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
2019 amerique-sud

6 maths questions

Q1A Exponential Distribution View
Let $T$ denote the random variable equal to the lifespan, in months, of a stopwatch and we assume it follows an exponential distribution with parameter $\lambda = 0.0555$.
  1. Calculate the average lifespan of a stopwatch (rounded to the nearest unit).
  2. Calculate the probability that a stopwatch has a lifespan between one and two years.
  3. A coach has not changed his stopwatch for two years. What is the probability that it will still be in working order for at least one more year?
Q1B Conditional Probability Bayes' Theorem with Production/Source Identification View
This club makes group orders of bearings for its members from two suppliers A and B.
  • Supplier A offers higher prices but the bearings it sells are defect-free with a probability of 0.97.
  • Supplier B offers more advantageous prices but its bearings are defective with a probability of 0.05.

A bearing is chosen at random from the club's stock and we consider the events: $A$: ``the bearing comes from supplier A'', $B$: ``the bearing comes from supplier B'', $D$: ``the bearing is defective''.
  1. The club buys $40\%$ of its bearings from supplier A and the rest from supplier B. a. Calculate the probability that the bearing comes from supplier A and is defective. b. The bearing is defective. Calculate the probability that it comes from supplier B.
  2. If the club wants less than $3.5\%$ of the bearings to be defective, what minimum proportion of bearings should it order from supplier A?
Q1C Normal Distribution Finding Unknown Standard Deviation from a Given Probability Condition View
The standard inner diameter of a bearing on a roller wheel is 8 mm. Let $X$ denote the random variable giving in mm the diameter of a bearing and we assume that $X$ follows a normal distribution with mean 8 and standard deviation 0.1.
A bearing is said to be compliant if its diameter is between $7.8 \mathrm{~mm}$ and $8.2 \mathrm{~mm}$.
  1. Calculate the probability that a bearing is compliant.
  2. Supplier $B$ sells its bearings in batches of 16 and claims that only $5\%$ of its bearings are non-compliant. The club president, who bought 30 batches from him, finds that 38 bearings are non-compliant. Does this check call into question supplier B's claim? An asymptotic fluctuation interval at the $95\%$ threshold may be used.
  3. The bearing manufacturer of this supplier decides to improve the production of its bearings. The adjustment of the machine that manufactures them is modified so that $96\%$ of the bearings are compliant. We assume that after adjustment the random variable $X$ follows a normal distribution with mean 8 and standard deviation $\sigma$. a. What is the distribution followed by $\frac{X - 8}{\sigma}$? b. Determine $\sigma$ so that the manufactured bearing is compliant with a probability equal to 0.96.
Q2 5 marks Applied differentiation Applied modeling with differentiation View
Vasopressin is a hormone that promotes the reabsorption of water by the body. The level of vasopressin in the blood is considered normal if it is less than $2.5 \mu\mathrm{g}/\mathrm{mL}$. This hormone is secreted as soon as blood volume decreases. In particular, vasopressin is produced following a hemorrhage.
The following model will be used:
$$f(t) = 3t\mathrm{e}^{-\frac{1}{4}t} + 2 \text{ with } t \geqslant 0$$
where $f(t)$ represents the level of vasopressin (in $\mu\mathrm{g}/\mathrm{mL}$) in the blood as a function of time $t$ (in minutes) elapsed after the start of a hemorrhage.
  1. a. What is the level of vasopressin in the blood at time $t = 0$? b. Justify that twelve seconds after a hemorrhage, the level of vasopressin in the blood is not normal. c. Determine the limit of the function $f$ as $t \to +\infty$. Interpret this result.
  2. We admit that the function $f$ is differentiable on $[0; +\infty[$.
    Verify that for every positive real number $t$,
    $$f^{\prime}(t) = \frac{3}{4}(4 - t)\mathrm{e}^{-\frac{1}{4}t}$$
  3. a. Study the direction of variation of $f$ on the interval $[0; +\infty[$ and draw the variation table of the function $f$ (including the limit as $t \to +\infty$). b. At what time is the level of vasopressin maximal? What is this level then? Give an approximate value to $10^{-2}$ near.
  4. a. Prove that there exists a unique value $t_0$ belonging to $[0; 4]$ such that $f(t_0) = 2.5$. Give an approximate value to $10^{-3}$ near. We admit that there exists a unique value $t_1$ belonging to $[4; +\infty[$ satisfying $f(t_1) = 2.5$. An approximate value of $t_1$ to $10^{-3}$ near is given: $t_1 \approx 18.930$. b. Determine for how long, in a person who has suffered a hemorrhage, the level of vasopressin remains above $2.5 \mu\mathrm{g}/\mathrm{mL}$ in the blood.
  5. Let $F$ be the function defined on $[0; +\infty[$ by $F(t) = -12(t + 4)\mathrm{e}^{-\frac{1}{4}t} + 2t$. a. Prove that the function $F$ is an antiderivative of the function $f$ and deduce an approximate value of $\int_{t_0}^{t_1} f(t)\,\mathrm{d}t$ to the nearest unit. b. Deduce an approximate value to 0.1 near of the average level of vasopressin, during a hemorrhagic accident during the period when this level is above $2.5 \mu\mathrm{g}/\mathrm{mL}$.
Q3 Vectors 3D & Lines Multi-Part 3D Geometry Problem View
We consider a cube $ABCDEFGH$. The point M is the midpoint of $[\mathrm{BF}]$, I is the midpoint of [BC], the point N is defined by the relation $\overrightarrow{\mathrm{CN}} = \frac{1}{2}\overrightarrow{\mathrm{GC}}$ and the point P is the center of the face ADHE.
Part A:
  1. Justify that the line (MN) intersects the segment [BC] at its midpoint I.
  2. Construct, on the figure provided in the appendix, the cross-section of the cube by the plane (MNP).

