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__asie_j1

4 maths questions

Q1 Vectors: Lines & Planes True/False or Verify a Given Statement View
Space is referred to an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We consider:
  • $\alpha$ any real number;
  • the points $\mathrm { A } ( 1 ; 1 ; 0 ) , \mathrm { B } ( 2 ; 1 ; 0 )$ and $\mathrm { C } ( \alpha ; 3 ; \alpha )$;
  • (d) the line with parametric representation:
$$\left\{ \begin{array} { l } x = 1 + t \\ y = 2 t , \quad t \in \mathbb { R } \\ z = - t \end{array} \right.$$ For each of the following statements, specify whether it is true or false, then justify the answer given. An answer without justification will not be taken into account. Statement 1: For all values of $\alpha$, the points $A , B$ and $C$ define a plane and a normal vector to this plane is $\vec { J } \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$. Statement 2: There exists exactly one value of $\alpha$ such that the lines ( $A C$ ) and (d) are parallel. Statement 3: A measure of the angle $\widehat { \mathrm { OAB } }$ is $135 ^ { \circ }$. Statement 4: The orthogonal projection of point $A$ onto the line (d) is the point $\mathrm { H } ( 1 ; 2 ; 2 )$. Statement 5: The sphere with center $O$ and radius 1 intersects the line $( d )$ at two distinct points. Recall that the sphere with center $\Omega$ and radius $r$ is the set of points in space at distance $r$ from $\Omega$.
Q2 Conditional Probability Sequential/Multi-Stage Conditional Probability View
A company that manufactures toys must perform conformity checks before their commercialization. In this exercise, we are interested in two tests performed by the toy company: a manufacturing test and a safety test. Following a large number of verifications, the company claims that:
  • $95 \%$ of toys pass the manufacturing test;
  • Among toys that pass the manufacturing test, $98 \%$ pass the safety test;
  • $1 \%$ of toys pass neither of the two tests.
A toy is chosen at random from the toys produced. We denote:
  • F the event: ``the toy passes the manufacturing test'';
  • S the event: ``the toy passes the safety test''.

Part A
  1. From the data in the statement, give the probabilities $P ( F )$ and $P _ { F } ( S )$.
  2. a. Construct a probability tree that illustrates the situation with the data available in the statement. b. Show that $P _ { \bar { F } } ( \bar { S } ) = 0.2$.
  3. Calculate the probability that the chosen toy passes both tests.
  4. Show that the probability that the toy passes the safety test is 0.97 rounded to the nearest hundredth.
  5. When the toy has passed the safety test, what is the probability that it passes the manufacturing test? Give an approximate value of the result to the nearest hundredth.

Part B
A batch of $n$ toys is randomly selected from the company's production, where $n$ is a strictly positive integer. We assume that this selection is made from a sufficiently large quantity of toys to be assimilated to a succession of $n$ independent draws with replacement. Recall that the probability that a toy passes the manufacturing test is equal to 0.95. Let $S _ { n }$ be the random variable that counts the number of toys that have passed the manufacturing test. We admit that $S _ { n }$ follows the binomial distribution with parameters $n$ and $p = 0.95$.
  1. Express the expectation and variance of the random variable $S _ { n }$ as a function of $n$.
  2. In this question, we set $n = 150$. a. Determine an approximate value to $10 ^ { - 3 }$ of $P \left( S _ { 150 } = 145 \right)$. Interpret this result in the context of the exercise. b. Determine the probability that at least $94 \%$ of the toys in this batch pass the manufacturing test. Give an approximate value of the result to $10 ^ { - 3 }$.
  3. In this question, the non-zero natural integer $n$ is no longer fixed.
Let $F _ { n }$ be the random variable defined by: $F _ { n } = \frac { S _ { n } } { n }$. The random variable $F _ { n }$ represents the proportion of toys that pass the manufacturing test in a batch of $n$ toys selected. We denote $E \left( F _ { n } \right)$ the expectation and $V \left( F _ { n } \right)$ the variance of the random variable $F _ { n }$. a. Show that $E \left( F _ { n } \right) = 0.95$ and that $V \left( F _ { n } \right) = \frac { 0.0475 } { n }$. b. We are interested in the following event $I$: ``the proportion of toys that pass the manufacturing test in a batch of $n$ toys is strictly between $93 \%$ and $97 \%$''. Using the Bienaymé-Chebyshev inequality, determine a value $n$ of the size of the batch of toys to select, from which the probability of event $I$ is greater than or equal to 0.96.
Q3 Sequences and series, recurrence and convergence Applied/contextual sequence problem View
A patient must take a dose of 2 ml of a medication every hour. We introduce the sequence $\left( u _ { n } \right)$ such that the term $u _ { n }$ represents the quantity of medication, expressed in ml, present in the body immediately after $n$ doses of medication. We have $u _ { 1 } = 2$ and for every strictly positive natural integer $n$: $u _ { n + 1 } = 2 + 0.8 u _ { n }$.
Part A Using this model, a doctor seeks to determine after how many doses of medication the quantity present in the patient's body is strictly greater than 9 mL.
  1. Calculate the value $u _ { 2 }$.
  2. Show by induction that: $$u _ { n } = 10 - 8 \times 0.8 ^ { n - 1 } \text { for every strictly positive natural integer } n.$$
  3. Determine $\lim _ { n \rightarrow + \infty } u _ { n }$ and give an interpretation of this result in the context of the exercise.
  4. Let $N$ be a strictly positive natural integer. Does the inequality $u _ { N } \geqslant 10$ have solutions? Interpret the result of this question in the context of the exercise.
  5. Determine after how many doses of medication the quantity of medication present in the patient's body is strictly greater than 9 mL. Justify your approach.

