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Subject of the **patent of maths 2017** blank where the exercises are independent from each other. Unless otherwise specified, all answers must be justified and calculations must be detailed. Classical geometry material is allowed.

**BREVET MATHEMATICS TEST**

Duration: 2 hours

The use of a calculator is allowed.

The use of the dictionary is not allowed.

**Exercise 1: (4 points)**

In this multiple choice questionnaire, for each question, three answers are proposed, and only one is correct. For each question, indicate on the copy the number of the question and copy the correct answer. No justification is expected.

**Exercise 2: (3 points) **

The questions 1) and 2) are independent.

1) a) Calculate , detailing the steps in the calculation.

b) Is A a decimal number? Justify.

2) For her herbarium, Heloise collects yellow, green and red leaves:

It has of green leaves and of red leaves.

To which fraction of the collection do the yellow leaves correspond?

**Exercise 3: (2 points)**

On October 14, 2012, Felix Baumgartner, made a jump from an altitude of 38,969.3 meters. The first part of his jump was done in free fall (closed parachute).

The second part was done with an open parachute. His goal was to be the first man to “break the sound barrier”. To “break the sound barrier” means to reach a speed greater than or equal to the speed of sound, i.e. .

The Fédération Aéronautique Internationale established that he had reached the maximum speed of during his free fall.

Did he achieve his goal? Justify your answer.

**Exercise 4: (4 points)**

The objective of switching to daylight saving time is to match the hours of activity with the hours of sunshine as closely as possible to limit the use of artificial lighting. The graph below represents the power consumed in megawatts (MW), according to the hours (h) of two days D1 and D2, D1 before the changeover to daylight saving time and D2 after the changeover to daylight saving time.

By graphical reading, answer the questions asked.

If necessary, the results will be rounded to the half-hour.

1) For the day D1, what is the power consumed at 7 am?

2) For the day D2, at what time(s) of the day is the power consumed 54,500 MW?

3) At what time of day does switching to daylight saving time save the most money?

4) What power consumption was saved at 7:30 p.m.?

**Exercise 5: (4 points) **

We consider a cone of revolution of height [AO] measuring 5 cm and whose base has radius 2 cm. Point A is the vertex of the cone and O is the center of its base.

1) Calculate the volume of the cone in .

Give the exact value and then round to the unit.

**Reminder : Volume of a cone of revolution**

where h denotes the height and r the radius of the cone of revolution.

2) A reduction of this cone by a factor of is performed. We obtain a new cone of vertex A whose base has the center B, middle of [AO].

Is it true that the volume of the small cone is half the volume of the original cone? Justify.

**Exercise 6: (4 points) **

To build a vertical wall, it is sometimes necessary to use formwork and shoring that will hold the vertical structure in place while the concrete dries (Figure 1).

This support can be represented by the diagram above (figure 2). The iron beams are cut and fixed so that :

- The segments [AB] and [AE] are perpendicular;
- C is located on the bar [AB]; D is located on the bar [BE];
- AB = 3.5m; AE = 2.625m and CD = 1.5m.

1) Calculate BE.

2) It is further assumed that (CD) and (AB) are perpendicular.

How far from B is point C?

**Exercise 7: (3 points)**

Be careful, the traced figures do not respect the length measures, nor the angle measures. Answer “true” or “false” or “can’t tell” to each of the following statements and explain your choice.

1) “Any triangle inscribed in a circle is a rectangle.”

2) “If a point M belongs to the perpendicular bisector of a segment [AB] then the triangle AMB is isosceles.”

3) “The quadrilateral ABCD opposite is a square.”

**Exercise 8: (4 points) **

Here are the characteristics of a pool that needs to be renovated:

1) The owner begins by emptying the pool with the drain pump.

This pool is filled to the brim. Will it be empty in less than 4 hours?

2) He then repaints the entire interior surface of this pool with resin paint.

What is the cost of the renovation?

**Exercise 9: (3 points) **

To prepare his trip to Marseille, Julien uses a website to choose the best itinerary. Here is the result of his search:

1) Knowing that road safety recommends at least a 10 to 20 minute break every two hours of driving, what should be the minimum duration that Julian should plan for his trip?

2) For this question, show on the copy the approach used. All traces of research will be taken into account during the evaluation even if the work is not completely finished.

Knowing that the tank of his car has a capacity of 60 L and that a liter of gasoline costs 1,42€, can he make the trip with only one tank of gasoline by relying on the data of the Internet site?

**Exercise 10: (5 points)**

What does this sprite do when the program is executed?

This 2017 math patent topic was written by a national education team.

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