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 antilles-guyane

4 maths questions

Q1 6 marks Stationary points and optimisation Count or characterize roots using extremum values View
Let $f$ be the function defined on the interval $]0; +\infty[$ by $f(x) = \ln x$. For every strictly positive real number $a$, we define on $]0; +\infty[$ the function $g_a$ by $g_a(x) = ax^2$. We denote by $\mathscr{C}$ the curve representing the function $f$ and $\Gamma_a$ that of the function $g_a$ in a coordinate system of the plane. The purpose of the exercise is to study the intersection of the curves $\mathscr{C}$ and $\Gamma_a$ according to the values of the strictly positive real number $a$.
Part A
We have constructed in appendix 1 the curves $\mathscr{C}, \Gamma_{0,05}, \Gamma_{0,1}, \Gamma_{0,19}$ and $\Gamma_{0,4}$.
  1. Name the different curves on the graph. No justification is required.
  2. Use the graph to make a conjecture about the number of intersection points of $\mathscr{C}$ and $\Gamma_a$ according to the values (to be specified) of the real number $a$.

Part B
For a strictly positive real number $a$, we consider the function $h_a$ defined on the interval $]0; +\infty[$ by $$h_a(x) = \ln x - ax^2.$$
  1. Justify that $x$ is the abscissa of a point $M$ belonging to the intersection of $\mathscr{C}$ and $\Gamma_a$ if and only if $h_a(x) = 0$.
  2. a. We admit that the function $h_a$ is differentiable on $]0; +\infty[$ and we denote by $h_a'$ the derivative of the function $h_a$ on this interval. The variation table of the function $h_a$ is given below. Justify, by calculation, the sign of $h_a'(x)$ for $x$ belonging to $]0; +\infty[$.
    $x$0$\frac{1}{\sqrt{2a}}$
    $h_a'(x)$+0-
    $\frac{-1 - \ln(2a)}{2}$
    $h_a(x)$

    b. Recall the limit of $\frac{\ln x}{x}$ as $x \to +\infty$. Deduce the limit of the function $h_a$ as $x \to +\infty$. We do not ask you to justify the limit of $h_a$ at 0.
  3. In this question and only in this question, we assume that $a = 0,1$. a. Justify that, in the interval $\left.]0; \frac{1}{\sqrt{0,2}}\right]$, the equation $h_{0,1}(x) = 0$ admits a unique solution. We admit that this equation also has only one solution in the interval $]\frac{1}{\sqrt{0,2}}; +\infty[$. b. What is the number of intersection points of $\mathscr{C}$ and $\Gamma_{0,1}$?
  4. In this question and only in this question, we assume that $a = \frac{1}{2\mathrm{e}}$. a. Determine the value of the maximum of $h_{\frac{1}{2\mathrm{e}}}$. b. Deduce the number of intersection points of the curves $\mathscr{C}$ and $\Gamma_{\frac{1}{2\mathrm{e}}}$. Justify.
  5. What are the values of $a$ for which $\mathscr{C}$ and $\Gamma_a$ have no intersection points? Justify.
Q2 Exponential Distribution View
Part A
We consider a random variable $X$ that follows the exponential distribution with parameter $\lambda$ where $\lambda > 0$. We recall that, for every strictly positive real number $a$, $$P(X \leqslant a) = \int_0^a \lambda \mathrm{e}^{-\lambda t} \,\mathrm{d}t$$ We propose to calculate the mathematical expectation of $X$, denoted $E(X)$, and defined by $$E(X) = \lim_{x \to +\infty} \int_0^x \lambda t \mathrm{e}^{-\lambda t} \,\mathrm{d}t$$ We admit that the function $F$ defined on $\mathbb{R}$ by $F(t) = -\left(t + \frac{1}{\lambda}\right)\mathrm{e}^{-\lambda t}$ is an antiderivative on $\mathbb{R}$ of the function $f$ defined on $\mathbb{R}$ by $f(t) = \lambda t \mathrm{e}^{-\lambda t}$.
  1. Let $x$ be a strictly positive real number. Verify that $$\int_0^x \lambda t \mathrm{e}^{-\lambda t} \,\mathrm{d}t = \frac{1}{\lambda}\left(-\lambda x \mathrm{e}^{-\lambda x} - \mathrm{e}^{-\lambda x} + 1\right)$$
  2. Deduce that $E(X) = \frac{1}{\lambda}$.

