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__amerique-sud_j1

4 maths questions

Q1 4 marks Conditional Probability Markov Chain / Day-to-Day Transition Probabilities View
A student eats every day at the university restaurant. This restaurant offers vegetarian and non-vegetarian dishes.
  • When on a given day the student has chosen a vegetarian dish, the probability that he chooses a vegetarian dish the next day is 0.9.
  • When on a given day the student has chosen a non-vegetarian dish, the probability that he chooses a vegetarian dish the next day is 0.7.

For any natural number $n$, we denote by $V _ { n }$ the event ``the student chose a vegetarian dish on the $n ^ { \mathrm { th } }$ day'' and $p _ { n }$ the probability of $V _ { n }$. On the first day of the semester, the student chose the vegetarian dish. Thus $p _ { 1 } = 1$.
  1. a. Indicate the value of $p _ { 2 }$. b. Show that $p _ { 3 } = 0.88$. You may use a probability tree. c. Given that on the 3rd day the student chose a vegetarian dish, what is the probability that he chose a non-vegetarian dish the previous day? Round the result to $10 ^ { - 2 }$.
  2. Copy and complete the probability tree.
  3. Justify that, for any natural number $n \geqslant 1 , p _ { n + 1 } = 0.2 p _ { n } + 0.7$.
  4. We wish to have the list of the first terms of the sequence $( p _ { n } )$ for $n \geqslant 1$. For this, we use a function called meals programmed in Python language, of which three versions are proposed below.

\begin{verbatim} Program 1 def meals(n): p=1 L= [p] for k in range(1,n): p = 0.2*p+0.7 L. append(p) return(L) \end{verbatim}
\begin{verbatim} Program 2 def meals(n): p=1 L= [p] for k in range(1,n+1): p = 0.2*p+0.7 L. append(p) return(L) \end{verbatim}
\begin{verbatim} Program 3 def meals(n): p=1 L=[p] for k in range(1,n): p = 0.2*p+0.7 L.append(p+1) return(L) \end{verbatim}
a. Which of these programs allows displaying the first $n$ terms of the sequence $\left( p _ { n } \right)$? No justification is required. b. With the program chosen in question a., give the result displayed for $n = 5$.
4. Prove by induction that, for any natural number $n \geqslant 1 , p _ { n } = 0.125 \times 0.2 ^ { n - 1 } + 0.875$.
5. Deduce the limit of the sequence $\left( p _ { n } \right)$.
Q2 Combinations & Selection Basic Combination Computation View
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
  1. Two football teams of 22 and 25 players shake hands at the end of a match. Each player from one team shakes hands once with each player from the other team.

Statement 1 47 handshakes were exchanged.
2. A race involves 18 competitors. The three first-place finishers are rewarded indiscriminately by offering the same prize to each.
Statement 2 There are 4896 possibilities for distributing these prizes.
3. An association organizes a hurdle race competition that will establish a podium (the podium consists of the three best athletes ranked in their order of arrival). Seven athletes participate in the tournament. Jacques is one of them.
Statement 3 There are 90 different podiums on which Jacques appears.
4. Let $X _ { 1 }$ and $X _ { 2 }$ be two random variables with the same distribution given by the table below:
$x _ { i }$- 2- 125
$P \left( X = x _ { i } \right)$0.10.40.30.2

We assume that $X _ { 1 }$ and $X _ { 2 }$ are independent and we consider $Y$ the random variable sum of these two random variables. Statement 4 $P ( Y = 4 ) = 0.25$.
5. A swimmer trains with the objective of swimming 50 metres freestyle in less than 25 seconds. Through training, it turns out that the probability of achieving this is 0.85. He performs, on one day, 20 timed 50-metre swims. We denote by $X$ the random variable that counts the number of times he swims this distance in less than 25 seconds on this day. We admit that $X$ follows the binomial distribution with parameters $n = 20$ and $p = 0.85$.
Statement 5 Given that he achieved his objective at least 15 times, an approximate value to $10 ^ { - 3 }$ of the probability that he achieved it at least 18 times is 0.434.
Q3 Differential equations Applied Modeling with Differential Equations View
We propose to study the concentration in the blood of a medication ingested by a person for the first time. Let $t$ be the time (in hours) elapsed since the ingestion of this medication. We admit that the concentration of this medication in the blood, in grams per litre of blood, is modelled by a function $f$ of the variable $t$ defined on the interval $[ 0 ; + \infty [$.
Part A: graphical readings

The graph above shows the representative curve of the function $f$. With the precision allowed by the graph, give without justification:
  1. The time elapsed from the moment of ingestion of this medication to the moment when the concentration of medication in the blood is maximum according to this model.
  2. The set of solutions to the inequality $f ( t ) \geqslant 1$.
  3. The convexity of the function $f$ on the interval $[ 0 ; 8 ]$.