Part B:
We equip space with the orthonormal coordinate system ($A; \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE}$).
  1. Justify that the vector $\vec{n}\left(\begin{array}{l}1\\2\\2\end{array}\right)$ is a normal vector to the plane (MNP).
    Deduce a Cartesian equation of the plane (MNP).
  2. Determine a system of parametric equations of the line (d) passing through G and orthogonal to the plane (MNP).
  3. Show that the line (d) intersects the plane (MNP) at the point K with coordinates $\left(\frac{2}{3}; \frac{1}{3}; \frac{1}{3}\right)$. Deduce the distance GK.
  4. We admit that the four points M, E, D and I are coplanar and that the area of the quadrilateral MEDI is $\frac{9}{8}$ square units. Calculate the volume of the pyramid GMEDI.
Q4 5 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
We consider the sequence $\left(u_n\right)$ defined for every integer $n \geqslant 0$ by: $\left\{\begin{array}{l}u_{n+1} = 3 - \dfrac{10}{u_n + 4}\\u_0 = 5\end{array}\right.$
Part A:
  1. Determine the exact value of $u_1$ and $u_2$.
  2. Prove by induction that for every natural integer $n$, $u_n \geqslant 1$.
  3. Prove that, for every natural integer $n$, $u_{n+1} - u_n = \dfrac{\left(1 - u_n\right)\left(u_n + 2\right)}{u_n + 4}$.
  4. Deduce the direction of variation of the sequence $(u_n)$.
  5. Justify that the sequence $\left(u_n\right)$ converges.

Part B:
We consider the sequence $(v_n)$ defined for every natural integer $n$ by $v_n = \dfrac{u_n - 1}{u_n + 2}$.
  1. a. Prove that $\left(v_n\right)$ is a geometric sequence whose common ratio and first term $v_0$ we will determine. b. Express $v_n$ as a function of $n$.
    Deduce that for every natural integer $n$, $v_n \neq 1$.
  2. Prove that for every natural integer $n$, $u_n = \dfrac{2v_n + 1}{1 - v_n}$.
  3. Deduce the limit of the sequence $(u_n)$.

Part C:
We consider the algorithm below.
$u \leftarrow 5$
$n \leftarrow 0$
While $u \geqslant 1.01$
$n \leftarrow n + 1$
$u \leftarrow 3 - \dfrac{10}{u + 4}$
End While

  1. After execution of the algorithm, what value is contained in the variable $n$?
  2. Using parts A and B, interpret this value.