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__metropole_j2

5 maths questions

Q1 Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
Exercise 1 — Part A
The centre offers people coming for a weekend an introductory roller skating formula consisting of two training sessions. We randomly choose a person among those who have subscribed to this formula. We denote by $A$ and $B$ the following events:
  • A: ``The person falls during the first session'';
  • B: ``The person falls during the second session''.
For any event $E$, we denote $P(E)$ its probability and $\bar{E}$ its complementary event. Observations allow us to assume that $P(A) = 0{,}6$. Furthermore, we observe that:
  • If the person falls during the first session, the probability that they fall during the second is 0.3;
  • If the person does not fall during the first session, the probability that they fall during the second is 0.4.

  1. Represent the situation with a probability tree.
  2. Calculate the probability $P(\bar{A} \cap \bar{B})$ and interpret the result.
  3. Show that $P(B) = 0{,}34$.
  4. The person does not fall during the second training session. Calculate the probability that they did not fall during the first session.
  5. We call $X$ the random variable which, for each sample of 100 people who have subscribed to the formula, associates the number of them who did not fall during either the first or the second session. We assimilate the choice of a sample of 100 people to a draw with replacement. We admit that the probability that a person does not fall during either the first or the second session is 0.24.
    1. [a.] Show that the random variable $X$ follows a binomial distribution whose parameters you will specify.
    2. [b.] What is the probability of having, in a sample of 100 people who have subscribed to the formula, at least 20 people who do not fall during either the first or the second session?
    3. [c.] Calculate the expectation $E(X)$ and interpret the result in the context of the exercise.
Q2 Linear combinations of normal random variables View
Exercise 1 — Part B
We randomly choose a person who came to the multisports centre on a weekend. We denote $T_1$ the random variable giving their total waiting time in minutes before access to sports activities during Saturday and $T_2$ the random variable giving their total waiting time in minutes before access to sports activities during Sunday. We admit that:
  • $T_1$ follows a probability distribution with expectation $E(T_1) = 40$ and standard deviation $\sigma(T_1) = 10$;
  • $T_2$ follows a probability distribution with expectation $E(T_2) = 60$ and standard deviation $\sigma(T_2) = 16$;
  • the random variables $T_1$ and $T_2$ are independent.
We denote $T$ the random variable giving the total waiting time before access to sports activities over the two days, expressed in minutes. Thus we have $T = T_1 + T_2$.
  1. Determine the expectation $E(T)$ of the random variable $T$. Interpret the result in the context of the exercise.
  2. Show that the variance $V(T)$ of the random variable $T$ is equal to 356.
  3. Using the Bienaymé-Chebyshev inequality, show that, for a person randomly chosen among those who came to the multisports centre on a weekend, the probability that their total waiting time $T$ is strictly between 60 and 140 minutes is greater than 0.77.
Q3 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Exercise 2 (5 points)
The space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:
  • the points $\mathrm{A}(-1; 2; 1)$, $\mathrm{B}(1; -1; 2)$ and $\mathrm{C}(1; 1; 1)$;
  • the line $d$ whose parametric representation is given by: $$d : \left\{ \begin{aligned} x &= \frac{3}{2} + 2t \\ y &= 2 + t \\ z &= 3 - t \end{aligned} \quad \text{with } t \in \mathbb{R}; \right.$$
  • the line $d'$ whose parametric representation is given by: $$d' : \left\{ \begin{aligned} x &= s \\ y &= \frac{3}{2} + s \\ z &= 3 - 2s \end{aligned} \quad \text{with } s \in \mathbb{R}. \right.$$

Part A
  1. Show that the lines $d$ and $d'$ intersect at the point $\mathrm{S}\left(-\frac{1}{2}; 1; 4\right)$.
    1. [a.] Show that the vector $\vec{n}\begin{pmatrix}1\\2\\4\end{pmatrix}$ is a normal vector to the plane (ABC).
    2. [b.] Deduce that a Cartesian equation of the plane (ABC) is: $$x + 2y + 4z - 7 = 0$$
  3. Prove that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and S are not coplanar.
    1. [a.] Prove that the point $\mathrm{H}(-1; 0; 2)$ is the orthogonal projection of S onto the plane (ABC).
    2. [b.] Deduce that there is no point $M$ in the plane (ABC) such that $\mathrm{S}M < \frac{\sqrt{21}}{2}$.

