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NATIONAL PATENT DIPLOMA

SESSION 2021

MATHEMATICS

Foreign centers

Exercise 1: (24 points)

In this exercise, each question is independent. No justification is required.

1) Decompose 360 into a product of prime factors.

2) From the triangle BEJ, rectangle isosceles in J, we obtained by tiling the figure below.

a) What is the image of triangle BEJ by the symmetry of axis (BD)?

b) What is the image of triangle AMH by the translation that transforms point E into B?

c) By what transformation do we go from the AIH triangle to the AMD triangle?

3) Calculate by detailing the steps:

The result will be given as an irreducible fraction.

4) For this question, the only correct answer should be indicated on the copy.

Knowing that the diameter of the Moon is about 3 474 km, the value that best approximates its

volume is :

5) Consider a triangle RST rectangular in S.

Complete the table given in the APPENDIX to be returned with the copy.

The value of the angles will be rounded to the unit.

Exercise 2: (21 points)

Part 1

In this first game, a well-balanced six-sided die numbered 1 to 6 is rolled,

then we note the number of the face above.

1) Give without justification the possible outcomes.

2) What is the probability of event A: “We get2 “?

3) What is the probability of event B: “An odd number is obtained”?

Part 2

In this second part, two well-balanced six-sided dice are rolled simultaneously, one

red and one green.

The sum of the numbers on each die is called the “score”.

1) What is the probability of event C: “the score is 13”? What do we call

such an event?

2) In the double entry table given in the APPENDIX, fill in each box with the

sum of the numbers obtained on each die.

a) Complete, without justification, the table given in the APPENDIX.

b) Give the list of possible scores.

3)

a) Determine the probability of the event D: “the score is 10”.

b) Determine the probability of the event E: “the score is a multiple of 4”.

c) Show that the score obtained has as much chance of being a prime number as a

number strictly greater than 7.

Exercise 3: (16 points)

A teacher proposes three calculation programs to his students, two of which are realized

with a programming software.

1)

a) Show that if we choose 1 as the starting number then the program A displays

for 2 seconds “We get 3″.

b) Show that if we choose 2 as the starting number then program B displays

for 2 seconds ” You get -15 “.

2) Let x be the starting number, what literal expression do we obtain at the end of the execution

of the C program?

3) a student states that with one of the three programs you always get triple the

selected number. Is he right?

4)

a) Solve the equation (x + 3)(x – 5) = O.

b) For what starting values does program B display “We get O”?

5) For what starting value(s) does the C program display the same result as the

program A ?

Exercise 4: (19 points)

Aurélie is cycling in England at the Hardknott Pass.

It started from an altitude of 251 meters and will arrive at the summit an altitude of 393

meters.

On the diagram below, which is not full-scale, the starting point is represented

through point A and the vertex through point E. Aurélie is currently at point D.

The lines (AB) and (DB) are perpendicular.

The lines (AC) and (CE) are perpendicular.

The points A, D and E are aligned. The points A, B and C are aligned.

AD = 51.25 m and DB = 11.25 m.

1) Justify that the height difference that Aurélie will have covered, i.e. the height EC, is equal to

142 m.

2)

a) Prove that the lines (DB) and (EC) are parallel.

b) Show that the distance Aurelie still has to travel, i.e. the length DE,

is about 596 m.

3) We will use for the length DE the value 596 m.

Knowing that Aurélie drives an average speed of 8 km/h, if she leaves at 9:55 a.m. from

point D, what time will it arrive at point E? Round up the minute.

Exercise 5: (20 points)

A ski resort offers its customers three packages for the winter season:

- Formula A : you pay 36,50€ per day of skiing.
- Formula B : you pay 90 € for a ” SkiPlus ” subscription for the season, then 18,50 €.

per day of skiing. - Formula C : you pay 448,50 € for a ” SkiTotal ” subscription which then allows a

free access to the resort during the whole season.

1) Marin wonders which formula to choose this winter.

He makes a table to calculate the amount to pay for each of the formulas according to the number of ski days.

Complete, without justification, the table provided in the APPENDIX to be returned with the copy.

2) In this question, x is the number of ski days.

Consider the three functions f, g and h defined by :

a) Which of these three functions represents a proportionality situation?

b) Associate, without justification, each of these functions with the corresponding formula A, B or C.

c) Calculate the number of ski days for which the amount paid with formulas A and B is identical.

3) The three functions have been graphed in the graph below.

Without justification and with the help of the graph:

a) Associate each graphical representation (d1), (d2) and (d3) with the corresponding function f, g or h.

b) Determine the maximum number of days Marin can ski

with a budget of 320 €, choosing the most advantageous formula.

c) Determine the number of days of skiing that it is advantageous to choose the

formula C.

**APPENDIX To be returned with the copy.**

Exercise 1 question 5):

Exercise 2 Part 2 question 2) a):

Exercise 5 question 1):

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