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**Exercise 1 : 13 points**

Damien has made three perfectly balanced but somewhat unusual six-sided dice.

On the faces of the first die are written the six smallest strictly positive even numbers: 2; 4; 6;

8; 10; 12.

On the faces of the second die are written the six smallest positive odd numbers.

On the sides of the third die are written the six smallest prime numbers.

After rolling a die, the number on the top face is noted.

1. What are the six numbers on the second die?

What are the six numbers on the third die?

2. Zoe chooses the third die and rolls it. It squares the resulting number. Leo chooses the first

and rolls it. He squares the resulting number.

a. Zoë got a square equal to 25. What was the number on the die she rolled?

b. What is the probability that Leo will get a square greater than the one obtained by Zoe?

3. Mohamed chooses one of the three dice and rolls it four times in a row. It multiplies the four numbers

obtained and gets 525.

a. Can we determine the numbers obtained in the four throws? Justify.

b. Can we determine which die Mohamed chose? Justify.

**Exercise 2 : 18 points**

“Turn 90” means to turn to the right.

Mathieu, Pierre and Elise want to draw the pattern below using their computer. They begin

all by the common script below, but write a different Motif script.

1. Draw Mathieu’s pattern taking as a scale: 1 cm for 10 pixels.

2. Which student has a script to achieve the desired pattern? No justification is required.

3. a. We use this pattern to obtain the figure below.

Which transformation of the plane allows you to go from pattern 1 to pattern 2, from pattern 2 to pattern 3 and from pattern 3 to pattern 4 at the same time?

b. Modify the common script from line 7 onwards to obtain the desired figure.

Only the modified part will be written on the copy.

You may use some or all of the following instructions:

4. A student draws the two figures A and B that you will find in ANNEX 1.1

Place on this appendix, which is to be returned with the copy, the center O of the central symmetry that

transforms figure A into figure B.

**Exercise 3: 17 points**

On July 1, 2018, the maximum speed limit on two-way roads, without

from 90 km/h to 80 km/h.

In 2016, 1,911 people were killed on two-way roads with no dividers

This represents about 55% of all road deaths in France.

Source : www.securite-routiere.gouv.fr

1. a. To show that in 2016, there were approximately 3475 deaths on all roads in France.

b. Experts have estimated that lowering the speed limit to 80 km/h would have saved 400

lives in 2016.

By what percentage would the number of deaths on all roads in France have been reduced?

lowered? Give an approximate value to the nearest 0.1%.

2. In September 2018, gendarmes conducted a series of checks on a road whose

The maximum speed limit is 80 km/h. The results were entered into a spreadsheet in the following order

increasing speeds. Unfortunately, the data in column B has been deleted.

a. Calculate the average speed of motorists checked who exceeded the speed limit

maximum allowed. Give an approximate value to the nearest 0.1 km/h.

b. Knowing that the range of the recorded speeds is equal to 27 km/h and that the median is equal to

at 82 km/h, what are the missing data in column B?

c. What formula should be entered in cell K2 to obtain the total number of motorists

controlled?

**Exercise 4: 10 points**

Leila is visiting Paris. Today, it is in the Champ de Mars where you can see the Eiffel Tower, whose

the total height BH is 324 m.

She puts her camera on the ground at a distance AB = 600 m from the monument and programs it to

take a picture (see drawing below).

1. What is the measure, to the nearest degree, of the angle ?

2. Knowing that Leila is 1.70 m tall, at what distance AL from her camera should she stand to

look as tall as the Eiffel Tower in her picture?

Give an approximate value of the result to the nearest centimeter.

**Exercise 5 : 22 points**

1. a. Show that, if we choose the number 5, the result of program A is 29.

b. What is the result of program B if we choose the number 5?

c. If we name the chosen number x, explain why the result of the program A can

can be written x² +4.

2. What is the result of program B if we name the chosen number x ?

3. Are the following statements true or false? Justify the answers and write the steps

any calculations:

a. “If we choose the number , the result of program B is . ”

b. “If we choose an integer, the result of program B is an odd integer.

”

c. “The result of program B is always a positive number.”

d. “For the same chosen integer, the results of programs A and B are either

both even integers, or both odd integers.”

**Exercise 6: 20 points**

To serve fruit juices, a restaurant owner can choose between two types of glasses: a cylindrical glass

A of height 10 cm and radius 3 cm and a conical glass B of height 10 cm and radius 5.2 cm.

The graph in ANNEX 1.2 represents the volume of fruit juice in each glass in

according to the height of the fruit juice they contain.

1. Answer the following questions using the graph in ANNEX 1.2 :

a. For which glass are the volume and height of the juice proportional? Justify.

b. For glass A, what is the volume of fruit juice if the height is 5 cm?

c. What is the height of the fruit juice if 50 is poured into glass B?

2. Show, by calculation, that the two glasses have the same total volume to the nearest 1 .

3. Calculate the height of the fruit juice served in glass A so that the volume of juice is equal to

200 . Give an approximate value to the nearest centimeter.

4. A restaurant owner serves his glasses so that the height of the juice in the glass is

equal to 8 cm.

a. By graphical reading, determine which type of glass the restaurant owner should choose to serve

as many glasses as possible with 1 L of fruit juice.

b. By calculation, determine the maximum number of glasses A that can be served with 1 L of

fruit juice.

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