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
2015 amerique-sud

8 maths questions

Q1A Curve Sketching Asymptote Determination View
In the plane equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, we denote by $\mathscr{C}_{u}$ the curve representing the function $u$ defined on the interval $]0; +\infty[$ by: $$u(x) = a + \frac{b}{x} + \frac{c}{x^2}$$ where $a, b$ and $c$ are fixed real numbers.
The curve $\mathscr{C}_{u}$ passes through the points $\mathrm{A}(1; 0)$ and $\mathrm{B}(4; 0)$ and the $y$-axis and the line $\mathscr{D}$ with equation $y = 1$ are asymptotes to the curve $\mathscr{C}_{u}$.
  1. Give the values of $u(1)$ and $u(4)$.
  2. Give $\lim_{x \rightarrow +\infty} u(x)$. Deduce the value of $a$.
  3. Deduce that, for all strictly positive real $x$, $u(x) = \frac{x^2 - 5x + 4}{x^2}$.
Q1B Stationary points and optimisation Construct or complete a full variation table View
Let $f$ be the function defined on the interval $]0; +\infty[$ by: $$f(x) = x - 5\ln x - \frac{4}{x}$$
  1. Determine the limit of $f(x)$ as $x$ tends to 0. You may use without proof the fact that $\lim_{x \rightarrow 0} x \ln x = 0$.
  2. Determine the limit of $f(x)$ as $x$ tends to $+\infty$.
  3. Prove that, for all strictly positive real $x$, $f'(x) = u(x)$, where $u(x) = \frac{x^2 - 5x + 4}{x^2}$.

