math certificate Report an error / Note? BREVET Math 2022

Session: May 2022

Duration of the test: 2 hours – 40 points.

The use of a calculator is allowed

Exercise 1: (6 points)

This exercise is a multiple choice questionnaire (MCQ).

For each of the five questions, three answers are proposed, only one of them is correct.

For each of the five questions, indicate on the copy the question number and the answer chosen. All answers must be justified. A wrong answer or the absence of an answer does not remove a point.

 Questions Answer A Answer B Answer C Q1 Consider the function f defined by The image of – 2 is : – 39 – 25 25 Q2 We consider the function The image of 2 by the function f is 1. The image of 1 by the function f is 2. 2 has no image by the function f. Q3 The reduced expanded expression of is :   Q4 Consider a 14 cm pen and its image on a screen through a converging lens. The pen and its image are parallel.  The height, in cm, of the pen image is calculated by doing :   Q5 A solution of the equation is : – 10 3 – 3

Exercise 2: (4 points)

To hit Averell’s hat, Lucky Luke will have to tilt his gun with precision. It is assumed that the two cowboys are standing perpendicular to the ground.

• Averell’s height: 2.13 m
• Distance from ground to gun: PS = 1 m
• Distance from the gun to Averell : PA = 6 m
• The triangle PAC is rectangular in A.

Calculate the angle of inclination formed by the trajectory of the ball and the horizontal (give the result to the nearest degree).

Exercise 3: (5 points)

Consider the homothety of center O and ratio k which transforms points A, B and C into A’, B’ and C’ respectively. 1) The center O does not appear on the figure. Explain on the copy how it can be found.

2) What is the sign of the ratio of this homothety? Justify your answer.

3) By taking the necessary measurements on the figure, determine the ratio of this homothety.

4) How many times is the area of triangle A’B’C’ greater than that of triangle ABC?

Exercise 4: (6.5 points)

After one of his running training sessions, Leo receives from his trainer the summary of his run, reproduced below. The average pace of the runner is the quotient of the duration of the race by the distance covered and is expressed in min/km.

Example: if Leo takes 18 minutes to cover 3 km, his pace is 6 minutes/km.

1) Leo is surprised that his average speed is not shown. Calculate this average speed in km/h.

2) Let f be the function defined for all x>0 by , where x is the pace in min/km and is the speed in km/h.

This function allows you to know the speed (in km/h) when you know the pace (in min/km).

1. a) Using this function, find the result obtained in question 1).
2. b) On his last run, Leo’s average pace was 5 min/km.

Compute the image of 5 by f. What does the result represent?

3) Answer the following questions using the graphical representation of the function f below:

1. a) Give an antecedent of 10 by the function f.
2. b) A pedestrian travels at about 14 min/km. Give an approximate value of its speed in km/h. Exercise 5: (5.5 points)

Alexandre wishes to prepare a cocktail for his birthday.

 Document 1: cocktail recipe for 6 people Ingredients for 6 people: – 6 dL of mango juice – 30 cL of pear juice – 120 mL lime juice – 15 cL of blackcurrant syrup Document 2: Alexander’s cylindrical container  We consider that the container has the shape of a cylinder of diameter 16 cm and height 20 cm.

Question :

Is the container chosen by Alexandre big enough to prepare the cocktail for 20 people?

Reminders:

• 1 L = 1,000 cm3
• Geometry Form:
 L = length ; l= width ; h = height ; r = radius Volume of a right block Volume of a cylinder Volume of a cone Volume of a ball    Exercise 6: (9 points)

Mathilde participates in a mountain bike rally on a marked course. The path is represented in solid lines.

The start of the rally is in A and the finish is in G.

The start of the rally is in A and the finish is in G.

The drawing is not to scale.

• The points A, B and C are aligned.
• Points C, D and E are aligned.
• The points B. D and F are aligned.
• The points E. F and G are aligned.
• The triangle BCD is rectangular in C.
• The triangle DEF is rectangular in E 1) Show that the length BD is equal to 2.5 km.

2) Justify that the lines (BC) and (EF) are parallel.

3) Show that the length DF is equal to 6.25 km.

4) Calculate the total length of the course.

5) Mathilde drives at an average speed of 16 km/h to get from point A to point B. How long will it take her to get from point A to point B? Give your answer in minutes and seconds.

Exercise 7: (3 points)

We wish to represent this frieze composed of six rectangles using the scratch programming software.

In this software, one step corresponds to one pixel. The drawing below is not to scale. 1) On your copy, give the value of the letters A, B and C so that the “pattern” block can draw a rectangle of width 30 pixels and length 150 pixels.

2) The script on the right must allow you to obtain the frieze, it uses the “motif” block.

The total length of the frieze is 220 pixels. On your copy, calculate the value that the letter D must take in order for the script to answer the problem posed. Justify the answer.

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