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Brevet des collèges

Reunion Island Antilles-Guyana

Make-up session – September 16, 2019

**Exercise 1 : 18 points**

Michel participates in a VIT rally on a marked course. The path is represented in solid lines.

The start of the rally is in A and the finish is in G.

1. Show that the length BD is equal to 2.5 km.

2. Justify that the lines (BC) and (EF) are parallel.

3. Calculate the length DF.

4. Calculate the total length of the course.

5. Michael drives at an average speed of 16 km/h to get from point A to point B.

How long will it take to get from point A to point B?

Give your answer in minutes and seconds.

**Exercise 2 : 14 points**

1. a. Determine the product of prime factors decomposition of 2,744.

b. Deduce the product of prime factors decomposition of .

c. Using this decomposition, find such that .

2. Let a and b be two integers greater than 2 such that .

a. Calculate b when a = 100.

b. Determine two integers a and b greater than 2 and less than 10 that satisfy

equality .

**Exercise 3: 17 points**

Human activities produce carbon dioxide (CO2) which contributes to global warming.

The following graph represents the evolution of the average atmospheric concentration

in CO2 (in ppm) as a function of time (in years).

1. Determine graphically the concentration of in ppm in 1995 and in 2005.

2. We want to model the evolution of the concentration of as a function of time using a

function g where g (x) is the concentration of in ppm as a function of the year .

a. Explain why an affine function seems appropriate to model the concentration

in as a function of time between 1995 and 2005.

b. Arnold and Billy each propose an expression for the function g :

Arnold proposes the expression ;

Billy suggests the expression .

Which expression best models the evolution of the concentration of ? Justify.

c. Using the function you chose in the previous question, indicate the year

for which the value of 450 ppm is reached.

3. In France, the forests, thanks to photosynthesis, capture about 70 megatons of per year,

which represents 15% of national carbon emissions (year 2016).

Calculate an approximate value to the nearest megaton of the mass M of emitted in France in

2016.

**Exercise 4 : 16 points**

For Dominique and Camille’s wedding, the pastry chef proposes two pieces of cake made of

cakes of different sizes and shapes.

All cakes were made from the recipe below which gives the quantity of

ingredients corresponding to 100 g of chocolate.

1. What is the ratio (mass of butter : mass of chocolate) ? Give the result as a fraction

irreducible.

2. Calculate the amount of flour needed for 250 g of dark chocolate according to the above recipe.

3. Calculate the length of the side of the base of the smallest cake in the square tower.

4. Which tower has the largest volume? Justify your answer by detailing the calculations.

We recall that the volume V of a cylinder of radius r and height h is given by the formula :

.

**Exercise 5: 15 points**

The following calculation program is given:

1. a. Show that if the number chosen is 4, the result is 20.

b. What is the result when we apply this calculation program to the number -3?

2. Zoë thinks that a starting number is chosen, the result is equal to the sum of this number

and its square.

a. Check that she is right when the number chosen at the beginning is 4, and also when we choose

-3.

b. Ishmael decides to use a spreadsheet to verify Zoe’s assertion on some examples.

He wrote formulas in B2 and B3 to automatically execute steps 2 and 3 of the

calculation program.

What right-copy formula did he write in cell B4 to perform the step

4?

c. Zoë observes the results, then confirms that for any number chosen, the result of the

calculation program is well . Demonstrate your answer.

d. Determine all the numbers for which the result of the program is 0.

**Exercise 6: 20 points**

Two friends Armelle and Basile play dice using well-balanced dice whose faces have

have been modified. Armelle plays with the A die and Basile plays with the B die.

During a game, each player rolls his or her dice and the one who gets the highest number wins a

point.

Here are the patterns of the two dice:

1. Can a game end in a draw?

2. a. If the result obtained with die A is 2, what is the probability that Basil wins a point?

b. If the result obtained with die B is 1, what is the probability that Armelle wins a point?

3. Players want to compare their chances of winning. They decide to simulate a game of

sixty thousand duels with the help of a computer program.

Here is a part of the program they have realized.

We specify that the expression (random number between 1 and 6) returns equiprobably a

number can be 1; 2; 3; 4; 5 or 6.

The variables Face A and Face B record the results of dice A and B. For example, the variable Face A

can take either the value 2 or 6, since these are the only numbers on the die A.

The variables Victory of A and Victory of B count the victories of the players.

1. When the subroutine “Throw the die A” is executed, what is the probability that the variable

Side A takes the value 2?

2. Copy line 7 of the main program and complete it.

3. Write a “Roll B” subroutine that simulates the roll of the B die and records the

number obtained in the variable Face B.

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