Part B Using the same modeling, the doctor is interested in the average quantity of medication present in the patient's body over time. For this purpose, the sequence ( $S _ { n }$ ) is defined for every strictly positive natural integer $n$ by $$S _ { n } = \frac { u _ { 1 } + u _ { 2 } + \cdots + u _ { n } } { n } .$$ We admit that the sequence ( $S _ { n }$ ) is increasing.
  1. Calculate $S _ { 2 }$.
  2. Show that for every strictly positive natural integer $n$, $$u _ { 1 } + u _ { 2 } + \cdots + u _ { n } = 10 n - 40 + 40 \times 0.8 ^ { n } .$$
  3. Calculate $\lim _ { n \rightarrow + \infty } S _ { n }$.
  4. The following mystery function is given, written in Python language: \begin{verbatim} def mystere(k) : n = 1 s =2 while sJustify that this value is strictly greater than 10.
Q4 5 marks Applied differentiation Full function study (variation table, limits, asymptotes) View
We consider the function $f$ defined on the interval $] 0 ; + \infty [$ by $$f ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } } { 2 \sqrt { x } }$$ and we call $\mathscr { C } _ { f }$ its representative curve in an orthonormal coordinate system.
  1. We define the function $g$ on the interval $] 0 ; + \infty \left[ \operatorname { by } g ( x ) = \mathrm { e } ^ { \sqrt { x } } \right.$. a. Show that $g ^ { \prime } ( x ) = f ( x )$ for all $x$ in the interval $] 0 ; + \infty [$. b. For all real $x$ in the interval $] 0 ; + \infty \left[ \right.$, calculate $f ^ { \prime } ( x )$ and show that: $$f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( \sqrt { x } - 1 ) } { 4 x \sqrt { x } } .$$
  2. a. Determine the limit of the function $f$ at 0. b. Interpret this result graphically.
  3. a. Determine the limit of the function $f$ at $+ \infty$. b. Study the direction of variation of the function $f$ on $] 0 ; + \infty [$. Draw the variation table of the function $f$ showing the limits at the boundaries of the domain of definition. c. Show that the equation $f ( x ) = 2$ has a unique solution on the interval $\left[ 1 ; + \infty \left[ \right. \right.$ and give an approximate value to $10 ^ { - 1 }$ of this solution.
  4. We set $I = \int _ { 1 } ^ { 2 } f ( x ) \mathrm { d } x$. a. Calculate $I$. b. Interpret the result graphically.
  5. We admit that the function $f$ is twice differentiable on the interval $] 0 ; + \infty [$ and that: $$f ^ { \prime \prime } ( x ) = \frac { \mathrm { e } ^ { \sqrt { x } } ( x - 3 \sqrt { x } + 3 ) } { 8 x ^ { 2 } \sqrt { x } } .$$ a. By setting $X = \sqrt { x }$, show that $x - 3 \sqrt { x } + 3 > 0$ for all real $x$ in the interval $] 0 ; + \infty [$. b. Study the convexity of the function $f$ on the interval $] 0 ; + \infty [$.