Part B
We consider a point $M$ belonging to the segment [CS]. We thus have $\overrightarrow{\mathrm{CM}} = k\overrightarrow{\mathrm{CS}}$ with $k$ a real number in the interval $[0; 1]$.
  1. Determine the coordinates of point $M$ as a function of $k$.
  2. Does there exist a point $M$ on the segment [CS] such that the triangle $(MAB)$ is right-angled at $M$?
Q4 Curve Sketching Multi-Statement Verification (Remarks/Options) View
Exercise 3
For each of the following statements, indicate whether it is true or false. Justify each answer. An unjustified answer earns no points.
  1. The sequence $(u_n)$ is defined for every natural integer $n$ by $$u_n = \frac{1 + 5^n}{2 + 3^n}$$ Statement 1: The sequence $(u_n)$ converges to $\frac{5}{3}$.
  2. We consider the sequence $(w_n)$ defined by: $$w_0 = 0 \text{ and, for every natural integer } n,\ w_{n+1} = 3w_n - 2n + 3.$$ Statement 2: For every natural integer $n$, $w_n \geqslant n$.
  3. We consider the function $f$ defined on $]0; +\infty[$ whose representative curve $\mathscr{C}_f$ is given in an orthonormal coordinate system in the figure (Fig. 1). We specify that:
    • $T$ is the tangent to $\mathscr{C}_f$ at point A with abscissa 8;
    • The x-axis is the horizontal tangent to $\mathscr{C}_f$ at the point with abscissa 1.
    Statement 3: According to the graph, the function $f$ is convex on its domain of definition.
  4. Statement 4: For every real $x > 0$, $\ln(x) - x + 1 \leqslant 0$, where $\ln$ denotes the natural logarithm function.
Q5 Second order differential equations Solving non-homogeneous second-order linear ODE View
Exercise 4
The purpose of this exercise is to study the stopping of a cart on a ride, from the moment it enters the braking zone at the end of the course. We denote $t$ the elapsed time, expressed in seconds, from the moment the cart enters the braking zone. We model the distance travelled by the cart in the braking zone, expressed in metres, as a function of $t$, using a function denoted $d$ defined on $[0; +\infty[$. We thus have $d(0) = 0$. Furthermore, we admit that this function $d$ is differentiable on its domain of definition. We denote $d'$ its derivative function.
Part A
In the figure (Fig. 2), we have drawn in an orthonormal coordinate system:
  • the representative curve $\mathscr{C}_d$ of the function $d$;
  • the tangent $T$ to the curve $\mathscr{C}_d$ at point A with abscissa 4.7;
  • the asymptote $\Delta$ to $\mathscr{C}_d$ at $+\infty$.

In this part, no justification is expected. With the precision that the graph allows, answer the questions below. According to this model:
  1. After how much time will the cart have travelled 15 m in the braking zone?
  2. What minimum length must be provided for the braking zone?
  3. What is the value of $d'(4{,}7)$? Interpret this result in the context of the exercise.

Part B
We recall that $t$ denotes the elapsed time, in seconds, from the moment the cart enters the braking zone. We model the instantaneous velocity of the cart, in metres per second ($\mathrm{m.s^{-1}}$), as a function of $t$, by a function $v$ defined on $[0; +\infty[$. We admit that:
  • the function $v$ is differentiable on its domain of definition, and we denote $v'$ its derivative function;
  • the function $v$ is a solution of the differential equation $$(E): \quad y' + 0{,}6\, y = \mathrm{e}^{-0{,}6t},$$ where $y$ is an unknown function and $y'$ is the derivative function of $y$.
We further specify that, upon arrival in the braking zone, the velocity of the cart is equal to $12\,\mathrm{m.s^{-1}}$, that is $v(0) = 12$.
    1. [a.] We consider the differential equation $$(E'): \quad y' + 0{,}6\, y = 0$$ Determine the solutions of the differential equation $(E')$ on $[0; +\infty[$.
    2. [b.] Let $g$ be the function defined on $[0; +\infty[$ by $g(t) = t\,\mathrm{e}^{-0{,}6t}$. Verify that the function $g$ is a solution of the differential equation $(E)$.
    3. [c.] Deduce the solutions of the differential equation $(E)$ on $[0; +\infty[$.
    4. [d.] Deduce that for every real $t$ belonging to the interval $[0; +\infty[$, we have: $$v(t) = (12 + t)\,\mathrm{e}^{-0{,}6t}$$
  2. In this question, we study the function $v$ on $[0; +\infty[$.
    1. [a.] Show that for every real $t \in [0; +\infty[$, $v'(t) = (-6{,}2 - 0{,}6t)\,\mathrm{e}^{-0{,}6t}$.
    2. [b.] By admitting that: $$v(t) = 12\,\mathrm{e}^{-0{,}6t} + \frac{1}{0{,}6} \times \frac{0{,}6t}{\mathrm{e}^{0{,}6t}}$$ determine the limit of $v$ at $+\infty$.
    3. [c.] Study the direction of variation of the function $v$ and draw up its complete variation table. Justify.
    4. [d.] Show that the equation $v(t) = 1$ has a unique solution $\alpha$, of which you will give an approximate value to the nearest tenth.
  3. When the velocity of the cart is less than or equal to 1 metre per second, a mechanical system is triggered allowing its complete stopping. Determine after how much time this system comes into action. Justify.

Part C
We recall that for every real $t$ belonging to the interval $[0; +\infty[$: $$v(t) = (12 + t)\,\mathrm{e}^{-0{,}6t}.$$ We admit that for every real $t$ in the interval $[0; +\infty[$: $$d(t) = \int_0^t v(x)\,\mathrm{d}x$$
  1. Using integration by parts, show that the distance travelled by the cart between times 0 and $t$ is given by: $$d(t) = \mathrm{e}^{-0{,}6t}\left(-\frac{5}{3}t - \frac{205}{9}\right) + \frac{205}{9}$$
  2. We recall that the stopping device is triggered when the velocity of the cart is less than or equal to 1 metre per second. Determine, according to this model, an approximate value to the nearest hundredth of the distance travelled by the cart in the braking zone before the triggering of this device.