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DIPLÔME NATIONAL DU BREVET

SESSION 2017

FIRST TEST

Part 1

MATHEMATICS

General series

Duration of the test: 2 hours – 50 points

Presentation of the copy and use of the French language 5 points

All answers must be justified unless otherwise indicated.

For each question, if the work is not completed, leave a record of the

research. It will be taken into account in the scoring.

** **

**COMMON THEME OF THE MATHEMATICS-SCIENCE TEST: WATER**

**Exercise 1 : 6 points**

An opaque bag contains 120 balls all indistinguishable to the touch, 30 of which are blue. The other balls are red or green.

Consider the following random experiment:

You draw a ball at random, look at its color, put the ball back in the bag and mix.

1. What is the probability of drawing a blue ball? Write the result as a fraction

irreducible.

2. Cecile performed this random experiment 20 times and got a green ball 8 times . Choose, among the following answers, the number of green balls contained in the bag (no justification is required):

a. 48 b. 70 c. We can’t know d. 25

3. The probability of drawing a red ball is 0.4.

a. What is the number of red balls in the bag?

b. What is the probability of drawing a green ball?

**Exercise 2 : 7 points**

To illustrate the exercise, the figure below was made freehand.

Points D, F, A and B are aligned, as are points E, G, A and C.

Moreover, the lines (DE) and (FG) are parallel.

1. Show that the triangle AFG is a right triangle.

2. Calculate the length of the segment [AD]. Deduce the length of the segment [FD].

3. Are the lines (FG) and (BC) parallel? Justify.

**Exercise 3 : 6 points**

Here are three different figures, none of which are to the scale shown in the exercise:

The program below contains a variable named “length”.

It is recalled that the instruction means that the pen is oriented to the right.

1. a. Draw the figure obtained with the “one turn” block given in the right frame above, for a

starting length equal to 30, being oriented to the right with the pen, at the beginning of the line.

We will take 1 cm for 30 units of length, i.e. 30 pixels.

b. How is the pen oriented after this line? (no justification is required)

2. Which of figures 1 or 3 does the above program produce? Justify your answer.

3. What modification must be made to the “one turn” block to obtain the figure 2 above?

**Exercise 4 : 9 points**

Mr. Chapuis wishes to change the tiles and baseboards(*) in the living room of his apartment. To do this, he has to buy tiles, glue and wooden skirting boards that will be nailed. It has the following documents:

1. a. Noting that the length GD is equal to 7m, determine the area of triangle BCH.

b. Show that the area of the room is 32 m².

2. In order not to run out of tiles or glue, the salesman advises Mr. Chapuis to provide a

area 10% larger than the area calculated in question 1.

Mr. Chapuis has to buy whole boxes and bags.

Determine the number of boxes of tile and the number of bags of glue to purchase.

3. The vendor also recommends a 10% margin on the length of the baseboards. Determine the total number of baseboards that Mr. Chapuis must buy to go around the room. We specify

that there is no baseboard on the door.

4. How much does Mr. Chapuis have to spend, knowing that he can make do with a pack of nails? Round the answer to the nearest euro.

**Exercise 5: 5 points**

For each statement, say whether it is true or false.

**Exercise 6: 5 points**

In a ski resort, the managers have to cover the slalom course with artificial snow. Artificial snow is produced with the help of snow cannons. The runway is modeled by a rectangle whose width is 25 m and length is 480 m.

Each snow gun uses 1 m3 of water to produce 2 m3 of snow.

Snow production rate: 30 m3 per hour per gun.

1. In order to prepare the slalom course properly, a 40 cm thick layer of artificial snow must be produced.

How much snow should be produced? How much water will be used?

2. On this ski slope, there are 7 snow guns that all produce the same amount of snow.

Determine the time required to operate the snow guns to produce the 4,800 m

3 of snow desired. Give the result to the nearest hour.

**Exercise 7: 7 points**

Legionella are bacteria present in drinking water. When the water temperature is

between 30°C and 45°C, these bacteria proliferate and can reach, in 2 or 3 days, concentrations

dangerous for humans.

Recall that “μm” is the abbreviation for micrometer. A micrometer is equal to one millionth of a meter.

1. The size of a legionella bacterium is 0.8 micrometer.

Express this size in m and give the result in scientific writing.

2. When the water temperature is 37°C, this population of legionella bacteria doubles every quarter of an hour.

A population of 100 legionella bacteria is placed under these conditions.

The following spreadsheet has been created to give the number of legionella bacteria in

depending on the number of quarter hours elapsed:

a. In cell B3, we want to enter a formula that we can stretch downwards in column B

to calculate the number of legionella bacteria corresponding to the number of quarter hours

elapsed. What is this formula?

b. What is the number of legionella bacteria after one hour?

c. Is the number of legionella bacteria proportional to the time elapsed?

d. After how many quarters of an hour does this population exceed ten thousand legionella bacteria?

3. We wish to test the effectiveness of an antibiotic against legionella bacteria. We introduce

the antibiotic in a container that contains 104 legionella bacteria at time = 0. The representation

graph, on the appendix, to be returned with the copy, gives the number of bacteria in the container in

function of time.

a. After 3 hours, approximately how many legionella bacteria remain in the container?

b. After approximately how long does 6000 legionella bacteria remain in the container?

c. It is estimated that an antibiotic will be effective on humans if it manages to reduce the number of

of bacteria in the container in less than 5 hours.

With the help of the graph, study the effectiveness of the antibiotic tested on humans.

**Appendix to be returned with the copy**

Show the lines justifying the answers in question 3. of Exercise 7.

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