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**patent 2021 in France**. The subject consists of five exercises dealing with statistics and the concepts of average and percentage. An exercise on arithmetic and the Thales theorem and the Pythagorean theorem. A geometry quiz with axial symmetry and homothety. A calculus program exercise with literal calculus and product equations.

NATIONAL PATENT DIPLOMA

FRANCE – SESSION 2021

MATHEMATICS

General series

Exercise 1 (20 points)

This spreadsheet shows the average monthly temperatures in Tours in 2019.

1) According to the table above, what was the average temperature in Tours in November

2019 ?

2) Determine the range of this series.

3) What formula should be entered in cell N2 to calculate the average annual temperature?

4) Verify that the average annual temperature is 13, 1 oc.

5) The average annual temperature in Tours in 2009 was 11.9°C.

Is the percentage increase between 2009 and 2019, rounded to the nearest whole number, : 7%; 10% or 13%? Justify the answer.

Exercise 2 (20 points)

The Futuroscope is a leisure park located in the Vienne. The year 2019 recorded 1.9 million

of visitors.

1) How many more visitors would have been needed in 2019 to reach 2 million visitors?

2) Is the statement “There were approximately 5,200 visitors per day in 2019” true? Justify the

answer.

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 wishes that each group

has the same number of girls and the same number of boys.

a) Decompose the numbers 126 and 90 into prime factors

b) Find all the integers that divide both the numbers 126 and 90

c) Deduce the largest number of groups that the teacher can

constitute. How many girls and boys will there be in each group?

4) Two 3rd grade students, Marie and Adrien, remember seeing in math 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 cnrncide

with ælle of the tower. After having made several measurements, Adrien makes the following 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.

Exercise 3 (20 points)

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:

A box contains 7 green chips, 4 red chips, 3 blue chips and 2 yellow chips. The tokens

Are indistinguishable to the touch. We draw a random chip from this urn.

PART B:

We consider the following figure, composed of twenty patterns numbered from 1 to 20, in which

– .

– pattern 11 is the image of pattern 1 by the homothety of center O and ratio 2.

Exercise 4 (20 points)

Here is a calculation program:

1) Verify that if we choose 4 as the starting number, we get 18.

2) Apply this calculation program to the number – 3.

3) You will find below a script, written with scratch.

Complete lines 5 and 6 on the APPENDIX to make this script match the calculation program.

4) We want to determine the number to choose at the start to obtain zero as the result.

a) We call x the starting number. Express the final result as a function of x.

b) Verify that this result can also be written as (x + 5)(x – 2).

c) What number(s) must be chosen at the start to obtain the number O at the finish?

Exercise 5 (20 points)

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%.

1) By how many tons did the annual production of waste per French person in 2017 decrease

compared to 2007?

2) To continue to reduce their waste production, many families use

now a composter.

One of these families chose the following model, composed of a right block and a right prism

(the figure of the is not to scale). The description indicates that it has a capacity of approximately .

We wish to verify this information.

a) In trapezoid ABCD, calculate the length CH.

b) Show that the length DH is equal to 45 cm.

c) Verify that the area of trapezoid ABCD is 2385 cm².

d) Calculate the volume of the composter.

Is the statement “it has a capacity of about 0.5 ” true? Justify.

APPENDIX to be returned with the copy

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