bac-s-maths

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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 centres-etrangers

6 maths questions

QExercise 2 6 marks Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
The purpose of this exercise is to study the sequence $(u_n)$ defined by the value of its first term $u_1$ and, for every natural number $n$ greater than or equal to 1, by the relation: $$u_{n+1} = (n+1)u_n - 1$$
Part A
  1. Verify, by detailing the calculation, that if $u_1 = 0$ then $u_4 = -17$.
  2. Copy and complete the algorithm below so that by first entering in $U$ a value of $u_1$ it calculates the terms of the sequence $(u_n)$ from $u_2$ to $u_{13}$.
    For $N$ going from 1 to 12 $$U \leftarrow$$ End For
  3. This algorithm was executed for $u_1 = 0.7$ then for $u_1 = 0.8$.
    Here are the values obtained.
    For $u_1 = 0.7$For $u_1 = 0.8$
    0.40.6
    0.20.8
    -0.22.2
    -210
    -1359
    -92412
    -7373295
    -663429654
    -66341296539
    -7297523261928
    -875702539143135
    -113841326508860754

    What appears to be the limit of this sequence if $u_1 = 0.7$? And if $u_1 = 0.8$?

Part B
We consider the sequence $(I_n)$ defined for every natural number $n$, greater than or equal to 1, by: $$I_n = \int_0^1 x^n \mathrm{e}^{1-x} \mathrm{~d}x$$ We recall that the number e is the value of the exponential function at 1, that is to say that $\mathrm{e} = \mathrm{e}^1$.
  1. Prove that the function $F$ defined on the interval $[0;1]$ by $F(x) = (-1-x)\mathrm{e}^{1-x}$ is an antiderivative on the interval $[0;1]$ of the function $f$ defined on the interval $[0;1]$ by $f(x) = x\mathrm{e}^{1-x}$.
  2. Deduce that $I_1 = \mathrm{e} - 2$.
  3. It is admitted that, for every natural number $n$ greater than or equal to 1, we have: $$I_{n+1} = (n+1)I_n - 1.$$ Use this formula to calculate $I_2$.
  4. a. Justify that, for every real number $x$ in the interval $[0;1]$ and for every natural number $n$ greater than or equal to 1, we have: $0 \leqslant x^n \mathrm{e}^{1-x} \leqslant x^n \mathrm{e}$. b. Justify that: $\int_0^1 x^n \mathrm{e} \, \mathrm{d}x = \dfrac{\mathrm{e}}{n+1}$. c. Deduce that, for every natural number $n$ greater than or equal to 1, we have: $0 \leqslant I_n \leqslant \dfrac{\mathrm{e}}{n+1}$. d. Determine $\lim_{n \rightarrow +\infty} I_n$.

