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**Exercise 1: (13 points)**

We consider a game composed of a rotating board and a ball.

Shown below, this board has 13 squares numbered from 0 to 12.

The ball is thrown on the board, and ends up randomly on a numbered square.

The ball has the same probability of stopping on each square.

1. What is the probability that the ball stops on the square numbered 8?

2. What is the probability that the number of the square on which the ball stops is an odd number?

3. What is the probability that the number of the square on which the ball stops is a number

first?

4. On the last two throws, the ball came to rest on the 9 square each time.

Is it now more likely that the ball will end up on the 9 square than on the

the box numbered 7?

Argue using a probability calculation.

**Exercise 2: (9 points)**

The paving shown in figure 1 is made from a pattern called pied-de-coq which is present

on many fabrics used in the manufacture of clothing.

The houndstooth pattern is represented by the polygon below right (figure 2) which can be realized

using a regular grid.

1. In figure 1, what type of geometric transformation is used to obtain pattern 2 from the

reason 1?

2. In this question, we consider that: AB = 1 cm (figure 2).

Determine the area of a houndstooth pattern.

3. Mary says “if I divide the lengths of a pattern by 2, its area will also be divided by 2”.

Is she right? Explain why.

**Exercise 3: (9 points)**

This exercise is a M.C.Q. M.C.Q. (Multiple Choice Questionnaire).

For each question, four answers are proposed and only one is correct. A response

false or absent does not remove a point.

For each of the three questions, write the question number and the letter corresponding to the correct answer on your copy.

**Exercise 4: (18 points)**

1. Corinne chooses the number 1 and applies the program A.

Explain in detail that the result of the calculation program is 4.

2. Tidjane chooses the number -5 and applies program B. What result does he obtain?

3. Lina wants to group the result of each program using a spreadsheet. It creates the

spreadsheet below. What formula, copied then to the right in cells C3 to H3,

entered in cell B3?

4. Zoë tries to find a starting number for which the two calculation programs give

the same result. To do this, it calls x the number chosen at the beginning and expresses the result of

each calculation program as a function of x.

a. Show that the result of the program A as a function of x can be written in expanded form

and reduced: x² -6x +9,

b. Write the result of program B.

c. Is there a starting number for which both programs give the same result

?

If so, which one?

**Exercise 5: (20 points)**

In the whole exercise the unit of length is the mm.

A dart is thrown at a square plate with a circular target on it (shown in gray in the figure). If the tip of the dart is on the edge of the target, the target is not reached.

We consider this experiment to be random and we are interested in the probability that the dart hits the target.

– The length of the side of the square plate is 200.

– The radius of the target is 100.

– The dart is represented by the point F of coordinates (x; y) where x and y are random numbers between -100 and 100.

1. In the above example, the dart F is located at the point of coordinates (72; 54).

Show that the distance OF, between the dart and the origin of the reference frame is 90.

2. In general, what number must not exceed the distance OF for the dart to

reach the target?

3. We realize a program which simulates several times the throw of this dart on the plate

square and counts the number of throws hitting the target. The programmer has created three

named variables: **OF square, distance and score**.

a. When this program is run, how many throws are simulated?

b. What is the role of the score variable?

c. Complete and copy on the copy only lines 5, 6 and 7 of the program so that it

works properly.

d. After one execution of the program, the score variable is equal to 102.

How often was the target achieved in this simulation?

Express the result as an irreducible fraction.

4. We admit that the probability of reaching the target is equal to the quotient: area of the target divided by

by area of the square plate.

Give an approximate value of this probability to the nearest hundredth.

**Exercise 6: (15 points)**

Chris is racing a mountain bike. The graph below represents his heart rate (in beats per minute) as a function of time during the race.

1. What is Chris’ heart rate at the start of his run?

2. What is the maximum heart rate that Chris reaches during his run?

3. Chris left at 9:33 a.m. from home and finished his run at 10:26 a.m.

What was the duration, in minutes, of his run?

4. Chris ran 11 km in this race.

Show that its average speed is about 12.5 km/h.

5. MHR (Maximum Heart Rate) is the maximum frequency that can be supported by the body.

the organization. Chris’ is FCM= 190 beats per minute.

While researching on specialized websites, he found the following table:

Estimate the length of time that Chris has been making a sustained effort during his

race.

**Exercise 7: (16 points)**

The figure below is not to scale.

We consider above a triangle ABC rectangle in A such that and AB = 7 cm. H is the foot

of the height from A.

1. Draw the figure in true size on the copy. Leave the construction lines visible on the

copy.

2. Show that AH = 3.5 cm.

3. Show that triangles ABC and HAC are similar.

4. Determine the reduction coefficient to go from triangle ABC to triangle HAC.

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