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MATHEMATICS PATENT

TEST DURATION: 2 hours

Exercise 1:

In this exercise, we will use the following calculation program:

**Calculation program:**

– Choose a number x,

– Remove 3 from the double of x,

– Take the square of the result,

– Subtract 16 from the result.

1) If we choose = 5, what is the final result? And for x = -3 ?

2) State which of the following expressions describes the given program:

a)

b)

c)

d)

e)

f)

3) We pose .

Show that .

4) For what values of x does the calculation program give the number 0 as the final result?

**Exercise 2:**

The figure below is not to scale.

We give:

– AC = 59 cm and AE= 76.7 cm.

– B is a point on the segment [AC] such that AB = 37 cm.

– D is a point on the segment [AE] such that AD = 48.1 cm.

1) Determine the PGCD of the numbers 481 and 767.

2) Simplify the fraction by detailing the calculations.

3) Using the previous result, show that the lines (BD) and (CE) are parallel.

**Exercise 3:**

Here are the distances between the Sun and three planets of the solar system:

Venus: km Mars: km Earth: km.

Among these three planets, which one is the farthest from the Sun? Justify.

**Exercise 4:**

In the Aléa family, we have found an original way to designate which of the three children will be the one to make the

dishes : we throw 2 times a 1€ coin :

– If the piece falls on its face twice, it will be Marc,

– If the coin falls twice on Tails, it will be Elise,

– If the coin falls on two different sides, it will be Leo.

Does this seem fair to you? Explain.

**Exercise 5:**

The diagram is not to scale.

A Guyanese crew, participating in a regatta, decides to redo the sails of its three-masted boat.

1. The small sail is represented by the triangle EFG rectangular in E with EG= 4,5 m and FG = 7,5 m.

a) Show that EF = 6 m.

b) Calculate the measure, rounded to the nearest degree, of the angle .

2. The average sail is represented by the triangle DEC rectangular in C with EC = 7.5 m.

a) Using the geometric configurations coded on the figure, show that the lines (DC) and

(EF) are parallel.

b) Calculate the distance DC.

3. For the main sail, represented by the BAC triangle, the crew already has the measurements that

are : AB = 24 m, BC = 7 m AC = 25 m .

Is the triangle ABC rectangular?

4. On the attached sheet, draw the triangles EFG and DCE using a scale of 1 cm for 2 m.

You will make this figure carefully leaving the construction lines.

**Exercise 6:**

On the figure provided in the Appendix are plotted the graphical representations of two functions f and g.

The curve C corresponds to the function f and the line D to the function g.

For the 4 following questions you will indicate by dotted lines on the figure provided in Appendix the

justifications of your graphic reading.

The questions are independent.

1. Read on the graph f(0), f(-3) and f(7)

2. What are the images of 1 and 4 by the function f?

3. What are the antecedents of 4 by the function f?

4. How many antecedents of -2 are there to the function f?

5. What about the function g? Determine its expression. You will justify your answers.

6. Draw on the figure provided the graphical representation of the function .

You will justify your plot.

**Exercise 7:**

In this exercise, any trace of research, even if incomplete, will be taken into account in the scoring.

For each statement, say whether it is true or false; justify the answer:

**Assertion 1:**

For any integer, the expression is always different from zero.

**Assertion 2:**

If you lower the price of an item by 20% and then raise the new price by 20% then

we come back to the starting price.

**Assertion 3:**

The product of by is 1.

**Assertion 4:**

-2 is a solution of the equation .

**APPENDICES**

Exercise 5: Draw below the figure requested in question 4

Exercise 6: Here is the graph to complete

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