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__metropole-juin_j1

5 maths questions

QA Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Main topics covered: Space geometry with respect to an orthonormal coordinate system; orthogonality in space
In an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$ we consider
  • the point A with coordinates $( 1 ; 3 ; 2 )$,
  • the vector $\vec { u }$ with coordinates $\left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right)$
  • the line $d$ passing through the origin O of the coordinate system and having $\vec { u }$ as its direction vector.

The purpose of this exercise is to determine the point on $d$ closest to point A and to study some properties of this point.
  1. Determine a parametric representation of the line $d$.
  2. Let $t$ be any real number, and $M$ a point on the line $d$, the point $M$ having coordinates $( t ; t ; 0 )$. a. We denote $AM$ the distance between points A and $M$. Prove that: $$AM ^ { 2 } = 2 t ^ { 2 } - 8 t + 14 .$$ b. Prove that the point $M _ { 0 }$ with coordinates $( 2 ; 2 ; 0 )$ is the point on the line $d$ for which the distance $AM$ is minimal. We will assume that the distance $AM$ is minimal when its square $AM ^ { 2 }$ is minimal.
  3. Prove that the lines $( A M _ { 0 } )$ and $d$ are orthogonal.
  4. We call $A ^ { \prime }$ the orthogonal projection of point $A$ onto the plane with Cartesian equation $z = 0$. The point $A ^ { \prime }$ therefore has coordinates $( 1 ; 3 ; 0 )$.
    Prove that the point $M _ { 0 }$ is the point of the plane $\left( A A ^ { \prime } M _ { 0 } \right)$ closest to point O, the origin of the coordinate system.
  5. Calculate the volume of the pyramid $O M _ { 0 } A ^ { \prime } A$.
    Recall that the volume of a pyramid is given by: $V = \frac { 1 } { 3 } \mathscr { B } h$, where $\mathscr { B }$ is the area of a base and $h$ is the height of the pyramid corresponding to this base.
QB Differential equations First-Order Linear DE: General Solution View
Main topics covered: Differential equations; exponential function.
We consider the differential equation
$$\text { (E) } y ^ { \prime } = y + 2 x \mathrm { e } ^ { x }$$
We seek the set of functions defined and differentiable on the set $\mathbb { R }$ of real numbers that are solutions to this equation.
  1. Let $u$ be the function defined on $\mathbb { R }$ by $u ( x ) = x ^ { 2 } \mathrm { e } ^ { x }$. We admit that $u$ is differentiable and we denote $u ^ { \prime }$ its derivative function. Prove that $u$ is a particular solution of $( E )$.
  2. Let $f$ be a function defined and differentiable on $\mathbb { R }$. We denote $g$ the function defined on $\mathbb { R }$ by: $$g ( x ) = f ( x ) - u ( x )$$ a. Prove that if the function $f$ is a solution of the differential equation $( E )$ then the function $g$ is a solution of the differential equation: $y ^ { \prime } = y$. We admit that the converse of this property is also true. b. Using the solution of the differential equation $y ^ { \prime } = y$, solve the differential equation (E).
  3. Study of the function $u$ a. Study the sign of $u ^ { \prime } ( x )$ for $x$ varying in $\mathbb { R }$. b. Draw the table of variations of the function $u$ on $\mathbb { R }$ (limits are not required). c. Determine the largest interval on which the function $u$ is concave.
Q1 4 marks Differentiating Transcendental Functions Higher-order or nth derivative computation View
Let $f$ be the function defined for all real numbers $x$ in the interval $] 0 ; + \infty [$ by:
$$f ( x ) = \frac { \mathrm { e } ^ { 2 x } } { x }$$
The expression of the second derivative $f ^ { \prime \prime }$ of $f$ is given, defined on the interval $] 0 ; + \infty [$ by:
$$f ^ { \prime \prime } ( x ) = \frac { 2 \mathrm { e } ^ { 2 x } \left( 2 x ^ { 2 } - 2 x + 1 \right) } { x ^ { 3 } } .$$
  1. The function $f ^ { \prime }$, the derivative of $f$, is defined on the interval $] 0 ; + \infty [$ by: a. $f ^ { \prime } ( x ) = 2 \mathrm { e } ^ { 2 x }$ b. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( x - 1 ) } { x ^ { 2 } }$ c. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( 2 x - 1 ) } { x ^ { 2 } }$ d. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( 1 + 2 x ) } { x ^ { 2 } }$.
  2. The function $f$: a. is decreasing on $] 0 ; + \infty [$ b. is monotonic on $] 0 ; + \infty [$ c. admits a minimum at $\frac { 1 } { 2 }$ d. admits a maximum at $\frac { 1 } { 2 }$.
  3. The function $f$ has the following limit as $x \to + \infty$: a. $+ \infty$ b. 0 c. 1 d. $\mathrm { e } ^ { 2 x }$.
  4. The function $f$: a. is concave on $] 0 ; + \infty [$ b. is convex on $] 0 ; + \infty [$ c. is concave on $] 0 ; \frac { 1 } { 2 } ]$ d. is represented by a curve admitting an inflection point.
Q2 Conditional Probability Bayes' Theorem with Diagnostic/Screening Test View
A manufacturing line produces mechanical parts. It is estimated that $5 \%$ of the parts produced by this line are defective.
An engineer has developed a test to apply to the parts. This test has two possible results: ``positive'' or ``negative''. This test is applied to a part chosen at random from the production of the line. We denote $p ( E )$ the probability of an event $E$. We consider the following events:
  • $D$: ``the part is defective'';
  • T: ``the part shows a positive test'';
  • $\bar { D }$ and $\bar { T }$ denote respectively the complementary events of $D$ and $T$.