Deduce the table of variations of the function $f$ by specifying the limits and particular values.
Q1C Areas Between Curves Compute Area Directly (Numerical Answer) View
  1. Determine the area $\mathscr{A}$, expressed in square units, of the shaded region in the graph of Part A (the region bounded by the curve $\mathscr{C}_u$ where $u(x) = \frac{x^2 - 5x + 4}{x^2}$ between its zeros $x=1$ and $x=4$).
  2. For all real $\lambda$ greater than or equal to 4, we denote by $\mathscr{A}_{\lambda}$ the area, expressed in square units, of the region formed by the points $M$ with coordinates $(x; y)$ such that $$4 \leqslant x \leqslant \lambda \quad \text{and} \quad 0 \leqslant y \leqslant u(x).$$ Does there exist a value of $\lambda$ for which $\mathscr{A}_{\lambda} = \mathscr{A}$?
Q2 Vectors: Lines & Planes True/False or Verify a Given Statement View
For each of the following statements, indicate whether it is true or false and justify the answer.
Space is equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. The points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ are defined by their coordinates: $$\mathrm{A}(3; -1; 4), \quad \mathrm{B}(-1; 2; -3), \quad \mathrm{C}(4; -1; 2).$$ The plane $\mathscr{P}$ has the Cartesian equation: $2x - 3y + 2z - 7 = 0$. The line $\Delta$ has the parametric representation $\left\{\begin{array}{rl} x &= -1 + 4t \\ y &= 4 - t \\ z &= -8 + 2t \end{array}, t \in \mathbb{R}\right.$.
Statement 1: The lines $\Delta$ and $(AC)$ are orthogonal.
Statement 2: The points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ determine a plane and this plane has the Cartesian equation $2x + 5y + z - 5 = 0$.
Statement 3: All points whose coordinates $(x; y; z)$ are given by $$\left\{\begin{array}{rl} x &= 1 + s - 2s' \\ y &= 1 - 2s + s' \\ z &= 1 - 4s + 2s' \end{array}\right., \quad s, s' \in \mathbb{R}$$ lie in the plane $\mathscr{P}$.
Statement 4: There exists a plane parallel to the plane $\mathscr{P}$ which contains the line $\Delta$.
Q3A Conditional Probability Conditional Probability as a Function of a Parameter View
Chikungunya is a viral disease transmitted from one human to another by bites of infected female mosquitoes. A test has been developed for the screening of this virus. The laboratory manufacturing this test provides the following characteristics:
  • the probability that a person affected by the virus has a positive test is 0.98;
  • the probability that a person not affected by the virus has a positive test is 0.01.
An individual is chosen at random from a target population. We call:
  • $M$ the event: "The chosen individual is affected by chikungunya"
  • $T$ the event: "The test of the chosen individual is positive"
We denote by $\bar{M}$ (respectively $\bar{T}$) the opposite event of the event $M$ (respectively $T$). We denote by $p$ $(0 \leqslant p \leqslant 1)$ the proportion of people affected by the disease in the target population.
  1. a. Copy and complete the probability tree. b. Express $P(M \cap T)$, $P(\bar{M} \cap T)$ then $P(T)$ as a function of $p$.
  2. a. Prove that the probability of $M$ given $T$ is given by the function $f$ defined on $[0;1]$ by: $$f(p) = \frac{98p}{97p + 1}$$ b. Study the variations of the function $f$.
  3. We consider that the test is reliable when the probability that a person with a positive test is actually affected by chikungunya is greater than 0.95. Using the results of question 2., from what proportion $p$ of sick people in the population is the test reliable?
Q3B Modelling and Hypothesis Testing View
In July 2014, the health surveillance institute of an island published that $15\%$ of the population is affected by the virus. To verify whether the actual proportion is higher, a sample of 1000 people chosen at random from this island is studied. The population is large enough to consider that such a sample results from draws with replacement.
We denote by $X$ the random variable which, for any sample of 1000 people chosen at random, corresponds to the number of people affected by the virus and by $F$ the random variable giving the associated frequency.
  1. a. Under the hypothesis $p = 0.15$, determine the distribution of $X$. b. In a sample of 1000 people chosen at random from the island, 197 people affected by the virus are counted. What conclusion can be drawn from this observation about the figure of $15\%$ published by the health surveillance institute? Justify. (You may use the calculation of a fluctuation interval at the $95\%$ threshold.)
  2. We now consider that the value of $p$ is unknown. Using the sample from question 1.b., propose a confidence interval for the value of $p$, at the $95\%$ confidence level.
Q3C Normal Distribution Finding Unknown Mean from a Given Probability Condition View
The incubation time, expressed in hours, of the virus can be modeled by a random variable $T$ following a normal distribution with standard deviation $\sigma = 10$. We wish to determine its mean $\mu$.
  1. a. Conjecture, using the graph of the probability density function, an approximate value of $\mu$. b. We are given $P(T < 110) = 0.18$. Shade on the graph a region whose area corresponds to the given probability.
  2. We denote by $T'$ the random variable equal to $\frac{T - \mu}{10}$. a. What distribution does the random variable $T'$ follow? b. Determine an approximate value to the nearest unit of the mean $\mu$ of the random variable $T$ and verify the conjecture from question 1.
Q4 Sequences and series, recurrence and convergence Applied/contextual sequence problem View
In a country with a constant population equal to 120 million, the inhabitants live either in rural areas or in cities. Population movements can be modeled as follows:
  • in 2010, the population consists of 90 million rural inhabitants and 30 million city dwellers;
  • each year, $10\%$ of rural inhabitants migrate to the city;
  • each year, $5\%$ of city dwellers migrate to rural areas.
For any natural integer $n$, we denote:
  • $u_n$ the population in rural areas, in the year $2010 + n$, expressed in millions of inhabitants;
  • $v_n$ the population in cities, in the year $2010 + n$, expressed in millions of inhabitants.
We have $u_0 = 90$ and $v_0 = 30$.
Part A
  1. Translate the fact that the total population is constant by a relation linking $u_n$ and $v_n$.
  2. What formulas can be entered in cells B3 and C3 which, copied downward, allow us to obtain the spreadsheet showing the evolution of $(u_n)$ and $(v_n)$?
  3. What conjectures can be made concerning the long-term evolution of this population?

Part B
Deduce from the recurrence relations that, for every natural integer $n$, $R_n = 50 \times 0.85^n + 40$ and determine the expression of $C_n$ as a function of $n$. b. Determine the limit of $R_n$ and of $C_n$ when $n$ tends towards $+\infty$. What can we conclude from this for the population studied? 6. a. Complete the algorithm so that it displays the number of years after which the urban population will exceed the rural population. b. By solving the inequality with unknown $n$, $$50 \times 0.85^n + 40 < 80 - 50 \times 0.85^n,$$ find again the value displayed by the algorithm.