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
2023 bac-spe-maths__amerique-sud_j2

4 maths questions

Q1 Conditional Probability Markov Chain / Day-to-Day Transition Probabilities View
A game offered at a fairground consists of making three successive shots at a moving target.
It has been observed that:
  • If the player hits the target on one shot then they miss it on the next shot in $65\%$ of cases;
  • If the player misses the target on one shot then they hit it on the next shot in $50\%$ of cases.

The probability that a player hits the target on their first shot is 0.6. For any event $A$, we denote $p(A)$ its probability and $\bar{A}$ the complementary event of $A$. We randomly choose a player for this shooting game. We consider the following events:
  • $A_1$: ``The player hits the target on the $1^{\text{st}}$ shot''
  • $A_2$: ``The player hits the target on the $2^{\mathrm{nd}}$ shot''
  • $A_3$: ``The player hits the target on the $3^{\mathrm{rd}}$ shot''.

Part A
  1. Copy and complete, with the corresponding probabilities on each branch, the probability tree below modelling the situation.

Let $X$ be the random variable that gives the number of times the player hits the target during the three shots.
2. Show that the probability that the player hits the target exactly twice during the three shots is equal to 0.4015.
3. The objective of this question is to calculate the expectation of the random variable $X$, denoted $E(X)$. a. Copy and complete the table below giving the probability distribution of the random variable $X$.
$X = x_i$0123
$p\left(X = x_i\right)$0.10.0735

b. Calculate $E(X)$. c. Interpret the previous result in the context of the exercise.
Part B
We consider $N$, a natural number greater than or equal to 1.
A group of $N$ people comes to this stand to play this game under identical and independent conditions.
A player is declared a winner when they hit the target three times. We denote $Y$ the random variable that counts among the $N$ people the number of players declared winners.
  1. In this question, $N = 15$. a. Justify that $Y$ follows a binomial distribution and determine its parameters. b. Give the probability, rounded to $10^{-3}$, that exactly 5 players win this game.
  2. By the method of your choice, which you will explain, determine the minimum number of people who must come to this stand so that the probability that there is at least one winning player is greater than or equal to 0.98.
Q2 Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
In an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ we consider the points: $$\mathrm{A}(1;1;-4), \quad \mathrm{B}(2;-1;-3), \quad \mathrm{C}(0;-1;-1) \text{ and } \Omega(1;1;2).$$
  1. Prove that the points $\mathrm{A}$, $\mathrm{B}$ and C define a plane.
  2. a. Prove that the vector $\vec{n}$ with coordinates $\left(\begin{array}{l}1\\1\\1\end{array}\right)$ is normal to the plane (ABC). b. Justify that a Cartesian equation of the plane (ABC) is $x + y + z + 2 = 0$.
  3. a. Justify that the point $\Omega$ does not belong to the plane (ABC). b. Determine the coordinates of the point H, the orthogonal projection of the point $\Omega$ onto the plane (ABC).

We admit that $\Omega\mathrm{H} = 2\sqrt{3}$. We define the sphere $S$ with centre $\Omega$ and radius $2\sqrt{3}$ as the set of all points M in space such that $\Omega\mathrm{M} = 2\sqrt{3}$.
4. Justify, without calculation, that any point N of the plane (ABC), distinct from H, does not belong to the sphere $S$. We say that a plane $\mathscr{P}$ is tangent to the sphere $S$ at a point K when the following two conditions are satisfied:
  • $\mathrm{K} \in \mathscr{P} \cap S$
  • $(\Omega\mathrm{K}) \perp \mathscr{P}$