Given the characteristics of the test, we know that:
  • The probability that a part shows a positive test given that it is defective is equal to 0.98;
  • the probability that a part shows a negative test given that it is not defective is equal to 0.97.

PART I
  1. Represent the situation using a probability tree.
  2. a. Determine the probability that a part chosen at random from the production line is defective and shows a positive test. b. Prove that: $p ( T ) = 0.0775$.
  3. The positive predictive value of the test is called the probability that a part is defective given that the test is positive. A test is considered effective if it has a positive predictive value greater than 0.95.

Calculate the positive predictive value of this test and specify whether it is effective.
PART II
A sample of 20 parts is chosen from the production line, treating this choice as a draw with replacement. Let $X$ be the random variable that gives the number of defective parts in this sample. Recall that: $p ( D ) = 0.05$.
  1. Justify that $X$ follows a binomial distribution and determine the parameters of this distribution.
  2. Calculate the probability that this sample contains at least one defective part.

Give a result rounded to the nearest hundredth.
3. Calculate the expected value of the random variable $X$ and interpret the result obtained.
Q3 Geometric Sequences and Series Applied Geometric Model with Contextual Interpretation View
Cécile has invited friends to lunch on her terrace. For dessert, she has planned an assortment of individual cakes that she bought frozen. She takes the cakes out of the freezer at $- 19 ^ { \circ } \mathrm { C }$ and brings them to the terrace where the temperature is $25 ^ { \circ } \mathrm { C }$. After 10 minutes, the temperature of the cakes is $1.3 ^ { \circ } \mathrm { C }$.
I- First model
We assume that the thawing rate is constant, that is, the temperature increase is the same minute after minute. According to this model, determine what the temperature of the cakes would be 25 minutes after they are taken out of the freezer. Does this model seem relevant?
II - Second model
We denote $T _ { n }$ the temperature of the cakes in degrees Celsius, after $n$ minutes following their removal from the freezer; thus $T _ { 0 } = - 19$. We assume that to model the evolution of temperature, we must have the following relation
$$\text { For all natural integers } n , T _ { n + 1 } - T _ { n } = - 0.06 \times \left( T _ { n } - 25 \right) \text {. }$$
  1. Justify that, for all integers $n$, we have $T _ { n + 1 } = 0.94 T _ { n } + 1.5$
  2. Calculate $T _ { 1 }$ and $T _ { 2 }$. Give values rounded to the nearest tenth.
  3. Prove by induction that, for all natural integers $n$, we have $T _ { n } \leqslant 25$.

Returning to the situation studied, was this result foreseeable?
4. Study the direction of variation of the sequence $( T _ { n } )$.
5. Prove that the sequence $( T _ { n } )$ is convergent. 6. We set for all natural integers $n$, $U _ { n } = T _ { n } - 25$. a. Show that the sequence $( U _ { n } )$ is a geometric sequence and specify its common ratio and first term $U _ { 0 }$. b. Deduce that for all natural integers $n$, $T _ { n } = - 44 \times 0.94 ^ { n } + 25$. c. Deduce the limit of the sequence $( T _ { n } )$. Interpret this result in the context of the situation studied. 7. a. The manufacturer recommends consuming the cakes after half an hour at room temperature following their removal from the freezer. What is then the temperature reached by the cakes? Give a value rounded to the nearest integer. b. Cécile is a regular customer of these cakes, which she likes to enjoy while still fresh, at a temperature of $10 ^ { \circ } \mathrm { C }$. Give a range between two consecutive integers of the time in minutes after which Cécile should enjoy her cake. c. The following program, written in Python language, must return after its execution the smallest value of the integer $n$ for which $T _ { n } \geqslant 10$.
\begin{verbatim} def seuil() : n=0 T= while T T= n=n+1 return \end{verbatim}
Copy this program onto your paper and complete the incomplete lines so that the program returns the expected value.