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Brevet de maths 2017 with a topic for 3rd graders that covers trigonometry in the right triangle, Thales’ theorem, functions and Scratch.

**WHITE PATENT**

**Mathematics**

**2 hours**

The use of calculators and geometry equipment is permitted.

Five points are awarded for presentation, writing, and adherence to units and scoring.

**Exercise 1:**

For her comfort, Elise wants to install a triangular shade sail in her garden.

The area of this one must be of 6 m² minimum.

Which of the following 3 models can she choose?

**Exercise 2:**

Cristo Redentor, a Brazilian symbol, is a large statue overlooking the city of Rio that stands at the top of

of the Corcovado mountain.

At the foot of the monument, Julien and Magali want to measure the height of the statue (including the base). Julien, who is 1.90 m tall, stands a few meters in front of the statue. Magali places her gaze at ground level so that she can see the top of the Cristo (S) and that of Julien’s head (T) aligned; she is then at 50 cm from Julian who is at 10 m from the statue.

The situation is modeled below by the figure which is not to scale.

Determine the height S of the statue assuming that the monument and Julian are perpendicular to the ground.

**Exercise 3:**

It’s spring, Mr. Durand has to change the 4 tires of his car.

He wants to order his new tires on a website that proposes 2 offers valid for the purchase

Goodyear brand tires only:

**Offer n°1 :** 20% discount on each tire purchased.

**Offer n°2 :** For 3 tires purchased, the 4th is offered.

Using the following information, determine how much he will have to pay for his 4 tires

if he chooses the most interesting offer.

**Information n°1 : The rates displayed on the website.**

**Information #2: What the information on a tire means**

**Information #3:**

Result of a test Mr. Durand, using the information you provided

for your vehicle, here are the tire sizes that are appropriate for your car:

**Exercise 4:**

For his birthday, Amir received an archery set.

He shoots an arrow. The trajectory of the tip of this arrow is shown below.

The curve gives the height in meters as a function of the horizontal distance in meters covered by the

arrow.

1°) In this part, the answers will be given thanks to graphic readings. No justification

is expected.

a) From what height is the arrow shot?

b) At what distance from Amir does the arrow fall to the ground?

c) What is the maximum height reached by the arrow?

2°) According to the graph, is the height proportional to the horizontal distance? Justify.

3°) In this part, the answers will be justified by calculations.

The curve above represents the function f defined by f(x) = -0.1x²+0.9x+1.

a) Calculate f(5).

b) Does the boom rise more than 3 m high?

**Exercise 5:**

For each of the following six statements, say whether it is true or false, carefully justifying the

answer and detailing your calculations.

**Assertion 1 :** is always a prime number whatever the value of the integer n

natural.

**Assertion 2:** Since the radius of a germanium atom measures meters and the radius of an aluminum atom measures meters then the radius of the

germanium is 10 times larger than the radius of the aluminum atom.

**Assertion 3:** Since the distance from the Earth to the Moon is about kilometers and the distance from the Earth to the Sun is about 150 million kilometers then the distance from the Earth to

the Moon is approximately 390 times smaller than the distance from the Earth to the Sun.

**Assertion 4 :** The expression 3x²-5x+1 is -11 when x = -3.

**Assertion 5:** If you increase a price by 20% and then decrease the resulting price by 20%, then you

gets the initial price.

**Assertion 6:** A cyclist travels 35 km in 2 h 30 min so her average speed is 32.2 km/h.

**Exercise 6:**

1°) Draw a reference mark of unit 1 cm and place the points A(-2 ;1), B(8 ;1) and I the middle of [AB].

2°) Construct the point C with negative ordinate such that AC = 2.8 cm and BC = 9.6 cm.

3) Show that ABC is a rectangle at C.

4°) Draw D, the symmetric of point C with respect to I.

5°) Show that ACBD is a rectangle.

**Exercise 7:**

The CDI of a college must be arranged in two distinct parts: a work room and a room of

research. We wish to cover the floor of the workroom with a whole number of square tiles

identical whose side measures a whole number of centimeters.

1°) What are the dimensions of the workroom?

2°) Can the college buy 14 cm square tiles? 20 cm on a side?

3°) To speed up the installation time, we want the slabs to be as large as possible.

Find their size knowing that it is between 20 cm and 30 cm.

4°) The tiles cost 13,50 € per m2. How much will it cost to cover the workroom floor?

**Exercise 8:**

Amélie has written three programs A, B and C with scratch which are shown below.

1°) With which of the three programs did she obtain the following drawing?

2°) How to modify the program chosen in question 1° to obtain this cross ?

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