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
2017 liban

5 maths questions

Q1 Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Consider a cube ABCDEFGH whose graphical representation in cavalier perspective is given below. The edges have length 1. Space is referred to the orthonormal coordinate system $( \mathrm { D } ; \overrightarrow { \mathrm { DA } } , \overrightarrow { \mathrm { DC } } , \overrightarrow { \mathrm { DH } } )$.
Part A
  1. Show that the vector $\overrightarrow { \mathrm { DF } }$ is normal to the plane (EBG).
  2. Determine a Cartesian equation of the plane (EBG).
  3. Deduce the coordinates of point I, the intersection of line (DF) and plane (EBG).

One would show in the same way that point J, the intersection of line (DF) and plane (AHC), has coordinates $\left( \frac { 1 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.
Part B
For any real number $x$ in the interval $[ 0 ; 1 ]$, we associate the point $M$ of segment $[ \mathrm{DF} ]$ such that $\overrightarrow { \mathrm { DM } } = x \overrightarrow { \mathrm { DF } }$. We are interested in the evolution of the measure $\theta$ in radians of angle $\widehat { \mathrm { EMB } }$ as point $M$ moves along segment [DF]. We have $0 \leqslant \theta \leqslant \pi$.
  1. What is the value of $\theta$ if point $M$ coincides with point D? with point F?
  2. a. Justify that the coordinates of point $M$ are $( x ; x ; x )$. b. Show that $\cos ( \theta ) = \frac { 3 x ^ { 2 } - 4 x + 1 } { 3 x ^ { 2 } - 4 x + 2 }$. For this, one may consider the dot product of vectors $\overrightarrow { M \mathrm { E } }$ and $\overrightarrow { M \mathrm {~B} }$.
  3. The table of variations of the function below has been constructed $$f : x \longmapsto \frac { 3 x ^ { 2 } - 4 x + 1 } { 3 x ^ { 2 } - 4 x + 2 }$$
    $x$0$\frac { 1 } { 3 }$$\frac { 2 } { 3 }$1
    \begin{tabular}{ c } Variations
    of $f$
    & $\frac { 1 } { 2 }$ & & & & & & 0 & \hline \end{tabular}
    For which positions of point $M$ on segment [DF]: a. is triangle $MEB$ right-angled at $M$? b. is angle $\theta$ maximal?
Q2 6 marks Exponential Distribution View
In this exercise, we study some characteristic quantities of the operation of parking lots in a city. Throughout the exercise, probabilities will be given with a precision of $10 ^ { - 4 }$.
Parts A, B, and C are independent
Part A - Waiting time to enter an underground parking lot
The waiting time is defined as the time that elapses between the moment the car arrives at the parking entrance and the moment it passes through the parking entrance barrier. The following table presents observations made over one day.
Waiting time in minutes$[ 0 ; 2 [$$[ 2 ; 4 [$$[ 4 ; 6 [$$[ 6 ; 8 [$
Number of cars7519105

  1. Propose an estimate of the average waiting time for a car at the parking entrance.
  2. We decide to model this waiting time by a random variable $T$ following an exponential distribution with parameter $\lambda$ (expressed in minutes). a. Justify that we can choose $\lambda = 0.5 \mathrm {~min}$. b. A car arrives at the parking entrance. What is the probability that it takes less than two minutes to pass through the barrier? c. A car has been waiting at the parking entrance for one minute. What is the probability that it passes through the barrier in the next minute?

Part B - Duration and parking rates in this underground parking lot
Once parked, the parking duration of a car is modeled by a random variable $D$ that follows a normal distribution with mean $\mu = 70 \mathrm {~min}$ and standard deviation $\sigma = 30 \mathrm {~min}$.
  1. a. What is the average parking duration for a car? b. A motorist enters and parks in the parking lot. What is the probability that their parking duration exceeds two hours? c. To the nearest minute, what is the maximum parking time for at least $99 \%$ of cars?
  2. The parking duration is limited to three hours. The table gives the rate for the first hour and each additional hour is charged at a single rate. Any hour started is charged in full.

