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The patent of maths 2023 with a blank subject for the DNB of the colleges in third (3ème).This subject contains 9 independent exercises.

The calculator is allowed. Duration of the test: 2 hours. Scoring out of 40.

The clarity of the presentation and the quality of the writing count for 4 points in the final grade.

All answers must be carefully justified unless otherwise indicated.

## Exercise 1: (5 pts)

1. Calculate the expressions A and B and give the result as a fraction

irreducible :

2. Calculate the expression C and give the scientific writing of the result:

3. Develop and reduce :

## Exercise 2: (5 pts)

The following calculation program is given:

**Program :**

**– Choose a number**

**– Add 2**

**– Calculate the square of this sum.**

**– Subtract 9 from the result.**

1. We choose 3 as the starting number.

Show that the result of the program is 16.

2. We choose (-1) as the starting number.

Calculate the result of the program.

3. We call x the starting number. Write the result of the calculation program as a function of x.

4. Factor this expression.

5. What number(s) must be chosen at the beginning for the result of the program to be zero?

## Exercise 3: (4 pts)

The following information concerns the salaries of men and women in the same company:

Salaries of each woman:

1 200 € ; 1 230 € ; 1 250 € ; 1 310 € ; 1 370 € ; 1 400 € ; 1 440 € ; 1 500 € ; 1 700 € ; 2 100 €

Men’s Wages:

Total number of employees: 20

Average: €1,769

Scope: 2 400 €.

Median: €2,000

Men’s salaries are all different.

1. Compare the average salary for men and women.

2. One person in the company is drawn at random. What is the probability that it is a woman?

3. The lowest salary in the company is €1,000. Which salary is the highest?

4. In this company, how many people earn more than 2 000 € ?

## Exercise 4: (4 pts)

In an urn there are 1 yellow ball (J), 4 blue balls (B) and 2 red balls (R), indistinguishable by touch. Two balls are drawn successively, with delivery (the first ball is put back before taking the second).

1. Construct the possibility tree describing the random experiment; place the probabilities on each branch.

2. What is the probability that the first ball is red and the second ball is blue?

3. We consider the event A: “the 2nd ball is yellow”.

a. What are all the possible outcomes?

b. Deduce the probability of the event A.

## Exercise 5: (4 pts)

1. Construct a triangle ABC such that: AB = 10.5 cm, AC = 6.3 cm and BC = 8.4cm.

Place the point E on the line (AB) such that: E [AB] and BE = 4.5 cm.

Draw the perpendicular to the line (BC) passing through point E. It intersects the line (BC) at F.

Place F.

2. Prove that triangle ABC is a rectangle.

3. Calculate the length BF.

4. Place the points M and N such that: M [AB], N [BC], BM = 5cm and BN = 4cm.

Are the lines (MN) and (AC) parallel ? Justify the answer.

## Exercise 6: (4 pts)

This exercise is a multiple choice questionnaire. For each question, only one answer is required

accurate. No points are deducted for no answer or a wrong answer.

Circle the selected answer. **
No justification is required
** .

## Exercise 7: (4 pts)

We used a spreadsheet to calculate the images of different values of x by a function f

and by another function g. A copy of the resulting screen is given below:

1. What is the image of -3 by f ?

2. Calculate f(7).

3. Give the expression for f (x).

4. We know that g (x) = x² + 4. A formula was entered in cell B3 and then copied to

To the right to complete the cell range C3: H3. What is this formula?

## Exercise 8: (3 pts)

1. Make the quotient irreducible using a PGCD calculation.

A merchant has 175 red Christmas ornaments and 126 blue ones. He chose to

make identical bags. He would like to have as many of them as possible by using

all the balls.

2. How many bags can he make?

3. How many balls of each color will be in each bag?

## Exercise 9: (3 pts)

Prior to the event, a map was given to the students participating in the race.

It is represented by the figure below:

We agree that:

– the lines (AE) and (BD) intersect at C.

– the lines (AB) and (DE) are parallel.

– ABC is a right triangle at A.

Calculate the actual length of the path ABCDE.

*If the work is not completed, leave a research trail. It will be taken in **taken into account in the scoring.*

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