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
2021 bac-spe-maths__polynesie

5 maths questions

QA Differential equations First-Order Linear DE: General Solution View
EXERCISE A Main topics covered: Exponential function, convexity, differentiation, differential equations
This exercise consists of three independent parts. Below is represented, in an orthonormal coordinate system, a portion of the representative curve $\mathscr { C }$ of a function $f$ defined on $\mathbb { R }$.
Consider the points $\mathrm { A } ( 0 ; 2 )$ and $\mathrm { B } ( 2 ; 0 )$.
Part 1
Knowing that the curve $\mathscr { C }$ passes through A and that the line (AB) is tangent to the curve $\mathscr { C }$ at point A, give by reading the graph:
  1. The value of $f ( 0 )$ and that of $f ^ { \prime } ( 0 )$.
  2. An interval on which the function $f$ appears to be convex.

Part 2
We denote $(E)$ the differential equation $$y ^ { \prime } = - y + \mathrm { e } ^ { - x }$$ It is admitted that $g : x \longmapsto x \mathrm { e } ^ { - x }$ is a particular solution of $(E)$.
  1. Give all solutions on $\mathbb { R }$ of the differential equation $( H ) : y ^ { \prime } = - y$.
  2. Deduce all solutions on $\mathbb { R }$ of the differential equation $(E)$.
  3. Knowing that the function $f$ is the particular solution of $(E)$ which satisfies $f ( 0 ) = 2$, determine an expression of $f ( x )$ as a function of $x$.

Part 3
It is admitted that for every real number $x , f ( x ) = ( x + 2 ) \mathrm { e } ^ { - x }$.
  1. Recall that $f ^ { \prime }$ denotes the derivative function of the function $f$. a. Show that for all $x \in \mathbb { R } , f ^ { \prime } ( x ) = ( - x - 1 ) \mathrm { e } ^ { - x }$. b. Study the sign of $f ^ { \prime } ( x )$ for all $x \in \mathbb { R }$ and draw up the table of variations of $f$ on $\mathbb { R }$. Neither the limit of $f$ at $- \infty$ nor the limit of $f$ at $+ \infty$ will be specified. Calculate the exact value of the extremum of $f$ on $\mathbb { R }$.
  2. Recall that $f ^ { \prime \prime }$ denotes the second derivative function of the function $f$. a. Calculate for all $x \in \mathbb { R } , f ^ { \prime \prime } ( x )$. b. Can we assert that $f$ is convex on the interval $[ 0 ; + \infty [$?
QB Differentiating Transcendental Functions Full function study with transcendental functions View
EXERCISE B Main topics covered: Natural logarithm function, differentiation
This exercise consists of two parts. Some results from the first part will be used in the second.
Part 1: Study of an auxiliary function
Let the function $f$ defined on the interval $[1 ; 4]$ by: $$f ( x ) = - 30 x + 50 + 35 \ln x$$
  1. Recall that $f ^ { \prime }$ denotes the derivative function of the function $f$. a. For every real number $x$ in the interval $[ 1 ; 4 ]$, show that: $$f ^ { \prime } ( x ) = \frac { 35 - 30 x } { x }$$ b. Draw up the sign table of $f ^ { \prime } ( x )$ on the interval $[ 1 ; 4 ]$. c. Deduce the variations of $f$ on this same interval.
  2. Justify that the equation $f ( x ) = 0$ has a unique solution, denoted $\alpha$, on the interval $[1;4]$ then give an approximate value of $\alpha$ to $10 ^ { - 3 }$ near.
  3. Draw up the sign table of $f ( x )$ for $x \in [ 1 ; 4 ]$.

