Answers to the mathematics exam 2021 in France

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Find the answers to the 2021 French patent exam.

Exercise 1:

The average temperature in Tours in November 2019 was 8.2°C.

2. The range of this series is the difference between the extreme values:
e = 22.6 – 4.4 = 18.2°C.

3. The formula to enter in cell N2 to calculate the annual average temperature is :
= AVERAGE(B2:M2) or SUM(B2:M2)/12.

4. The average annual temperature is :


5. The percentage increase between 2009 and 2019, rounded to the nearest whole number, is :


Exercise 2:

To reach 2 million visitors, 0.1 million visitors are missing, i.e. 100,000 visitors.

2.2019 has 365 days so on average the number of visitors per day (assuming the park is open every day) is :


So the statement is false, if we look for an approximate value to the unit.

3. A teacher organizes a field trip to the Futuroscope for his third grade students. He wants to divide the 126 boys and 90 girls into groups. He wants each group to have the same number of girls and the same number of boys.
3. a. Decompose the numbers 126 and 90 into products of prime factors.

126=2\times  \,3^2\times  \,7\\90=2\times  \,3^2\times  \,5

b. The integers that divide both the numbers 126 and 90 are: 1; 2; 3; 6; 9 and 18.

3. c. Deduce the largest number of groups that the teacher can form. How many girls and boys will there be in each group?
– The number of groups n must be a common divisor of 126 and 90, and we are looking for the largest,
it will be the PGCD of 126 and 90 or :
126\,=\,18\,\times  \,7\,;90\,=\,18\,\times  \,5
n = P GCD(126, 90) = 18
Each group will have 7 boys and 5 girls.

4. Two students of 3ème, Marie and Adrien, remember having seen in mathematics that the inaccessible heights could be determined with the shadow. They want to calculate the height of the Futuroscope Gyrotour. Mary places herself as shown in the figure below, so that her shadow coincides with that of the tower. After making several measurements, Adrien makes the diagram below (the diagram is not to scale), on which points A, E and B and points A, D and C are aligned.

Calculate the height BC of the Gyrotour.

The points A, D, C and A, E, B are aligned and the lines (ED) and (BC) are parallel.
We can therefore apply the theorem of Thales:


The point D belongs to the segment [AD] so :
AC = AD + DC = 2 + 54.25 = 56.25 m
\frac{2}{56,25}=\frac{1,6}{BC} so BC=\frac{56,25\times  \,1,6}{2}=45\,m.

Exercise 3:
This exercise is a multiple choice questionnaire (MCQ). No justification is required. For each question, three answers (A, B and C) are proposed. Only one answer is correct. Copy the question number and the answer on the copy.
Part A
1. The correct answer is C.

2. The correct answer is A.

Part B

3. The correct answer is A.

4. The correct answer is B.

5. The correct answer is B.

Exercise 4:

Here is a calculation program:
– Choose a number.
– Take the square of the starting number.
– Add three times the number of starts.
– Subtract 10 from the result.


Exercise 5:

  1. The annual production of waste per French person was 5.2 tons per capita in 2007. Between 2007 and 2017, it decreased by 6.5% so that’s a decrease of :

5,2\times  \,\frac{6,5}{100}=0,338\,tonne.

2.a. ABHD is a rectangle so HB = DA = 39 cm. And since the point H belongs to the segment [CB] we have :
CH = CB – HB = 67 – 39 = 28 cm.

b. In the triangle HDC rectangular in H, according to the Pythagorean theorem we have :

Now HD is positive since it is a length, so the only possible solution is :
HD = 45 cm

c. The area of trapezoid ABCD is :

The volume of the compositor is the sum of that of the right prism V1 and that of the right block V2.

The height of the right block is :
h = 1.1 m – AB = 110 – 45 = 65 cm

So we have:
V1\,=\,Aire(ABCD)\,\times  \,70\,=\,166\,\,\,950\,\,cm^3\,\\V2\,=\,70\,\times  \,67\,\times  \,(110-\,45)\,=\,\,\,304\,\,\,850\,\,cm^3


The volume of the composter was shown to be V = 471 800 cm^3
. We still have to convert it into m^3.
. V = 471 800 cm^3 = 0, 471 800 m^3.
We can therefore say that the assertion is true, but we would have to specify the degree of approximation desired.

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