    \setcounter{enumi}{4}
  1. Let the plane $\mathscr{P}$ with Cartesian equation $x + y - z - 6 = 0$ and the point K with coordinates $\mathrm{K}(3;3;0)$. Prove that the plane $\mathscr{P}$ is tangent to the sphere $S$ at point K.
  2. We admit that the planes (ABC) and $\mathscr{P}$ intersect along a line ($\Delta$). Determine a parametric equation of the line ($\Delta$).
Q3 Proof by induction Prove a sequence bound or inequality by induction View
Let the sequence $(u_n)$ defined by $u_0 = 0$ and, for all $n \in \mathbb{N}$, $$u_{n+1} = 5u_n - 8n + 6.$$
  1. Calculate $u_1$ and $u_2$.
  2. Let $n$ be a natural number. Copy and complete the function \texttt{suite\_u} with argument \texttt{n} below, written in Python language, so that it returns the value of $u_n$. \begin{verbatim} def suite_u(n) : u = ... for i in range(1,n+1) : u = ... return u \end{verbatim}
  3. a. Prove by induction that, for all $n \in \mathbb{N}$, $u_n \geqslant 2n$. b. Deduce the limit of the sequence $(u_n)$. c. Let $p \in \mathbb{N}^*$. Why can we assert that there exists at least one integer $n_0$ such that, for all natural integers $n$ satisfying $n \geqslant n_0$, $u_n \geqslant 10^p$?
  4. Prove that the sequence $(u_n)$ is increasing.
  5. We consider the sequence $(v_n)$, defined for all $n \in \mathbb{N}$, by $v_n = u_n - 2n + 1$. a. Below the function \texttt{suite\_u} above, we have written the function \texttt{suite\_v} below: \begin{verbatim} def suite_v(n): L = [] for i in range(n+1) : L.append(suite_u(i) - 2*i + 1) return L \end{verbatim} The command ``L.append'' allows us to add, in the last position, an element to the list $L$. When we enter \texttt{suite\_v(5)} in the console, we obtain the following display: $$\begin{aligned} & \ggg \text{suite\_v}(5) \\ & [1, 5, 25, 125, 625, 3125] \end{aligned}$$ Conjecture, for all natural integer $n$, the expression of $v_{n+1}$ as a function of $v_n$. Prove this conjecture. b. Deduce, for all natural integer $n$, the explicit form of $u_n$ as a function of $n$.
Q4 Differentiating Transcendental Functions Full function study with transcendental functions View
Let the function $f$ defined on $\mathbb{R}$ by $$f(x) = \ln\left(1 + \mathrm{e}^{-x}\right) + \frac{1}{4}x.$$ We denote $\mathscr{C}_f$ the representative curve of the function $f$ in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$ of the plane.
Part A
  1. Determine the limit of $f$ at $+\infty$.
  2. We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function. a. Show that, for all real $x$, $f'(x) = \dfrac{\mathrm{e}^x - 3}{4\left(\mathrm{e}^x + 1\right)}$. b. Deduce the variations of the function $f$ on $\mathbb{R}$. c. Show that the equation $f(x) = 1$ admits a unique solution $\alpha$ in the interval $[2;5]$.

Part B
We will admit that the function $f'$ is differentiable on $\mathbb{R}$ and for all real $x$, $$f''(x) = \frac{\mathrm{e}^x}{\left(\mathrm{e}^x + 1\right)^2}.$$ We denote $\Delta$ the tangent line to the curve $\mathscr{C}_f$ at the point with abscissa 0. In the graph below, we have represented the curve $\mathscr{C}_f$, the tangent line $\Delta$, and the quadrilateral MNPQ such that M and N are the two points of the curve $\mathscr{C}_f$ with abscissas $\alpha$ and $-\alpha$ respectively, and Q and P are the two points of the line $\Delta$ with abscissas $\alpha$ and $-\alpha$ respectively.
  1. a. Justify the sign of $f''(x)$ for $x \in \mathbb{R}$. b. Deduce that the portion of the curve $\mathscr{C}_f$ on the interval $[-\alpha; \alpha]$ is inscribed in the quadrilateral MNPQ.
  2. a. Show that $f(-\alpha) = \ln\left(\mathrm{e}^{-\alpha} + 1\right) + \dfrac{3}{4}\alpha$. b. Prove that the quadrilateral MNPQ is a parallelogram.