Part 2: Optimisation
A company sells fruit juice. For $x$ thousand litres sold, with $x$ a real number in the interval $[ 1 ; 4 ]$, the analysis of sales leads to modelling the profit $B ( x )$ by the expression given in thousands of euros by: $$B ( x ) = - 15 x ^ { 2 } + 15 x + 35 x \ln x$$
  1. According to the model, calculate the profit made by the company when it sells 2500 litres of fruit juice. Give an approximate value to the nearest euro of this profit.
  2. For all $x$ in the interval $[ 1 ; 4 ]$, show that $B ^ { \prime } ( x ) = f ( x )$ where $B ^ { \prime }$ denotes the derivative function of $B$.
  3. a. Using the results from part 1, give the variations of the function $B$ on the interval $[1;4]$. b. Deduce the quantity of fruit juice, to the nearest litre, that the company must sell in order to achieve maximum profit.
Q1 5 marks Geometric Sequences and Series Prove a Transformed Sequence is Geometric View
Consider the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 10000$ and for every natural number $n$ :
$$u _ { n + 1 } = 0,95 u _ { n } + 200 .$$
  1. Calculate $u _ { 1 }$ and verify that $u _ { 2 } = 9415$.
  2. a. Prove, using proof by induction, that for every natural number $n$ : $$u _ { n } > 4000$$ b. It is admitted that the sequence $(u _ { n })$ is decreasing. Justify that it converges.
  3. For every natural number $n$, consider the sequence $\left( v _ { n } \right)$ defined by: $v _ { n } = u _ { n } - 4000$. a. Calculate $v _ { 0 }$. b. Prove that the sequence $(v _ { n })$ is geometric with common ratio equal to 0.95. c. Deduce that for every natural number $n$ : $$u _ { n } = 4000 + 6000 \times 0,95 ^ { n }$$ d. What is the limit of the sequence $\left( u _ { n } \right)$? Justify your answer.
  4. In 2020, an animal species numbered 10000 individuals. The evolution observed in previous years leads to the estimate that from 2021 onwards, this population will decrease by $5\%$ at the beginning of each year. To slow down this decline, it was decided to reintroduce 200 individuals at the end of each year, starting from 2021.
    A representative of an association supporting this strategy claims that: ``the species should not become extinct, but unfortunately, we will not prevent a loss of more than half the population''. What do you think of this statement? Justify your answer.
Q2 5 marks Conditional Probability Bayes' Theorem with Diagnostic/Screening Test View
A test is developed to detect a disease in a country. According to the health authorities of this country, $7\%$ of the inhabitants are infected with this disease. Among infected individuals, $20\%$ test negative. Among healthy individuals, $1\%$ test positive.
A person is chosen at random from the population. We denote:
  • $M$ the event: ``the person is infected with the disease'';
  • $T$ the event: ``the test is positive''.

  1. Construct a probability tree modelling the proposed situation.
  2. a. What is the probability that the person is infected with the disease and that their test is positive? b. Show that the probability that their test is positive is 0.0653.
  3. It is known that the test of the chosen person is positive. What is the probability that they are infected? Give the result as an approximation to $10 ^ { - 2 }$ near.
  4. Ten people are chosen at random from the population. The size of the population of this country allows us to treat this sample as a draw with replacement. Let $X$ be the random variable that counts the number of individuals with a positive test among the ten people. a. What is the probability distribution followed by $X$? Specify its parameters. b. Determine the probability that exactly two people have a positive test. Give the result as an approximation to $10 ^ { - 2 }$ near.
  5. Determine the minimum number of people to test in this country so that the probability that at least one of these people has a positive test is greater than $99\%$.
Q3 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
In space, consider the cube ABCDEFGH with edge length equal to 1. We equip the space with the orthonormal coordinate system (A ; $\overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } }$). Consider the point M such that $\overrightarrow { \mathrm { BM } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { BH } }$.
  1. By reading the graph, give the coordinates of points $\mathrm { B } , \mathrm { D } , \mathrm { E } , \mathrm { G }$ and H.
  2. a. What is the nature of triangle EGD? Justify your answer. b. It is admitted that the area of an equilateral triangle with side $c$ is equal to $\frac { \sqrt { 3 } } { 4 } c ^ { 2 }$. Show that the area of triangle EGD is equal to $\frac { \sqrt { 3 } } { 2 }$.
  3. Prove that the coordinates of M are $\left( \frac { 2 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.
  4. a. Justify that the vector $\vec { n } ( - 1 ; 1 ; 1 )$ is normal to the plane (EGD). b. Deduce that a Cartesian equation of the plane (EGD) is: $- x + y + z - 1 = 0$. c. Let $\mathscr { D }$ be the line perpendicular to the plane (EGD) and passing through point M. Show that a parametric representation of this line is: $$\mathscr { D } : \left\{ \begin{aligned} x & = \frac { 2 } { 3 } - t \\ y & = \frac { 1 } { 3 } + t , t \in \mathbb { R } \\ z & = \frac { 1 } { 3 } + t \end{aligned} \right.$$
  5. The purpose of this question is to calculate the volume of the pyramid GEDM. a. Let K be the foot of the height of the pyramid GEDM from point M. Prove that the coordinates of point K are $\left( \frac { 1 } { 3 } ; \frac { 2 } { 3 } ; \frac { 2 } { 3 } \right)$. b. Deduce the volume of the pyramid GEDM. Recall that the volume $V$ of a pyramid is given by the formula $V = \frac { b \times h } { 3 }$ where $b$ denotes the area of a base and h the associated height.