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**Middle school math 2018 patent subject in North America on Tuesday, June 5, 2018.**

** Indication of the whole subject.**

**All answers must be justified unless otherwise indicated.**

**For each question, if the work is not completed, leave a trace of the research;****It will be taken into account in the grading.**

EXERCISE 1 : 14 POINTS

The table below was created using a spreadsheet.

It shows the number of high-speed and ultra-high-speed Internet subscriptions between 2014 and 2016, on

fixed network, in France. (Sources: Arcep and Statistica).

1. How many very high speed Internet subscriptions, in millions, were counted for

the year 2016?

2. Verify that in 2016, there were 817,000 broadband and ultra-broadband Internet subscriptions from

more than in 2015.

3. What formula could have been entered in cell B4 and then copied to the right, to the

cell D4?

4. In 2015, only 5.6 percent of superfast Internet subscriptions used fiber.

How many broadband Internet subscriptions did this represent?

EXERCISE 2 : 14 POINTS

The figure below is not full size.

The following information is given:

– The triangle ADE has dimensions :

AD = 7 cm, AE = 4.2 cm and DE = 5.6 cm.

– F is the point of [AD] such that AF = 2.5 cm.

– B is the point on [AD) and C is the point on [AE)

such that: AB = AC = 9 cm.

– The line (FG) is parallel to the line (DE).

1. Make a full-scale figure.

2. Prove that ADE is a right triangle at E.

3. Calculate the length FG.

EXERCISE 3 : 15 POINTS

Two urns contain numbered balls that are indistinguishable by touch.

The diagram below represents the contents of each of the ballot boxes.

A two-digit integer is formed by randomly drawing a ball from each urn:

– the tens digit is the number of the ball from the D box;

– the number of units is the number of the ball coming from the urn U.

Example : by drawing the 1 ball from the D urn and then the 5 ball from the U urn, we form the

number 15.

1. Is it more likely to form an even number than an odd number?

2. a. Without justification, indicate the prime numbers that can be formed in this experiment.

b. Show that the probability of forming a prime number is equal to .

3. Define an event whose probability of occurrence is equal to .

EXERCISE 4 : 14 POINTS

In this exercise, no justification is expected.

Simon is working on a program. Here are copies of his screen:

1. He obtains the drawing below.

a. According to the main script, what is the side length of the smallest square drawn?

b. According to the main script, what is the side length of the largest square drawn?

2. In the main script, where can we insert the instruction

add 2 to the size of the pen to get the drawing below?

3. The main script is now modified to obtain the one shown below:

Which of the following drawings do you get?

EXERCISE 5 : 6 POINTS

Gaspard works with a dynamic geometry software to build a frieze.

He constructed a triangle ABC isosceles at C (pattern 1) and then obtained the rhombus ACBD (pattern 2).

Here are the screenshots of his work.

1. Specify a transformation to complete pattern 1 to obtain pattern 2.

2. Once the 2 motif was constructed, Gaspard repeatedly applied a translation.

He thus obtains the frieze below.

Specify which translation it is.

EXERCISE 6 : 16 POINTS

Mrs. Martin wishes to build a concrete terrace in front of her bay window.

She makes the drawing below.

To facilitate the drainage of rainwater, the floor of the terrace must be sloped.

The terrace has the shape of a right prism whose base is the quadrilateral

ABCD and the height is the segment [CG].

P is the point on the segment [AD] such that BCDP is a rectangle.

1. The dABP angle should be between 1° and 1.5°.

Does Ms. Martin’s project meet this condition?

2. Mrs. Martin wishes to have the concrete necessary for the construction of her terrace delivered.

She calls on a specialized company.

Using the information in the table below, determine the amount of the

invoice issued by the company.

We remind you that any trace of research, even incomplete, can be taken into account in

evaluation.

EXERCISE 7 : 15 POINTS

The following three questions are independent.

1. A = 2x(x -1)-4(x -1).

Expand and reduce the expression A.

2. Show that the number -5 is a solution of the equation (2x +1)×(x -2) = 63.

3. Consider the function f defined by f (x) = -3x +1.5.

a. Which of the two graphs below represents the function f?

b. Justify your choice.

EXERCISE 8 : 6 POINTS

We consider the download window below.

If the download speed remains constant, will it take more than one minute and twenty-five seconds

for the download to complete?

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