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DIPLÔME NATIONAL DU BREVET

SESSION 2019

MATHEMATICS

General series

Length of test: 2 h 00 100 points

**Exercise 1 : ( 10 points )**

The captain of a ship has a treasure consisting of 69 diamonds, 1,150 pearls and 4,140 gold coins.

1. Decompose 69; 1150 and 4140 into products of prime factors.

2. The captain shares the treasure equally among the sailors.

How many sailors are there knowing that all the coins, pearls and diamonds have been distributed?

**Exercise 2 : ( 19 points )**

In this exercise, if necessary, the results will be approximated to the nearest hundredth.

To build the set for a play (Figure 1), Joanna has a rectangular plate ABCD measuring 4 m by 2 m from which she must cut out the three triangles of the set before placing them on top of each other.

It proposes a cutting of the plate (Figure 2).

The ADM triangle satisfies the following conditions:

- The triangle ADM is rectangular in A
- AD=2 m

1. Show that [AM] is approximately 3.46 m tall.

2. The unused portion of the plate is shown in a grid pattern in Figure 2. Calculate a value approximating to the hundredth of the proportion of the plate that is not used.

3. To make the triangles overlap smoothly, Joanna wants the three triangles AMD, PNM and PDN to be similar.

Demonstrate that this is the case.

4. Joanna would like the magnification factor to go from the PDN triangle to the AMD triangle to be smaller than 1.5.

Is this the case? Justify.

**Exercise 3 : ( 17 points )**

Questions 1 and 2 are independent.

An hourglass is composed of :

– Two cylinders C₁ and C₂ of height 4.2 cm and diameter 1.5 cm

– A C3 cylinder

– Two half-spheres S₁ and S₂ of diameter 1,5 cm

We recall the volume of a cylinder of base area and height ℎ :

.

1.

a. Initially, the sand fills the cylinder C₂ two-thirds of the way. Show that the volume of sand is

approximately 4.95 .

b. We turn over the hourglass. Assuming that the sand flow rate is constant and equal to

1.98 , calculate the time in minutes and seconds that the sand will take to flow into

the lower cylinder.

2. In reality, the flow rate of the same hourglass is not constant.

In a factory where hourglasses like this one are made, you take an hourglass at random and

test the flow time in this hourglass several times . Here are the different times summarized

in the following table:

a. How many tests were performed in total?

b. An hourglass is offered for sale if it meets the three conditions below, otherwise it is eliminated.

– The time range is less than 20 s

– The median time is between 2 min 29 s and 2 min 31 s

– The average time is between 2 min 28 s and 2 min 32 s

Will the tested hourglass be eliminated?

**Exercise 4 : ( 19 points )**

We want to make a drawing made of two types of elements (dashes and squares) put together.

Each script on the right draws an element, and moves the pen.

Recall that “orienting at 90” means orienting the pen to the right.

1. Taking 1 cm for 2 pixels, represent the figure obtained if you run the Square script.

Specify the starting and ending positions of the pen on your figure.

To draw the complete design, we have made 2 scripts that use the “Square” and “Dash” blocks

above:

We run the two scripts and we obtain the two drawings below.

2. Assign to each script the figure drawn. Justify your choice.

3. We run the script 2.

a. What is the probability that the first element drawn is a square?

b. What is the probability that the first two elements are squares?

4. In script 2, we would like the color of the different elements, dashes or squares, to be random,

with each time a 50% chance to have a black element and a 50% chance to have a

red.

Write the sequence of instructions to be created and specify where to insert it in the script 2.

**Exercise 5 : ( 18 points )**

Olivia bought a painting to decorate her living room wall.

This table, represented below, is made up of four identical rectangles named 1, 2, 3 and 4 drawn inside a large rectangle ABCD of area equal to 1.215 m².

The length : width ratio is 3 : 2 for each of the five rectangles.

1. Copy and complete the following sentences. No justification is required.

a. Rectangle … is the image of rectangle … by the translation that transforms C into E.

b. Rectangle 3 is the image of rectangle … by the rotation of center F and angle 90° in the

clockwise.

c. Rectangle ABCD is the image of rectangle … by the homothety of center … and ratio 3.

(There are several possible answers, only one is required).

2. What is the area of a small rectangle?

3. What are the length and width of rectangle ABCD?

**Exercise 6 : ( 17 points )**

Here are two calculation programs.

1. Check that if we choose 5 as the starting number,

– The result of program 1 is 16.

– The result of program 2 is 28

We call the result of program 1 according to the number chosen at the beginning.

The function gives the result of program 2 according to the number chosen at the beginning.

2.

a. Express as a function of .

b. Determine the number that must be chosen at the beginning to obtain 0 as the result of the program 1.

3. Expand and reduce the expression :

4.

a. Show that .

b. What numbers must be chosen at the start for program 1 and program 2 to give the same result? Explain the process.

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