Part B: determination of the function $\boldsymbol { f }$
We consider the differential equation
$$( E ) : \quad y ^ { \prime } + y = 5 \mathrm { e } ^ { - t }$$
of unknown $y$, where $y$ is a function defined and differentiable on the interval $[ 0 ; + \infty [$. We admit that the function $f$ is a solution of the differential equation $( E )$.
  1. Solve the differential equation $\left( E ^ { \prime } \right) : y ^ { \prime } + y = 0$.
  2. Let $u$ be the function defined on the interval $\left[ 0 ; + \infty \left[ \operatorname { by } u ( t ) = a t \mathrm { e } ^ { - t } \right. \right.$ with $a \in \mathbb { R }$.

Determine the value of the real number $a$ such that the function $u$ is a solution of equation $( E )$.
3. Deduce the set of solutions of the differential equation $( E )$.
4. Since the person has not taken this medication before, we admit that $f ( 0 ) = 0$.
Determine the expression of the function $f$.
Part C: study of the function $\boldsymbol { f }$
In this part, we admit that $f$ is defined on the interval $\left[ 0 ; + \infty \left[ \operatorname { by } f ( t ) = 5 t \mathrm { e } ^ { - t } \right. \right.$.
  1. Determine the limit of $f$ at $+ \infty$.

Interpret this result in the context of the exercise.
2. Study the variations of $f$ on the interval $[ 0 ; + \infty [$ then draw up its complete variation table.
3. Prove that there exist two real numbers $t _ { 1 }$ and $t _ { 2 }$ such that $f \left( t _ { 1 } \right) = f \left( t _ { 2 } \right) = 1$.
Give an approximate value to $10 ^ { - 2 }$ of the real numbers $t _ { 1 }$ and $t _ { 2 }$.
4. For a medication concentration greater than or equal to 1 gram per litre of blood, there is a risk of drowsiness. What is the duration in hours and minutes of the drowsiness risk when taking this medication?
Part D: average concentration
The average concentration of the medication (in grams per litre of blood) during the first hour is given by:
$$T _ { m } = \int _ { 0 } ^ { 1 } f ( t ) \mathrm { d } t$$
where $f$ is the function defined on $\left[ 0 ; + \infty \left[ \operatorname { by } f ( t ) = 5 t \mathrm { e } ^ { - t } \right. \right.$. Calculate this average concentration. Give the exact value then an approximate value to 0.01.
Q4 Vectors: Lines & Planes Find Parametric Representation of a Line View
Space is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We consider the points
$$\mathrm { A } ( 2 \sqrt { 3 } ; 0 ; 0 ) , \quad \mathrm { B } ( 0 ; 2 ; 0 ) , \quad \mathrm { C } ( 0 ; 0 ; 1 ) \quad \text { and } \quad \mathrm { K } \left( \frac { \sqrt { 3 } } { 2 } ; \frac { 3 } { 2 } ; 0 \right) .$$
  1. Justify that a parametric representation of the line (CK) is:

$$\left\{ \begin{aligned} x & = \frac { \sqrt { 3 } } { 2 } t \\ y & = \frac { 3 } { 2 } t \quad ( t \in \mathbb { R } ) \\ z & = 1 - t \end{aligned} \right.$$
    \setcounter{enumi}{1}
  1. Let $\mathrm { M } ( t )$ be a point on the line (CK) parametrized by a real number $t$. Establish that $\mathrm { OM} ( t ) = \sqrt { 4 t ^ { 2 } - 2 t + 1 }$.
  2. Let $f$ be the function defined and differentiable on $\mathbb { R }$ by $f ( t ) = \mathrm { OM } ( t )$. a. Study the variations of the function $f$ on $\mathbb { R }$. b. Deduce the value of $t$ for which $f$ reaches its minimum.
  3. Deduce that the point $\mathrm { H } \left( \frac { \sqrt { 3 } } { 8 } ; \frac { 3 } { 8 } ; \frac { 3 } { 4 } \right)$ is the orthogonal projection of point O onto the line (CK).
  4. Prove, using the dot product tool, that point H is the orthocenter (intersection of the altitudes of a triangle) of triangle ABC.
  5. a. Prove that the line $( \mathrm { OH } )$ is orthogonal to the plane $( \mathrm { ABC } )$. b. Deduce an equation of the plane (ABC).
  6. Calculate, in square units, the area of triangle ABC.