Brevet de maths 2019 in North America: subject and answers

math certificate

EXERCISE 1 : 14 POINTS
Consider the figure below, made freehand and not to scale.

The following information is given:
– the lines (ER) and (FT) are secant at A;
– AE = 8 cm, AF = 10 cm, EF = 6 cm;
– AR = 12cm, AT= 14cm.

1. Prove that the triangle AEF is right-angled at E.
2. Deduce a measure of the angle $\widehat{EAF}$ to the nearest degree.
3. Are the lines (EF) and (RT) parallel?

EXERCISE 2 : 17 POINTS
Here are four statements. For each one, say whether it is true or false.

Remember that the answer must be justified.
1. Assertion 1: $\frac{3}{5}+\frac{1}{2}=\frac{3+1}{5+2}$
2. Consider the function f : $x,\mapsto ,5-3x$ .
Assertion 2: The image of -1 by f is -2.

3. We consider two randomized experiments:
– experiment 1: choose at random a whole number between 1 and 11 (1 and 11 included).
– experiment 2: roll a balanced six-sided die numbered 1 to 6 and announce the
number that appears on the top side.

Assertion 3:

It is more likely to choose a prime number in experiment #1 than
to obtain an even number in experiment n° 2.
4. Assertion 4:

For any number x, .

EXERCISE 3 : 12 POINTS
The chart below shows the amount of food wasted (in kg) for six countries per
inhabitant in 2010.

1. Approximate the amount of food wasted by an inhabitant of country D in
2010.
2. Can we say that the food wasted by an inhabitant of country F is about one
fifth of the food waste of a person in country A?
3. We want to report on the amount of food wasted for other countries. We realize
then the table below using a spreadsheet.

Reminder: 1 ton = 1000kg.

a. What is the total amount of food wasted by people in country X in 2010?
b. Here are three proposals for formulas, copy on your copy the one we have entered in the
D2 cell before stretching it to D4.

EXERCISE 4 : 10 POINTS
We programmed a game. The goal of the game is to get out of the maze.

At the beginning of the game, the sprite stands at the starting point. When the leprechaun touches a wall, represented by a
thick black line, it returns to the starting point.

The background is made up of a reference frame of origin O with points spaced 30 units vertically
and horizontally.
In this exercise, we will consider that only the walls of the labyrinth are black.
Here is the program:

The block corresponds to a subroutine that makes the sprite say “Win!” when it is located at the exit point; the game then stops.

1. Copy and complete the instruction of the program to bring the elf back to the
starting point if the black color is touched.
2. What is the minimum distance the sprite must travel from the starting point to the end point?
exit?
3. The program is launched by clicking on the flag. The leprechaun is at the starting point. We press
Briefly press the ↑ (“up arrow”) key and then the→(“right arrow”) key. What are
all the actions performed by the sprite?

EXERCISE 5 : 10 POINTS
In this exercise, no justification is expected.
Consider the hexagon ABCDEF of center O shown below.

1. Among the following propositions, copy the one which corresponds to the image of the CDEO quadrilateral
by the symmetry of center O.

2. What is the image of the segment [AO] by the symmetry of axis (CF) ?
3. Consider the rotation of center O that transforms the triangle OAB into the triangle OCD.
What is the image of triangle BOC by this rotation?

The figure above represents a paving stone whose basic pattern has the same shape as the hexagon above. Some of these hexagons have been numbered.

4. What is the image of hex 14 by the translation that transforms hex 2 into hex 12?

EXERCISE 6 :12 POINTS
The two parts A and B are independent.

Part A: absorption of the active ingredient of a drug
When a drug is taken, whether orally or not, the amount of active ingredient in
This drug in the bloodstream changes with time. This amount is measured in milligrams
per liter of blood.
The graph below represents the amount of active ingredient of a drug in the blood, in
depending on the time elapsed since the medication was taken.

1. What is the amount of active ingredient in the blood, thirty minutes after taking this medication
?
2. How long after taking this medication is the amount of active ingredient the most
high?

Part B: comparison of alcohol masses in two drinks
The following data is provided:

Question: the drink contains more alcohol than the beverage ?

EXERCISE 7 : 15 POINTS
To store cannonballs, sixteenth-century soldiers often used a type of pyramidal stack with a square base, as shown in the following drawings:

1. How many balls does the 2-level stack contain?
2. Explain why the 3-level stack contains 14 balls.
3. We arrange 55 cannonballs according to this method. How many levels does the resulting stack have?
4. These balls are made of cast iron; the density of this cast iron is 7 300 kg/m3.
We model a cannonball by a ball of radius 6 cm.
Show that the 3-level stack of these balls weighs 92 kg, to the nearest kg.

Reminders:
– volume of a ball = $\frac{4}{3}\times ,\pi\times ,rayon\times ,rayon\times ,rayon$ .
– a density of $7,300\,kg/m^3$ means that 1,m^3 weighs 7300 kg.

EXERCISE 8 : 10 POINTS
In a class of Terminale, eight students take an entrance exam to a school of higher education.
To be admitted, you must obtain a grade of 10 or higher.
A score is assigned with a precision of half a point (for example: 10; 10.5; 11; . . .)

The following information is available:

1. Explain why it is impossible for one of the two notes designated by or or 16.
2. Is it possible that the two notes designated by and or 12.5 and 13.5?

Consult the online answer key

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