Subject of the patent of maths France 2016

Brevet de maths
France 2016

Exercise 1: (4 points)
A company markets electronic components that it manufactures in two factories. During a
quality control, 500 components are taken from each plant and examined to determine
whether they are “good” or “bad”.
Results obtained for all 1000 components sampled:

1) If we take a component at random from factory A, what is the probability that it
is defective?
2) If we take a component at random from among those that are defective, what is the probability that it
comes from factory A?
3) The inspection is considered satisfactory if the percentage of defective components is less than 7% in
each plant. Is this control satisfactory?

Exercise 2: (4.5 points)
We consider the two calculation programs below.

1) Verify that choosing 2 at the beginning with program A gives 9.
2) What number must be chosen initially with program B to obtain 9?
3) Can we find a number for which both calculation programs give the same result?

Exercise 3: (5 points)
Three coded figures are given below. They are not drawn in full size.
For each, determine the length AB to the nearest millimeter.
In this exercise, a written demonstration is not expected. It is sufficient to briefly explain the
reasoning and to present the calculations clearly.

Exercise 4: (5 points)
During the sales, a merchant decides to apply a 30% discount on all the items in the
his store.
1) One of the items costs 54 € before the discount. Calculate its price after the discount.
2) The merchant uses the spreadsheet below to calculate the prices of the sale items.

a) To calculate the reduction, what formula could he enter in cell B2 before stretching it over the
line 2 ?
b) To obtain the balance price, what formula can he enter in cell B3 before stretching it over the
line 3 ?
3) The sale price of an article is 42,00 €. What was the original price?

Exercise 5: (5.5 points)
The PRC figure below represents a piece of land owned by a municipality.
The points P, A and R are aligned.
The points P, S and C are aligned.
It is planned to develop on this land

• a “children’s play area” on the PAS side;
• a “skatepark” on the RASC section.

The following dimensions are known:
PA = 30 m; AR = 10 m; AS = 18 m.

1) The municipality wishes to plant grass in the “children’s play area”. She decides to buy
5 kg bags of grass seed mix at 13,90 € each. Each bag covers one
How much money does this community need to budget to be able to seed the entire “turf area” with grass?
children’s games”?
2) Calculate the area of the “skatepark”.

Exercise 6: (7 points)
With strings of 20 cm, we build polygons as below:

Part 1:
In this section, a string is cut in step 1 so that “piece #1” is 8 cm long.
1) Draw the two polygons in real size.
2) Calculate the area of the resulting square.
3) Estimate the area of the resulting equilateral triangle by measuring on the drawing.

Part 2:
In this part, we now want to study the area of the two polygons obtained in step 3 by
depending on the length of “piece no. 1”.
1) Propose a formula to calculate the area of the square as a function of the length of the
“piece no. 1”.
2) On the graph below:
– curve A represents the function that gives the area of the square as a function of the length of the
“piece no. 1”;
– curve B represents the function that gives the area of the equilateral triangle as a function of the
length of the “piece nº 1”.

Using this graph, answer the following questions. No justification is expected.
a) What is the length of “piece #1” to obtain an equilateral triangle
of area 14 cm² ?
b) What is the length of “piece #1” that yields two polygons of area
equal?

Exercise 7: (5 points)
Antoine creates decorative objects with vases, marbles and colored water.
For his new creation, he decided to use the vase and the balls having the following characteristics:

He puts 150 marbles in the vase. Can he add a liter of colored water without risking an overflow?
Recall that the volume of the ball is given by the formula :

.

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