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
2024 bac-spe-maths__suede

4 maths questions

Q1 4 marks Differential equations Verification that a Function Satisfies a DE 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.
Statement 1: Let (E) be the differential equation: $y ^ { \prime } - 2 y = - 6 x + 1$. The function $f$ defined on $\mathbb { R }$ by: $f ( x ) = \mathrm { e } ^ { 2 x } - 6 x + 1$ is a solution of the differential equation (E).
Statement 2: Consider the sequence $\left( u _ { n } \right)$ defined on $\mathbb { N }$ by $$u _ { n } = 1 + \frac { 3 } { 4 } + \left( \frac { 3 } { 4 } \right) ^ { 2 } + \cdots + \left( \frac { 3 } { 4 } \right) ^ { n }$$ The sequence $(u _ { n })$ has limit $+ \infty$.
Statement 3: Consider the sequence $(u _ { n })$ defined in Statement 2. The instruction suite(50) below, written in Python language, returns $u _ { 50 }$. \begin{verbatim} def suite(k): S=0 for i in range(k): S=S+(3/4)**k return S \end{verbatim}
Statement 4: Let $a$ be a real number and $f$ the function defined on $] 0 ; + \infty [$ by: $$f ( x ) = a \ln ( x ) - 2 x$$ Let $C$ be the representative curve of the function $f$ in a coordinate system $(O ; \vec { \imath } , \vec { \jmath })$. There exists a value of $a$ for which the tangent to $C$ at the point with abscissa 1 is parallel to the horizontal axis.
Q2 5 marks Conditional Probability Markov Chain / Day-to-Day Transition Probabilities View
During a training session, a volleyball player practises serving. The probability that he succeeds on the first serve is equal to 0.85.
We further assume that the following two conditions are satisfied:
  • if the player succeeds on a serve, then the probability that he succeeds on the next one is equal to 0.6;
  • if the player fails a serve, then the probability that he fails the next one is equal to 0.6.

For any non-zero natural number $n$, we denote by $R _ { n }$ the event ``the player succeeds on the $n$-th serve'' and $\overline { R _ { n } }$ the complementary event.
Part A We are interested in the first two serves of the training session.
  1. Represent the situation with a probability tree.
  2. Prove that the probability of event $R _ { 2 }$ is equal to 0.57.
  3. Given that the player succeeded on the second serve, calculate the probability that he failed the first one.
  4. Let $Z$ be the random variable equal to the number of successful serves during the first two serves. a. Determine the probability distribution of $Z$ (you may use the probability tree from question 1). b. Calculate the mathematical expectation $\mathrm { E } ( Z )$ of the random variable $Z$.
Interpret this result in the context of the exercise.
Part B We now consider the general case. For any non-zero natural number $n$, we denote by $x _ { n }$ the probability of event $R _ { n }$.
  1. a. Give the conditional probabilities $P _ { R _ { n } } \left( R _ { n + 1 } \right)$ and $P _ { \overline { R _ { n } } } \left( \overline { R _ { n + 1 } } \right)$. b. Show that, for any non-zero natural number $n$, we have: $x _ { n + 1 } = 0.2 x _ { n } + 0.4$.
  2. Let the sequence $(u _ { n })$ be defined for any non-zero natural number $n$ by: $u _ { n } = x _ { n } - 0.5$. a. Show that the sequence $(u _ { n })$ is a geometric sequence. b. Determine the expression of $x _ { n }$ as a function of $n$. Deduce the limit of the sequence $\left( x _ { n } \right)$. c. Interpret this limit in the context of the exercise.
Q3 7 marks Integration by Parts Definite Integral Evaluation by Parts View
A certification body is commissioned to evaluate two heating devices, one from brand A and the other from brand B.
Parts 1 and 2 are independent.
Part 1: device from brand A Using a probe, the temperature inside the combustion chamber of a brand A device was measured. Below is a representation of the temperature curve in degrees Celsius inside the combustion chamber as a function of time elapsed, expressed in minutes, since the combustion chamber was ignited.
By reading the graph:
  1. Give the time at which the maximum temperature is reached inside the combustion chamber.
  2. Give an approximate value, in minutes, of the duration during which the temperature inside the combustion chamber exceeds $300 ^ { \circ } \mathrm { C }$.
  3. We denote by $f$ the function represented on the graph. Estimate the value of $\frac { 1 } { 600 } \int _ { 0 } ^ { 600 } f ( t ) \mathrm { d } t$. Interpret the result.

Part 2: study of a function Let the function $g$ be defined on the interval $[0 ; + \infty [$ by: $$g ( t ) = 10 t \mathrm { e } ^ { - 0.01 t } + 20 .$$
  1. Determine the limit of $g$ at $+ \infty$.
  2. a. Show that for all $t \in \left[ 0 ; + \infty \left[ , \quad g ^ { \prime } ( t ) = ( - 0.1 t + 10 ) \mathrm { e } ^ { - 0.01 t } \right. \right.$. b. Study the variations of the function $g$ on $[0 ; + \infty [$ and construct its variation table.
  3. Prove that the equation $g ( t ) = 300$ has exactly two distinct solutions on $[0 ; + \infty [$. Give approximate values to the nearest integer.
  4. Using integration by parts, calculate $\int _ { 0 } ^ { 600 } g ( t ) \mathrm { d } t$.

Part 3: evaluation For a brand B device, the temperature in degrees Celsius inside the combustion chamber $t$ minutes after ignition is modelled on $[0 ; 600]$ by the function $g$.
The certification body awards one star for each criterion validated among the following four:
  • Criterion 1: the maximum temperature is greater than $320 ^ { \circ } \mathrm { C }$.
  • Criterion 2: the maximum temperature is reached in less than 2 hours.
  • Criterion 3: the average temperature during the first 10 hours after ignition exceeds $250 ^ { \circ } \mathrm { C }$.
  • Criterion 4: the temperature inside the combustion chamber must not exceed $300 ^ { \circ } \mathrm { C }$ for more than 5 hours.
Does each device obtain exactly three stars? Justify your answer.
Q4 4 marks Vectors 3D & Lines MCQ: Relationship Between Two Lines View
A passage of an aerial acrobatics show in a duo is modelled as follows:
  • we place ourselves in an orthonormal coordinate system $(O ; \vec { \imath } , \vec { \jmath } , \vec { k })$, where one unit represents one metre;
  • plane no. 1 must travel from point O to point $A(0 ; 200 ; 0)$ along a straight trajectory, at the constant speed of $200 \mathrm {~m/s}$;
  • plane no. 2 must travel from point $B(-33 ; 75 ; 44)$ to point $C(87 ; 75 ; -116)$ also along a straight trajectory, and at the constant speed of $200 \mathrm {~m/s}$;
  • at the same instant, plane no. 1 is at point O and plane no. 2 is at point B.

  1. Justify that plane no. 2 will take the same time to travel segment $[BC]$ as plane no. 1 to travel segment $[OA]$.
  2. Show that the trajectories of the two planes intersect.
  3. Is there a risk of collision between the two planes during this passage?