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
2018 pondichery

7 maths questions

Q1A Arithmetic Sequences and Series Sequence Defined by Recurrence with AP Connection View
In a factory, a kiln bakes ceramics at a temperature of $1000^{\circ}\mathrm{C}$. At the end of baking, it is turned off and cools down. The temperature of the kiln is expressed in degrees Celsius (${}^{\circ}\mathrm{C}$). The kiln door can be opened safely for the ceramics as soon as its temperature is below $70^{\circ}\mathrm{C}$.
For a natural integer $n$, we denote $T_n$ the temperature in degrees Celsius of the kiln after $n$ hours have elapsed from the moment it was turned off. We therefore have $T_0 = 1000$. The temperature $T_n$ is calculated by the following algorithm:
\begin{verbatim} T←1000 For i going from 1 to n T←0.82 x T+3.6 End For \end{verbatim}
  1. Determine the temperature of the kiln, rounded to the nearest unit, after 4 hours of cooling.
  2. Prove that, for every natural integer $n$, we have: $T_n = 980 \times 0.82^n + 20$.
  3. After how many hours can the kiln be opened safely for the ceramics?
Q1B 6 marks Laws of Logarithms Determine Parameters of a Logarithmic Function View
In a factory, a kiln bakes ceramics at a temperature of $1000^{\circ}\mathrm{C}$. At the end of baking, it is turned off and cools down. The temperature of the kiln is expressed in degrees Celsius (${}^{\circ}\mathrm{C}$). The kiln door can be opened safely for the ceramics as soon as its temperature is below $70^{\circ}\mathrm{C}$.
In this part, we denote $t$ the time (in hours) elapsed since the moment the kiln was turned off. The temperature of the kiln (in degrees Celsius) at time $t$ is given by the function $f$ defined, for every positive real number $t$, by: $$f(t) = a\mathrm{e}^{-\frac{t}{5}} + b,$$ where $a$ and $b$ are two real numbers. We admit that $f$ satisfies the following relation: $f'(t) + \frac{1}{5}f(t) = 4$.
  1. Determine the values of $a$ and $b$ knowing that initially, the temperature of the kiln is $1000^{\circ}\mathrm{C}$, that is $f(0) = 1000$.
  2. For the following, we admit that, for every positive real number $t$: $$f(t) = 980\mathrm{e}^{-\frac{t}{5}} + 20.$$ a. Determine the limit of $f$ as $t$ tends to $+\infty$. b. Study the variations of $f$ on $[0; +\infty[$. Deduce its complete table of variations. c. With this model, after how many minutes can the kiln be opened safely for the ceramics?
  3. The average temperature (in degrees Celsius) of the kiln between two times $t_1$ and $t_2$ is given by: $\frac{1}{t_2 - t_1}\int_{t_1}^{t_2} f(t)\,\mathrm{d}t$. a. Using the graphical representation of $f$, give an estimate of the average temperature $\theta$ of the kiln over the first 15 hours of cooling. Explain your approach. b. Calculate the exact value of this average temperature $\theta$ and give its value rounded to the nearest degree Celsius.
  4. In this question, we are interested in the temperature drop (in degrees Celsius) of the kiln over the course of one hour, that is between two times $t$ and $(t+1)$. This drop is given by the function $d$ defined, for every positive real number $t$, by: $d(t) = f(t) - f(t+1)$. a. Verify that, for every positive real number $t$: $d(t) = 980\left(1 - \mathrm{e}^{-\frac{1}{5}}\right)\mathrm{e}^{-\frac{t}{5}}$. b. Determine the limit of $d(t)$ as $t$ tends to $+\infty$. What interpretation can be given to this?
Q2 4 marks Complex numbers 2 Geometric Interpretation in the Complex Plane View
The plane is equipped with an orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). The points $\mathrm{A}$, $\mathrm{B}$ and C have affixes respectively $a = -4$, $b = 2$ and $c = 4$.
  1. We consider the three points $\mathrm{A}^{\prime}$, $\mathrm{B}^{\prime}$ and $\mathrm{C}^{\prime}$ with affixes respectively $a^{\prime} = \mathrm{j}a$, $b^{\prime} = \mathrm{j}b$ and $c^{\prime} = \mathrm{j}c$ where j is the complex number $-\frac{1}{2} + \mathrm{i}\frac{\sqrt{3}}{2}$. a. Give the trigonometric form and the exponential form of j. Deduce the algebraic and exponential forms of $a^{\prime}$, $b^{\prime}$ and $c^{\prime}$. b. The points $\mathrm{A}$, $\mathrm{B}$ and C as well as the circles with center O and radii 2, 3 and 4 are represented on the graph provided in the Appendix. Place the points $\mathrm{A}^{\prime}$, $\mathrm{B}^{\prime}$ and $\mathrm{C}^{\prime}$ on this graph.
  2. Show that the points $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ are collinear.
  3. We denote M the midpoint of segment $[\mathrm{A}^{\prime}\mathrm{C}]$, N the midpoint of segment $[\mathrm{C}^{\prime}\mathrm{C}]$ and P the midpoint of segment $[\mathrm{C}^{\prime}\mathrm{A}]$. Prove that triangle MNP is isosceles.
Q3A Normal Distribution Direct Probability Calculation from Given Normal Distribution View
A company packages white sugar from two farms $U$ and $V$ in 1 kg packets of different qualities. Extra fine sugar is packaged separately in packets bearing the label ``extra fine''. Throughout the exercise, results should be rounded, if necessary, to the nearest thousandth.
To calibrate the sugar according to the size of its crystals, it is passed through a series of three sieves. Sugar crystals with a size less than $0.2\,\mathrm{mm}$ are found in the sealed bottom container at the end of calibration. They will be packaged in packets bearing the label ``extra fine sugar''.
  1. A sugar crystal is randomly selected from farm U. The size of this crystal, expressed in millimeters, is modeled by the random variable $X_{\mathrm{U}}$ which follows the normal distribution with mean $\mu_{\mathrm{U}} = 0.58\,\mathrm{mm}$ and standard deviation $\sigma_{\mathrm{U}} = 0.21\,\mathrm{mm}$. a. Calculate the probabilities of the following events: $X_{\mathrm{U}} < 0.2$ and $0.5 \leqslant X_{\mathrm{U}} < 0.8$. b. 1800 grams of sugar from farm $U$ is passed through the series of sieves. Deduce from the previous question an estimate of the mass of sugar recovered in the sealed bottom container and an estimate of the mass of sugar recovered in sieve 2.
  2. A sugar crystal is randomly selected from farm V. The size of this crystal, expressed in millimeters, is modeled by the random variable $X_{\mathrm{V}}$ which follows the normal distribution with mean $\mu_{\mathrm{V}} = 0.65\,\mathrm{mm}$ and standard deviation $\sigma_{\mathrm{V}}$ to be determined. During the calibration of a large quantity of sugar crystals from farm V, it is observed that $40\%$ of these crystals end up in sieve 2. What is the value of the standard deviation $\sigma_{\mathrm{V}}$ of the random variable $X_{\mathrm{V}}$?
Q3B Conditional Probability Bayes' Theorem with Production/Source Identification View
A company packages white sugar from two farms $U$ and $V$ in 1 kg packets. We admit that $3\%$ of the sugar from farm $U$ is extra fine and that $5\%$ of the sugar from farm V is extra fine. A packet of sugar is randomly selected from the company's production and we consider the following events:
  • $U$: ``The packet contains sugar from farm U'';
  • $V$: ``The packet contains sugar from farm V'';
  • $E$: ``The packet bears the label `extra fine' ''.