Part B
The lifetime, expressed in years, of an electronic component can be modeled by a random variable denoted $X$ following the exponential distribution with parameter $\lambda$ where $\lambda > 0$.
  1. On the graph of appendix 2 (to be returned with the answer sheet): a. Represent the probability $P(X \leqslant 1)$. b. Indicate where the value of $\lambda$ can be read directly.
  2. We assume that $E(X) = 2$. a. What does the value of the mathematical expectation of the random variable $X$ represent in the context of the exercise? b. Calculate the value of $\lambda$. c. Calculate $P(X \leqslant 2)$. Give the exact value then the value rounded to 0.01. Interpret this result. d. Given that the component has already functioned for one year, what is the probability that its total lifetime is at least three years? Give the exact value.

Part C
An electronic circuit is composed of two identical components numbered 1 and 2. We denote by $D_1$ the event ``component 1 fails before one year'' and we denote by $D_2$ the event ``component 2 fails before one year''. We assume that the two events $D_1$ and $D_2$ are independent and that $P(D_1) = P(D_2) = 0,39$. Two possible configurations are considered:
  1. When the two components are connected ``in parallel'', circuit A fails only if both components fail at the same time. Calculate the probability that circuit A fails before one year.
  2. When the two components are connected ``in series'', circuit B fails as soon as at least one of the two components fails. Calculate the probability that circuit B fails before one year.
Q3 4 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
We denote by $\mathbb{C}$ the set of complex numbers. In the complex plane equipped with an orthonormal coordinate system $(O; \vec{u}, \vec{v})$ we have placed a point $M$ with affixe $z$ belonging to $\mathbb{C}$, then the point $R$ intersection of the circle with center $O$ passing through $M$ and the half-axis $[O; \vec{u})$.
Part A
  1. Express the affixe of point $R$ as a function of $z$.
  2. Let the point $M'$ with affixe $z'$ defined by $$z' = \frac{1}{2}\left(\frac{z + |z|}{2}\right).$$ Reproduce the figure on the answer sheet and construct the point $M'$.

Part B
We define the sequence of complex numbers $(z_n)$ by a first term $z_0$ belonging to $\mathbb{C}$ and, for every natural integer $n$, by the recurrence relation: $$z_{n+1} = \frac{z_n + |z_n|}{4}$$ The purpose of this part is to study whether the behavior at infinity of the sequence $(|z_n|)$ depends on the choice of $z_0$.
  1. What can we say about the behavior at infinity of the sequence $(|z_n|)$ when $z_0$ is a negative real number?
  2. What can we say about the behavior at infinity of the sequence $(|z_n|)$ when $z_0$ is a positive real number?
  3. We now assume that $z_0$ is not a real number. a. What conjecture can we make about the behavior at infinity of the sequence $(|z_n|)$? b. Prove this conjecture, then conclude.
Q4 Sequences and series, recurrence and convergence Algorithm and programming for sequences View
(Candidates who have not followed the specialization course)
Part A
We consider the following algorithm:
\begin{tabular}{l} Variables:
Input: Processing:
Output:
&
$k$ and $p$ are natural integers
$u$ is a real number
Ask for the value of $p$
Assign to $u$ the value 5
For $k$ varying from 1 to $p$
Assign to $u$ the value $0,5u + 0,5(k-1) - 1,5$
End for
Display $u$
\hline \end{tabular}
Run this algorithm for $p = 2$ by indicating the values of the variables at each step. What number do we obtain as output?
Part B
Let $(u_n)$ be the sequence defined by its first term $u_0 = 5$ and, for every natural integer $n$ by $$u_{n+1} = 0,5u_n + 0,5n - 1,5.$$
  1. Modify the algorithm from the first part to obtain as output all the values of $u_n$ for $n$ varying from 1 to $p$.
  2. Using the modified algorithm, after entering $p = 4$, we obtain the following results:
    $n$1234
    $u_n$1$-0,5$$-0,75$$-0,375$

    Can we assert, based on these results, that the sequence $(u_n)$ is decreasing? Justify.
  3. Prove by induction that for every natural integer $n$ greater than or equal to 3, $u_{n+1} > u_n$. What can we deduce about the monotonicity of the sequence $(u_n)$?
  4. Let $(v_n)$ be the sequence defined for every natural integer $n$ by $v_n = 0,1u_n - 0,1n + 0,5$. Prove that the sequence $(v_n)$ is geometric with ratio 0,5 and express $v_n$ as a function of $n$.
  5. Deduce that, for every natural integer $n$, $u_n = 2^{1-n} \cdot 5 + n - 5$ (or the equivalent closed form expression for $u_n$ as a function of $n$).