Part C
In this part, we denote by $n!$ the number defined, for every natural number $n$ greater than or equal to 1, by: $1! = 1$, $2! = 2 \times 1$, and if $n \geqslant 3$: $n! = n \times (n-1) \times \ldots \times 1$. And, more generally: $(n+1)! = (n+1) \times n!$
  1. Prove by induction that, for every natural number $n$ greater than or equal to 1, we have: $$u_n = n! \left(u_1 - \mathrm{e} + 2\right) + I_n$$ We recall that, for every natural number $n$ greater than or equal to 1, we have: $$u_{n+1} = (n+1)u_n - 1 \quad \text{and} \quad I_{n+1} = (n+1)I_n - 1.$$
  2. It is admitted that: $\lim_{n \rightarrow +\infty} n! = +\infty$. a. Determine the limit of the sequence $(u_n)$ when $u_1 = 0.7$. b. Determine the limit of the sequence $(u_n)$ when $u_1 = 0.8$.
QExercise 3 5 marks Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View
The plane is equipped with a direct orthonormal coordinate system $(\mathrm{O}; \vec{u}, \vec{v})$.
The purpose of this exercise is to determine the non-zero complex numbers $z$ such that the points with affixes $1$, $z^2$ and $\dfrac{1}{z}$ are collinear. On the graph provided in the appendix, point A has affix 1.
Part A: study of examples
1. A first example
In this question, we set $z = \mathrm{i}$. a. Give the algebraic form of the complex numbers $z^2$ and $\dfrac{1}{z}$. b. Plot the points $N_1$ with affix $z^2$, and $P_1$ with affix $\dfrac{1}{z}$ on the graph given in the appendix. We note that in this case the points $\mathrm{A}$, $N_1$ and $P_1$ are not collinear.
2. An equation
Solve in the set of complex numbers the equation with unknown $z$: $z^2 + z + 1 = 0$.
3. A second example
In this question, we set: $z = -\dfrac{1}{2} + \mathrm{i}\dfrac{\sqrt{3}}{2}$. a. Determine the exponential form of $z$, then those of the complex numbers $z^2$ and $\dfrac{1}{z}$. b. Plot the points $N_2$ with affix $z^2$ and $P_2$, with affix $\dfrac{1}{z}$ on the graph given in the appendix. We note that in this case the points $\mathrm{A}$, $N_2$ and $P_2$ are collinear.
Part B
Let $z$ be a non-zero complex number. We denote by $N$ the point with affix $z^2$ and $P$ the point with affix $\dfrac{1}{z}$.
  1. Establish that, for every complex number different from 0, we have: $$z^2 - \frac{1}{z} = \left(z^2 + z + 1\right)\left(1 - \frac{1}{z}\right)$$
  2. We recall that if $\vec{U}$ is a non-zero vector and $\vec{V}$ is a vector with affixes respectively $z_{\vec{U}}$ and $z_{\vec{V}}$, the vectors $\vec{U}$ and $\vec{V}$ are collinear if and only if there exists a real number $k$ such that $z_{\vec{V}} = k z_{\vec{U}}$. Deduce that, for $z \neq 0$, the points $\mathrm{A}$, $N$ and $P$ defined above are collinear if and only if $z^2 + z + 1$ is a real number.
  3. We set $z = x + \mathrm{i}y$, where $x$ and $y$ denote real numbers. Justify that: $z^2 + z + 1 = x^2 - y^2 + x + 1 + \mathrm{i}(2xy + y)$.
  4. a. Determine the set of points $M$ with affix $z \neq 0$ such that the points $\mathrm{A}$, $N$ and $P$ are collinear. b. Trace this set of points on the graph given in the appendix.
Q1 1 marks Binomial Distribution MCQ Selecting a Binomial Probability Expression or Value View
A statistical study established that one in four clients practises surfing.
In a cable car accommodating 80 clients of the resort, the probability rounded to the nearest thousandth that there are exactly 20 clients practising surfing is: a. 0.560 b. 0.25 c. 1 d. 0.103
Q2 1 marks Normal Distribution Symmetric Interval / Confidence-Style Bound View
The maximum thickness of an avalanche, expressed in centimetres, can be modelled by a random variable $X$ which follows a normal distribution with mean $\mu = 150 \mathrm{~cm}$ and unknown standard deviation. We know that $P ( X \geqslant 200 ) = 0.025$. What is the probability $P ( X \geqslant 100 )$ ? a. We cannot b. 0.025 c. 0.95 d. 0.975 answer because there are missing elements in the problem statement.
Q3 1 marks Exponential Distribution View
In a snowy corridor, the time interval separating two successive avalanches, called the occurrence time of an avalanche, expressed in years, is modelled by a random variable $T$ which follows an exponential distribution. It has been established that an avalanche is triggered on average every 5 years. Thus $E ( T ) = 5$. The probability $P ( T \geqslant 5 )$ is equal to: a. 0.5 b. $1 - \mathrm{e}^{-1}$ c. $\mathrm{e}^{-1}$ d. $\mathrm{e}^{-25}$
Q4 1 marks Confidence intervals Determine minimum sample size for a desired interval width View
The tourist office wishes to conduct a survey to estimate the proportion of clients satisfied with the services offered at the ski resort. For this, it uses a confidence interval of length 0.04 with a confidence level of 0.95. The number of clients to interview is: a. 50 b. 2500 c. 25 d. 625