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Subject of mathematics of the brevet des collèges

NEW CALEDONIA

December 9, 2016

Duration: 2 hours

Calculator allowed

Quality of writing, spelling and editing count for 4 points.

**Exercise 1: Multiple choice questionnaire 5 points**

This exercise is a multiple choice questionnaire (MCQ).

For each question, only one of the three proposed answers is correct.

On the copy, indicate the number of the question and the chosen answer.

No justification is required.

No points will be deducted for incorrect answers.

**Exercise 2: Video game 4 points**

In a video game, in order to gain experience points and develop your character, you must participate in battles.

Each victory earns a fixed number of points.

It is the same for every defeat.

Gabriel has already accumulated 1,350 points with 21 wins and 9 losses.

His brother Nathaniel had 12 wins to 18 losses and totaled 900 points.

How many points are earned in this game if you win? In case of defeat?

We will write the calculations that allow us to justify the answers.

**Exercise 3: Amedee Lighthouse 3 points**

During the vacations, Robin went to visit the Amédée lighthouse.

During a nap on the beach he noticed that the top of a beach umbrella was in perfect alignment with the top of the lighthouse, so Robin took some measurements and decided to make a diagram of the situation in the sand to find an estimate of the height of the lighthouse.

The points B, J and R are aligned.

(SB)and(BR)are perpendicular.

(PJ)and(BR)are perpendicular.

What height, rounded to the nearest meter, will he find with the help of his plan? Justify the answer.

**Exercise 4: Short walk 3 points**

Thomas and Hugo decide to go walking together. Thomas takes 0.7 meter steps at a rate of 5 steps every 3 seconds, while Hugo takes 0.6 meter steps at a rate of 7 steps in 4 seconds.

Which one moves faster? Explain the answer.

*In this exercise, any trace of research, even incomplete or unsuccessful, will be taken into account in the evaluation.*

**Exercise 5: Programming 3 points**

Here are two calculation programs:

1. We choose -8 as the starting number.

(a)Prove by calculation that the result obtained with program A is 12.

(b)Calculate the final result with program B.

2. Sandro says: “If we choose the same starting number for both programs, the result of program A

is always higher than that of program B.”

Prove him wrong.

3. Anne says: “With program B I found a result equal to my starting number”.

What was its starting number?

**Exercise 6: 3-point candlestick**

For their wedding, a couple wants to decorate the room with candle holders decorated with gold and silver candles.

The bride and groom ordered on a website an end of stock and received 180 gold candles and 108 silver candles.

They want to prepare as many identical candlesticks as possible without waste. That is:

– The number of golden candles is the same in all the candle holders.

– The number of silver candles is also the same in all candlesticks.

– All candles must be used.

How many candlesticks should they buy? Justify the answer.

How many candles of each color will be on each candlestick?

In this exercise, any trace of research, even if incomplete or unsuccessful, will be taken into account in the evaluation.

**Exercise 7: Pizza delivery 8 points**

Three young friends decide to work at night after school to earn some money. Since they have a driver’s license, they are oriented towards pizza delivery. They were able to get jobs at three different pizza places.

– David will receive a fixed salary of 70 000 F per month.

– Guillaume will have a monthly salary composed of a fixed part of 50,000 F to which 100 F will be added for each delivery made.

– Angelo will be paid each month 200 F per delivery.

1. if during a month the pizzerias receive very few orders, who should earn the most money?

For this question, use the table below.

(a)Complete the table.

(b)In one month, how many deliveries does William have to make to earn the same salary as David?

In this question, x is the number of deliveries made in a month. f , g and h are three defined functions

by:

– f (x) = 70000

– g(x) = 200x

– h(x) = 100x+50000

(a)Associate each of these functions with one of the three salaries.

(b) In the reference frame in Appendix 2, write the name of the function corresponding to each line.

(c)Using the graph below, determine the number of deliveries at which Angelo will be the one receiving the largest monthly salary.

**Exercise 8:At the table 3 points**

Alexis has a square table of 2 meters on each side. At the store, the only tablecloth he likes is a round tablecloth of 2.5 meters in diameter.

Will this tablecloth be big enough to cover the entire table (obviously, Alexis will not cut the tablecloth)?

Justify the answer.

*In this exercise, any trace of research, even if incomplete or unsuccessful, will be taken into account in the evaluation.*

**Exercise 9: Treasure hunt 4 points**

We want to organize a treasure hunt throughout New Caledonia.

Beacons will be hidden in each of the three provinces of New Caledonia.

Some of them will contain a key.

Here is their distribution:

– In the Southern Province, 7 beacons are located, 4 of which have a key,

– In the Northern Province there are 5 beacons, 3 of which have a key,

– In the Province of the Islands are located 3 beacons, 2 of which have a key.

The Notous team discovered a beacon in the Northern Province.

What is the probability that a key is inside?

2.the Notous team has found a key in this first beacon. They discover a second beacon in the Northern Province. What is the probability that it contains a key?

The Cagous team discovered two beacons in the Province of the Islands.

What is the probability that this team has found at least one key?

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