Subject 2018 France of the brevet de maths des collèges

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General series
Duration of the test: 2 h 00 out of 100 points

The evaluation takes into account the clarity and precision of the reasoning as well as, more broadly, the quality of the writing. It takes into account the tests and the steps taken, even if they are not successful.

Exercise 1 (11 points)
The big crystal globe is a trophy awarded to the winner of the ski world cup. This trophy weighs 9 kg and is 46 cm high.
1. French biathlete Martin Fourcade won the sixth big crystal globe of his career in 2017 in Pyeongchang, South Korea.

Give the approximate latitude and longitude of this location on the map below.

2. We consider that this globe is composed of a cylinder in crystal of
diameter 6 cm, topped with a crystal ball. See diagram below.

Show that an approximate value of the volume of the ball of this trophy is 6371 cm^3.
3. Mary states that the volume of the crystal ball is about 90% of the total volume of the trophy. Is she right?


– volume of a ball of radius R : V=\frac{4}{3}\pi\times  \,R^3

– volume of a cylinder of radius r and height h : V=\pi\times  ,r^2\times  ,h

Exercise 2 (14 points)
Among the many air pollutants, fine particles are regularly monitored.
PM10 are fine particles with a diameter of less than 0.01 mm.
In January 2017, the cities of Lyon and Grenoble experienced an episode of fine particle pollution. The following is data for the period January 16-25, 2017:

1. Which of these two cities had the highest average PM10 concentration between January 16 and 25?
2. Calculate the range of the PM10 series in Lyon and Grenoble. Which of these two cities has had the greater extent? Interpret this last result.
3. Is the following statement correct? Justify your answer.
“From January 16 to 25, the alert threshold of 80 μg/m^3 per day was exceeded at least 5 times in Lyon”.

Exercise 3 (12 points)
In his audio player, Theo has downloaded 375 pieces of music. Among them, there are 125 pieces of rap music. He presses the “shuffle” button to listen to a randomly selected song from all the available songs.

1. What is the probability that he listens to rap music?
2. The probability that he listens to rock music is equal to \frac{7}{15}.
How many rock songs does Theo have in his audio player?

3. Alice has 40% of rock songs in her audio player.
If Theo and Alice both press the “shuffle” button on their audio player, which one is more likely to listen to a rock song?

Exercise 4 (14 points)
The figure below is not shown at full size.
The points , and are aligned.
The triangle is rectangular in .
The triangle is rectangular in .

1. Show that the length is equal to 4 cm.
2. Show that the triangles and are similar.
3. Sophie says that the angle \widehat{BFE} is a right angle. Is she right?
4. Max says that the angle \widehat{ACD} is a right angle. Is he right?

Exercise 5 (16 points)
Here is a calculation program.

1. Verify that if we choose the number -1, this program gives 8 as the final result.
2. The program gives 30 as the final result, what is the number chosen at the beginning?
In the rest of the exercise, we name the number chosen at the beginning.
3. The expression A=2(4x+8) gives the result of the previous calculation program for a given number x.
We pose B=(4+x)^2-x^2.
Prove that the expressions and B are equal for all values of x.
4. For each of the following statements, indicate whether it is true or false. Remember that answers must be justified.

Assertion 1: This program gives a positive result for all values of x.
Assertion 2: If the number x is a whole number, the result is a multiple of 8.

Exercise 6 (16 points)
The lengths are in pixels.
The expression “turn 90” means to turn to the right.

1. We take as scale 1 cm for 50 pixels.
a. Show on your copy the figure obtained if the program is run up to and including line 7.
b. What are the coordinates of the pen after the execution of line 8?
2. We run the complete program and we obtain the figure below which has a vertical axis of symmetry.

Copy and complete line 9 of the program to obtain this figure.
3. a. Among the following transformations, translation, homothety, rotation, axial symmetry, which is the geometrical transformation which allows to obtain the small square from the big square ? Specify the reduction ratio.
b. What is the ratio of the areas between the two squares drawn?

Exercise 7 (17 points)
The hand-spinner is a kind of flat top that spins on itself.

The hand-spinner is given an initial speed of rotation at time t = 0, then, over time, its speed of rotation decreases until the hand-spinner comes to a complete stop. Its rotation speed is then equal to 0.
Using a measuring device, the speed of rotation expressed in revolutions per second was recorded.
On the graph below, we have represented this speed as a function of time expressed in seconds:

1. Are the time and speed of rotation of the hand-spinner proportional? Justify.
2. By graphic reading, answer the following questions:
a. What is the initial rotation speed of the hand-spinner (in revolutions per second)?
b. What is the speed of rotation of the hand spinner (in revolutions per second) after 1 minute and 20 seconds?
c. How long will it take for the hand-spinner to stop?
3. To calculate the rotation speed of the hand-spinner as a function of time, noted (), we use the following function:
V(t)=-0,214\times  ,t+V_{initiale}
t is the time (expressed in s) that has elapsed since the hand-spinner started rotating
V_{initiale} is the rotation speed at which the hand-spinner was launched at the beginning.

a. The hand-spinner is launched at an initial speed of 20 revolutions per second.

Its rotation speed is therefore given by the formula: V(t)=-0,214\times  ,t+20.

Calculate its rotation speed after 30 s.
b. How long will it take for the hand-spinner to stop? Justify with a calculation.
c. Is it true that, in general, if you spin the hand-spinner twice as fast at the start, it will spin twice as long? Justify.

Answers to the mathematics patent Consult the online answer key

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