The subject of the patent of maths 2017 in Pondicherry.
The event took place on Tuesday, May 2, 2017 in Pondicherry, India.
DIPLÔME NATIONAL DU BREVET
Duration of the test: 2 hours – 50 points
(including 5 points for the presentation of the copy and the use of the French language)
The subject consists of seven independent exercises.
The candidate may address them in any order.
All answers must be justified unless otherwise stated.
For each question, if the work is not completed, still leave a record of the research. It will be taken into account in the scoring.
COMMON THEME OF THE MATHEMATICS-SCIENCE TEST: ENERGY
Exercise 1 (5 points)
Consider the expression .
1. Develop E.
2. Factorize E and check that E=2F, where .
3. Determine all numbers x such that .
Exercise 2 (6 points)
A bag contains 20 balls each with the same probability of being drawn.
These 20 balls are numbered from 1 to 20. One ball is drawn at random from the bag.
All results will be given as irreducible fractions.
1. What is the probability of drawing the numbered ball 13?
2. What is the probability of drawing an even-numbered ball?
3. Is it more likely to get a ball with a number that is a multiple of 4 than a ball with a number that is a divisor of 4?
4. What is the probability of drawing a numbered ball that is a prime number?
Exercise 3 (7 points)
Consider the calculation program below in which x, Step 1,
Step 2 and Result are four variables.
1. a. Julie ran this program by choosing the number 5.
Check that what it says at the end is: “I finally get 20”.
b. What does the program say if Julie runs it by choosing the number 7 at the beginning?
2. Julie runs the program, and what it says at the end is: “I finally get 8”.
What number did Julie choose at the beginning?
3. If we call the number chosen at the beginning, write in function of the expression obtained at the end of the
program, then reduce this expression as much as possible.
4. Maxime uses the calculation program below:
Can we choose a number for which the result obtained by Maxime is the same as the one obtained by Julie?
Exercise 4 (7 points)
For his 32nd birthday, Denis bought an exercise bike to train during the winter.
Heart rate (HR) is the number of beats of the heart per minute.
1. Denis wants to estimate his heart rate: in fifteen seconds, he has counted 18 beats. What heart rate, expressed in beats per minute, does this correspond to?
2. His bike is equipped with a heart rate monitor that allows him to optimize his effort by recording all the pulsations of his heart in this heart rate monitor. At one point, the heart rate monitor measured an interval of 0.8 seconds between two pulses. Calculate the heart rate that will be displayed by the heart rate monitor.
3. After a workout, the heart rate monitor gave him the following information:
a. What is the range of recorded heart rates?
b. Denis did not time the duration of his training. How long did it take?
4. Denis wants to know his maximum recommended heart rate (MHR) so that he doesn’t exceed it and thus spare his heart. The CSCF of an individual depends on his age, expressed in years, it can be obtained through the following formula established by Astrand and Ryhming:
We note f (a) the CSCF as a function of age, so we have f(a)=220-a .
a. Verify that Denis’ BMI is equal to 188 beats/minute.
b. Compare Denis’ CSCF with the CSCF of a 15-year-old.
5. After some research, Denis finds another formula that allows him to obtain his CSCF more precisely. If denotes the age of an individual, its CSCF can be calculated using the Gellish formula:
We note g(a) the CSCF as a function of age, so we have
Denis uses a spreadsheet to compare the results obtained with the two formulas:
What formula should be inserted into cell C2 and copied down, in order to complete the column “FCMC g(a) (Gellish)”?
Exercise 5 (8 points)
A TeraWatt-hour is noted: 1 TWh.
Geothermal energy allows the production of electrical energy thanks to the heat of underground water.
The graph below represents the electricity production by different energy sources in
France in 2014.
1. a. Calculate the total electricity production in France in 2014.
b. Show that the proportion of electricity produced by “Other energies (including geothermal)” is approximately equal to 5.7%.
2. The following table shows the electricity production by the different energy sources, in France, in 2013 and 2014.
Alice and Tom discussed which energy source increased its electricity production the most. Tom thinks it is “Other energy (including geothermal)” and Alice thinks it is “Nuclear”. What is the reasoning behind each of them?
3. The geothermal power plant in Rittershoffen (Bas Rhin) was inaugurated on June 7, 2016. We dug a
well to collect hot water under pressure, at a depth of 2500 m, at a temperature of 170 degrees Celsius.
This well has the shape of a truncated cone.
The proportions are not respected.
The volume of a truncated cone is calculated using the following formula:
where h denotes the height of the truncated cone, R the radius of the large base and r the
radius of the small base.
a. Check that the volume of the well is approximately 225 .
b. The soil is compacted when it is in the ground. When extracted, it is not
more compact and its volume increases by 30%.
Calculate the final volume of soil to be stored after drilling the well.
Exercise 6 (7 points)
The slope of a road is obtained by calculating the quotient of the difference in elevation (i.e. the vertical displacement) by the corresponding horizontal displacement. A slope is expressed as a percentage.
In the example below, the slope of the road is :
Rank the following slopes in descending order,
i.e. from the steepest slope to the least steep.
Road going down from the castle of Adhemar, in Montélimar.
Section of a road coming down from the Grand Colombier pass (Ain).
Section of a road descending from the Alto de l’Angliru (Asturias region, Spain).
Exercise 7 (5 points)
Alban wants to apply for a job in a company. He/she must send in one envelope: 2 copies of his/her cover letter and 2 copies of his/her Curriculum Vitae (CV). Each copy is written on an A4 sheet.
1. He wishes to send his mail by priority mail. To determine the price of the stamp, he obtains the following postage rate table on the Internet:
Is the postage rate proportional to the weight of a letter?
2. In order to choose the right postage rate, it gathers the following information:
- Weight of its package of 50 envelopes: 175 g.
- Dimensions of an A4 sheet: 21 cm wide and 29.7 cm long.
- Grammage of an A4 sheet: 80 (grammage is the mass per m² of sheet).
- 1 m² = 104 cm².
Which postage rate should he choose?
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