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Brevet des collèges

Mathematics

Polynesia June 23, 2022

Duration: 2 hours

Exercise 1: 20 points.

For each of the following four statements, say whether it is true or false and explain

carefully the answer.

1. Adriana must perform the following calculation:

Assertion 1: The result she gets as an irreducible fraction is .

2. In the figure below, which is not to scale, the points G, A and R are aligned and the

points E, A and M are aligned.

Assertion 2: Lines (GE) and (MR) are parallel.

3. Assertion 3: The prime factor decomposition of 126 is 2×7×9.

4. In Thomas’ salad dressing recipe, the volumes of mustard, vinegar

and oil are in the ratio of 1 : 3 : 7.

5. Assertion 4: To make 330 mL of salad dressing, 210 mL of oil must be used.

Exercise 2: 16 points.

The graph below represents the two prices charged in a gym, according to the

number of hours worked :

– the line (d1) is the graphical representation of the “freedom” tariff

– the line (d2) is the graphical representation of the “subscriber” rate

1. Is the price paid with the “freedom” rate proportional to the number of hours worked

in the gym? Explain the answer.

2. We call:

– f is the function which, to the number of hours worked, associates the price paid in euro

with the “freedom” rate

– g is the function which, to the number of hours worked, associates the price paid in euro

with the “subscriber” rate

Answer the following questions by graphical reading:

a. What is the image of 5 by the function f?

b. What is the antecedent of 10 by the function g?

3. Using the graph, indicate which of the two proposed rates is more advantageous for an individual depending on the number of hours he or she wishes to work in the

gym.

4. Determine the price paid with the “freedom” rate for 15 hours worked.

Explain the process, even if it is not complete.

Exercise 3: 23 points.

*The two parts of this exercise are independent.*

A company produces and sells fruit juices contained in cardboard boxes that have

the shape of a right block.

PART A: Apple juice cartons.

These cartons are made to hold 350 mL of apple juice.

During an inspection, 24 bricks are randomly taken and analyzed.

The table below gives the volume of apple juice (in mL) contained in these cartons:

1. Determine the median of the volumes in this series. Interpret this result.

2. Calculate the range of this series.

3. We take at random a brick among those controlled, what is the probability

that it contains exactly 350 mL of apple juice?

4. When the volume of apple juice in a carton is between 345 mL

and 355 mL, this brick can be sold.

What is the percentage of bricks that the company can sell among the bricks

controlled?

PART B: Grape juice cartons.

The company wants to market a new brick in the shape of a straight block for grape juice. Its base is a rectangle of length 6.4 cm and width 5 cm.

1. Calculate the area of the base of this brick

2. How high must this brick be for its volume to be 400 ?

Exercise 4: 20 points.

We consider the following calculation program:

1. a. If you choose the number 7, check that you get 49 at the end of the program

b. If we choose the number -4, what result do we get at the end of the program?

2. We note the number chosen at the beginning

a. Express the result as a function of .

b. Expand and reduce (x +5)(x -5).

c. Sarah says, “With this calculation program, no matter what number is chosen at the beginning, the result is always the square of the starting number.

What do you think about it?

Exercise 5: 23 points.

The Pompidou Center is a museum of contemporary art in Paris. To access the floors, a set of external escalators called “caterpillar” must be used.

The caterpillar is composed of 5 escalators all identical (thick lines on the figure below) and 6 horizontal walkways all identical (thin horizontal lines on the figure below).

1. Using the figure below:

a. Check that the depth of each escalator is equal to 12 m.

b. Calculate the height of each escalator.

2. Using the RST triangle below:

a. Prove that the ST length of an escalator is 13.6 m.

b. Show that the measure of the angle formed by the escalator with the horizontal (i.e.

the angle rounded to the degree is 28°.

2. Sabine wants to represent the caterpillar with the Scratch software.

She wrote the program that is given on the APPENDIX.

We specify that: 1 step of the software corresponds to 1 m in reality.

Complete lines 6, 7, 9, and 10, on the APPENDIX (to be returned with the copy), to obtain

the following plot of the caterpillar:

Reminder: “Orienting at 90” means that you are oriented to the right

APPENDIX

To be completed and returned with the copy

Exercise 5 question 3:

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