France 2017: subject of the patent of maths with its corrected answers

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Part 1
General series
Duration of the test: 2 hours – 50 points


Exercise 1 (4 points)
In an urn containing green and blue balls, we draw a ball at random and look at its color. The ball is then returned to the urn and the balls are mixed.
The probability of getting a green ball is \frac{2}{5}.
1. Explain why the probability of getting a blue ball is equal to\frac{3}{5}.
2. Paul made 6 draws and got a green ball each time.
On the 7th draw, will he have a better chance of getting a blue ball than a green ball?
3. Determine the number of blue balls in this urn knowing that there are 8 green balls.

Exercise 2 (6 points)
The following program is given to draw several equilateral triangles of different sizes.
This program has a variable named “side”. The lengths are given in pixels.
It is recalled that the instruction means that we are heading to the right.

1. What are the coordinates of the starting point of the plot?
2. How many triangles are drawn by the script ?

3. a. What is the length (in pixels) of the side of the second triangle drawn?
b. Draw freehand the shape of the figure obtained when this script is executed.
4. We modify the initial script to obtain the figure below.

Indicate the number of a script instruction after which you can place
the instruction to obtain this new figure.

Exercise 3 (4 points)
A capacitor is an electronic component that stores electrical energy for later release.
The following graph shows the evolution of the voltage measured at the terminals of a capacitor as a function of time when it is in charge.
1. Is this a situation of proportionality? Justify.
2. What is the voltage measured after 0.2 s?
3. After how long will the voltage across the capacitor have reached 60% of the maximum voltage, which is estimated to be 5 V?

Exercise 4 (8 points)
Photovoltaic panels allow the production of electricity from solar radiation.
A common unit for measuring electrical energy is the kilowatt-hour, abbreviated as kWh.
1. Most often, the electricity produced is not used directly, but sold for distribution to the collective electrical network. The purchase price of the kWh, given in euro cents, depends on the type
of installation and its total power, as well as the date of installation of the photovoltaic panels. This purchase price per kWh is given in the table below.

In May 2015, a Type B solar power plant with a capacity of 28 kW was installed.
Verify that the purchase price of 31,420 kWh is approximately €4,383.

2. A person wants to install photovoltaic panels on the south-facing part of the roof of his house. This part is grayed out on the figure opposite. It is called the south side of the roof.
The electricity production of the solar panels depends on the slope of the roof.
Determine, to the nearest degree, the angle \widehat{ABC} that this south roof slope forms with the horizontal.

3. a. Show that the length AB is approximately equal to 5 m.
b. The photovoltaic panels have the shape of a square of 1 m side.
The owner plans to install 20 panels.
What percentage of the total area of the south roof slope will then be covered by the solar panels? We will give an approximate value of the result to the nearest 1%.
c. The installation instructions indicate that the panels must be placed close together and that a border of at least 30 cm wide must be left free for the fastening system all around the panels.
Can the owner install all 20 panels?

Exercise 5 (8 points)
1. At the 2016 Rio Olympics, Denmark’s Pernille Blume won the 50m freestyle in 24.07 seconds.
Did she swim faster than a person walking fast, i.e. 6 km/h?
2. We give the expression (3x,+,8)^2-,64,=,0.
a. Develop E.
b. Show that E can be written in factorized form:3x(3x,+,16).
c. Solve the equation (3x,+,8)^2,-,64,=,0.
3. The braking distance of a vehicle depends on its speed and the condition of the road.
It can be calculated using the following formula:
d=k\times  ,V^2 with d: braking distance in m , V: vehicle speed in m/s, k: coefficient depending on the road condition.
k= 0,14 on wet road
k = 0.08 on dry road.
What is the speed of a vehicle whose braking distance on a wet road is equal to 15 m?

Exercise 6 (8 points)

1. At a company, during a medical check-up, a doctor calculates the BMI of six of the employees.
He uses a spreadsheet to do this:

a. How many employees are overweight or obese in this company?
b. Which of these formulas was written in cell B3 and copied to the right to calculate BMI?
Copy the correct formula on the copy.

2. The doctor took stock of the BMI of each of the 41 employees of this company. He postponed the
The information collected in the following table in which BMIs have been rounded to the nearest unit.

a. Calculate an approximate value, rounded to the nearest integer, for the average BMI of employees in this
b. What is the median BMI? Interpret this result.
c. Some magazines state: “It is estimated that at least 5% of the world’s population is overweight
or is obese”. Is this the case for the employees of this company?

Exercise 7 (7 points)
Leo picked strawberries to make jam.
1. He uses his grandmother’s proportions: 700 g of sugar for 1 kg of strawberries.
He picked up 1.8 kg of strawberries. How much sugar does it need?
2. After cooking, Léo obtained 2.7 liters of jam.
He pours the jam into cylindrical jars of 6 cm in diameter and 12 cm high, which he fills to 1 cm from the top edge.
How many pots will he be able to fill?

Reminders: 1 liter = 1000 cm^3

Volume of a cylinder =\pi\times  ,R^2\times  ,h
3. He then sticks a rectangular label with a white background on his jars that covers the entire side surface of the jar.
a. Show that the length of the label is approximately 18.8 cm.
b. Draw the label to scale \frac{1}{3}.

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