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DIPLÔME NATIONAL DU BREVET

SESSION 2021

MATHEMATICS

General series

Duration of the test: 2 h 00 – 100 points

**EXERCISE 1 : MCQ (18 points)**

This exercise is a multiple choice questionnaire (MCQ). For each question, only one of the three proposed answers is correct.

On the copy, indicate the question number and the answer A, B or C chosen.

No justification is required.

No points will be deducted for incorrect answers.

**EXERCISE 2: The invoice (8 points)**

A price including all taxes is obtained by adding the tax called TGC (General Tax on Consumption) to the price excluding taxes.

In New Caledonia, there are four rates of CGT depending on the case: 22%, 11%, 6% and 3%.

Alexis has just had her car repaired at a body shop.

Here is an excerpt from his invoice which has been stained by paint.

Columns B, D and E indicate prices in francs.

1. What is the TGC amount for the bumper?

2. What percentage of the CGT is applied to labor?

3. The invoice was made using a spreadsheet.

What formula was entered in cell E6 to obtain the total to pay?

**EXERCISE 3: Calculation programs (11 points)**

The following two calculation programs are given:

1. Alice chooses the number 4 and applies the program A.

Show that it will get – 4.

2. Lucy chooses the number – 3 and applies program B.

What result will it achieve?

Tom wants to find a number for which both calculation programs will give the same result. He chooses x as the starting number for both programs.

3. Show that the result of the program A can be written ²-5.

4. Express the result obtained with program B as a function of x.

5. What number is Tom looking for?

Any trace of research, even if not completed, will be taken into account in the scoring.

**EXERCISE 4: The regatta (16 points)**

In the following figure, we give the distances in meters: AB = 400, AC = 300, BC = 500 and CD = 700.

1. Calculate the length DE.

2. Show that triangle ABC is a rectangle.

3. Calculate the measure of the angle . Round to the nearest degree.

During a race, competitors must complete several laps of the course shown above. They start from point A, then pass through points B, C, D and E in that order, then back through point C and then back to point A.

Mattéo, the winner, took 1 h 48 min to complete the 5 laps of the course. The distance covered to make a turn is 2,880 m.

4. Calculate the total distance traveled to complete the 5 laps of the course.

5. Calculate the average speed of Mattéo. Round off to the nearest whole number.

**EXERCISE 5: The rope (7 points)**

The triangle ABC rectangle in B below is such that AB = 5 m and AC = 5.25 m.

1. Calculate, in m, the length BC. Round to the tenth.

A 10.5 m long non-elastic rope is fixed to the ground by its two ends between two posts 10 m apart.

2. Could Melvin, who is 1.55 m tall, pass under this rope without bending down?

by lifting it from the middle?

Any trace of research, even if not completed, will be taken into account in the scoring.

**EXERCISE 6: Labels (14 points)**

1. Justify that the number 102 is divisible by 3.

2. We give the decomposition into products of prime factors of 85: 85=5 × 17. Decompose 102 into products of prime factors.

3. Give 3 non-prime divisors of the number 102.

A bookseller has a 85 cm x 102 cm sheet of cardboard. He wants to cut out square labels from it, using the whole sheet. The sides of these labels all have the same measurement.

4. Can the labels be 34 cm square? Justify.

5. The bookseller cuts labels of 17 cm on each side. How many labels will he be able to cut out in this case?

**EXERCISE 7: Housing (15 points)**

Nolan wants to build a home. He hesitates between a box and a house in the shape of a right prism. The box is represented by a right cylinder of axis (OO’) surmounted by a cone of revolution of summit S.

The dimensions are given in the following figures. x represents both the diameter of the square and the length AB of the right prism.

**Part 1:**

In this part, we consider that x = 6 m.

1. Show that the exact volume of the cylindrical part of the box is .

2. Calculate the volume of the conical part. Round off to the nearest whole number.

3. Deduce that the total volume of the box is about 66 .

**Part 2:**

In this section, the diameter is expressed in meters, the volume in m3. On the appendix , we have represented the function that gives the total volume of the box as a function of its diameter x.

1. By graphical reading, give an approximate value of the volume of a 7 m diameter box.

Draw dotted lines for reading.

The function that gives the volume of the right prism house is defined by V(x) = 12.5 x.

2. Calculate the image of 8 by the function V.

3. What is the nature of the function V?

4. On the appendix, draw the graph of the function V.

For practical reasons, the maximum value of x is 6 m.

Nolan wants to choose the construction that will give him the most volume.

5. Which construction should he choose? Justify.

**EXERCISE 8: Scratch (11 points)**

The following script allows you to draw a square of side 50 units.

1. On the appendix, complete the script to obtain an equilateral triangle of side 80 units.

We ran the following script:

2. Circle on the appendix, the figure obtained with this script.

**APPENDIX TO BE RETURNED WITH THE COPY**

**Exercise 7:**

**Part 2: Question 1 and 3.**

**Exercise 8:**

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