  1. Give the coordinates of points I and J.
  2. Show that the vector $\overrightarrow{\mathrm{EJ}}$ is normal to the plane (FHI).
  3. Show that a Cartesian equation of the plane (FHI) is $-2x - 2y + z + 1 = 0$.
  4. Determine a parametric representation of the line (EJ).
    1. [a.] We denote K the orthogonal projection of point E onto the plane $(\mathrm{FHI})$. Calculate its coordinates.
    2. [b.] Show that the volume of the pyramid EFHI is $\frac{1}{6}\mathrm{~cm}^3$.
    We may use the point L, midpoint of segment $[\mathrm{EF}]$. We admit that this point is the orthogonal projection of point I onto the plane (EFH).
    1. [c.] Deduce from the two previous questions the area of triangle FHI.
Q4 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Part A
We consider the function $f$ defined on the interval $[0; +\infty[$ by $$f(x) = \sqrt{x+1}.$$ We admit that this function is differentiable on this same interval.
  1. Prove that the function $f$ is increasing on the interval $[0; +\infty[$.
  2. Prove that for every real number $x$ belonging to the interval $[0; +\infty[$: $$f(x) - x = \frac{-x^2 + x + 1}{\sqrt{x+1} + x}.$$
  3. Deduce from this that on the interval $[0; +\infty[$ the equation $f(x) = x$ admits as unique solution: $$\ell = \frac{1+\sqrt{5}}{2}.$$

Part B
We consider the sequence $(u_n)$ defined by $u_0 = 5$ and for every natural number $n$, by $u_{n+1} = f(u_n)$ where $f$ is the function studied in part A. We admit that the sequence with general term $u_n$ is well defined for every natural number $n$.
  1. Prove by induction that for every natural number $n$, we have $$1 \leqslant u_{n+1} \leqslant u_n.$$
  2. Deduce from this that the sequence $(u_n)$ converges.
  3. Prove that the sequence $(u_n)$ converges to $\ell = \frac{1+\sqrt{5}}{2}$.
  4. We consider the Python script below: \begin{verbatim} from math import * def seuil(n): u=5 i=0 while abs(u-l)>=10**(-n): u=sqrt(u+1) i=i+1 return(i) \end{verbatim} We recall that the command $\mathbf{abs}(\mathbf{x})$ returns the absolute value of $x$.
    1. [a.] Give the value returned by \texttt{seuil(2)}.
    2. [b.] The value returned by \texttt{seuil(4)} is 9. Interpret this value in the context of the exercise.