\begin{tabular}{ c } Parking
duration
& Less than 15 min & Between 15 min and 1 h &
Additional
hour
\hline Rate in euros & Free & 3.5 & $t$ \hline \end{tabular}
Determine the rate $t$ for the additional hour that the parking manager must set so that the average parking price for a car is 5 euros.
Part C - Waiting time to park in a city center parking lot
The parking duration of a car in a city center parking lot is modeled by a random variable $T ^ { \prime }$ that follows a normal distribution with mean $\mu ^ { \prime }$ and standard deviation $\sigma ^ { \prime }$. It is known that the average parking time in this lot is 30 minutes and that $75 \%$ of cars have a parking time less than 37 minutes. The parking manager aims for the objective that $95 \%$ of cars have a parking time between 10 and 50 minutes. Is this objective achieved?
Q3 3 marks Stationary points and optimisation Geometric or applied optimisation problem View
Let $k$ be a strictly positive real number. Consider the functions $f _ { k }$ defined on $\mathbb { R }$ by: $$f _ { k } ( x ) = x + k \mathrm { e } ^ { - x } .$$ We denote by $\mathscr { C } _ { k }$ the representative curve of function $f _ { k }$ in a plane with an orthonormal coordinate system.
For every strictly positive real number $k$, the function $f _ { k }$ admits a minimum on $\mathbb { R }$. The value at which this minimum is attained is the abscissa of the point denoted $A _ { k }$ on the curve $\mathscr { C } _ { k }$. It would seem that, for every strictly positive real number $k$, the points $A _ { k }$ are collinear. Is this the case?
Q4A 5 marks Laws of Logarithms Analyze a Logarithmic Function (Limits, Monotonicity, Zeros, Extrema) View
Exercise 4 (Candidates who have not followed the specialization course)
The common spruce is a species of coniferous tree that can measure up to 40 meters in height and live more than 150 years. The objective of this exercise is to estimate the age and height of a spruce based on the diameter of its trunk measured at $1.30 \mathrm {~m}$ from the ground.
Part A - Modeling the age of a spruce
For a spruce whose age is between 20 and 120 years, the relationship between its age (in years) and the diameter of its trunk (in meters) measured at $1.30 \mathrm {~m}$ from the ground is modeled by the function $f$ defined on the interval $] 0 ; 1 [$ by: $$f ( x ) = 30 \ln \left( \frac { 20 x } { 1 - x } \right)$$ where $x$ denotes the diameter expressed in meters and $f ( x )$ the age in years.
  1. Prove that the function $f$ is strictly increasing on the interval $] 0 ; 1 [$.
  2. Determine the values of the trunk diameter $x$ such that the age calculated in this model remains consistent with its validity conditions, that is, between 20 and 120 years.

Part B
The average height of spruces in representative samples of trees aged 50 to 150 years was measured. The following table, created using a spreadsheet, groups these results and allows calculation of the average growth rate of a spruce.
ABCDEFGHIJKLM
1Ages (in years)507080859095100105110120130150
2Heights (in meters)11.215.618.0519.320.5521.82324.225.427.629.6533
3Growth rate (in meters per year)0.220.2450.25

  1. a. Interpret the number 0.245 in cell D3. b. What formula should be entered in cell C3 to complete line 3 by copying cell C3 to the right?
  2. Determine the expected height of a spruce whose trunk diameter measured at $1.30 \mathrm {~m}$ from the ground is 27 cm.
  3. The quality of the wood is better when the growth rate is maximal. a. Determine an age interval during which the wood quality is best by explaining the approach. b. Is it consistent to ask loggers to cut trees when their diameter measures approximately 70 cm?
Q4B 5 marks Number Theory Modular Arithmetic Computation View
Exercise 4 (Candidates who have followed the specialization course)
A bank card number is of the form: $$a _ { 1 } a _ { 2 } a _ { 3 } a _ { 4 } a _ { 5 } a _ { 6 } a _ { 7 } a _ { 8 } a _ { 9 } a _ { 10 } a _ { 11 } a _ { 12 } a _ { 13 } a _ { 14 } a _ { 15 } c$$ where $a _ { 1 } , a _ { 2 } , \ldots , a _ { 15 }$ and $c$ are digits between 0 and 9. The first fifteen digits contain information about the card type, the bank, and the bank account number. $c$ is the validation key for the number. This digit is calculated from the other fifteen. The following algorithm allows validation of the conformity of a given card number.
Initialization: $I$ takes the value 0 $P$ takes the value 0 $R$ takes the value 0 Processing: For $k$ going from 0 to 7: $R$ takes the value of the remainder of the Euclidean division of $2 a _ { 2 k + 1 }$ by 9 $I$ takes the value $I + R$ End For For $k$ going from 1 to 7: $P$ takes the value $P + a _ { 2 k }$ End For $S$ takes the value $I + P + c$ Output: If $S$ is a multiple of 10