  1. In this question, we admit that the company manufactures $30\%$ of its packets with sugar from farm U and the others with sugar from farm V, without mixing sugars from the two farms. a. What is the probability that the selected packet bears the label ``extra fine''? b. Given that a packet bears the label ``extra fine'', what is the probability that the sugar it contains comes from farm U?
  2. The company wishes to modify its supply from the two farms so that among the packets bearing the label ``extra fine'', $30\%$ of them contain sugar from farm U. How should it supply itself from farms U and V? Any working will be valued in this question.
Q3C Z-tests (known variance) View
A company packages white sugar from two farms $U$ and $V$ in 1 kg packets. Throughout the exercise, results should be rounded, if necessary, to the nearest thousandth.
  1. The company announces that $30\%$ of the packets of sugar bearing the label ``extra fine'' that it packages contain sugar from farm U. Before validating an order, a buyer wants to verify this announced proportion. He randomly selects 150 packets from the company's production of packets labeled ``extra fine''. Among these packets, 30 contain sugar from farm U. Does he have reason to question the company's announcement?
  2. The following year, the company declares that it has modified its production. The buyer wishes to estimate the new proportion of packets of sugar from farm U among the packets bearing the label ``extra fine''. He randomly selects 150 packets from the company's production of packets labeled ``extra fine''. Among these packets $42\%$ contain sugar from farm U. Give a confidence interval, at the $95\%$ confidence level, for the new proportion of packets labeled ``extra fine'' containing sugar from farm U.
Q4 5 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
In space equipped with the orthonormal coordinate system ($\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k}$) with unit 1 cm, we consider the points $\mathrm{A}$, $\mathrm{B}$, C and D with coordinates respectively $(2; 1; 4)$, $(4; -1; 0)$, $(0; 3; 2)$ and $(4; 3; -2)$.
  1. Determine a parametric representation of the line (CD).
  2. Let $M$ be a point on the line (CD). a. Determine the coordinates of the point $M$ such that the distance $BM$ is minimal. b. We denote H the point on the line $(\mathrm{CD})$ with coordinates $(3; 3; -1)$. Verify that the lines $(\mathrm{BH})$ and $(\mathrm{CD})$ are perpendicular. c. Show that the area of triangle BCD is equal to $12\,\mathrm{cm}^2$.
  3. a. Prove that the vector $\vec{n}\begin{pmatrix}2\\1\\2\end{pmatrix}$ is a normal vector to the plane (BCD). b. Determine a Cartesian equation of the plane (BCD).