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**A blank maths 2017 patent topic with probability, functions, programming with scratch.statistics and a qcm in 3rd.**

**WHITE PATENT OF MATHEMATICS 2017**

**Duration of the test: 2 hours**

The quality of the writing and presentation will be taken into account (5 points).

The use of a calculator is allowed.

The exercises can be done in any order.

All answers must be justified unless otherwise indicated.

For each question, if the work is not completed, leave a trace of the research; it will be taken into account in the marking.

Exercises 2 and 4 deal with the theme “WIND ENERGY”.

**Exercise 1 (4 points):**

Charlotte has between 400 and 450 pounds. She decides to sell them on the internet to buy others. She observes that she can group all of her books in packs of 3. She finds that she can also do it in packs of 5 and also in packs of 7.

Exactly how many books does Charlotte own?

**Exercise 2 (6.5 points):**

Questions 1 and 2 can be answered independently of each other.

The diagram below is that of a wind turbine whose maximum effective height, noted H, is 90 m and whose hub center is placed at 60 m from the ground (distance noted h).

The theoretical maximum power of a wind turbine is calculated, in watts, by the formula :

where S is the surface in m² of the disk swept by the blades and v is the wind speed in m/s.

1. We are interested here in the full-size wind turbine:

a. Show that the area S of the disk swept by the blades is equal to .

b. Deduce the theoretical maximum power, to the nearest kilowatt, for a speed of 20 m/s.

2. We now want to make a scale model of the wind turbine :

a. Calculate the length of a blade of the model.

b. Calculate the theoretical maximum power of the model, to the nearest watt, for a speed of 10 m/s.

**Exercise 3 (4 points):**

In the first jar we put 5 dark chocolate candies and 3 white chocolate candies. In the second pot we put 7 dark chocolate candies and 4 white chocolate candies. The candies are indistinguishable to the touch.

If you choose a candy from one of the two jars without looking, which one is more likely to be white chocolate? Justify the answer.

**Exercise 4 (4.5 points):**

The graph below shows the power (expressed in kW) delivered by a wind turbine according to the wind speed (expressed in m/s).

Answer the following questions with the accuracy allowed by the graph.

1. For what wind speeds does the wind turbine produce electricity?

2. What is the maximum power delivered by the wind turbine?

3. The wind speed increases to 100 km/h.

Explain in a sentence what is happening.

**Exercise 5 (5 points):**

Nina is in Les Estables for a “ski trip” with her class. It is at the bottom of the TELESKI “Chalet 2” where nobody is waiting.

It is 4:50 pm and his teacher has given an appointment at the foot of the slopes at 5 pm precisely for the return. Nina’s average speed is 15 km/h.

Does she have time for a final run if she takes one of these two tracks?

**Exercise 6 (7 points):**

Here are two programs made with scratch.

1. Show that the sprite will say 15 in both cases if the starting question in each of the two programs is answered 6.

2. What number(s) will it say if the starting question for each of the two programs is answered (-3)?

3. For any number chosen at the beginning, show that the two programs give identical results.

** **

**Exercise 7 (7 points):**

An electricity supplier wants to change its pricing. The amount of a customer’s annual bill is the sum of the price of the subscription chosen and the electricity consumption. The supplier wants to lower the price per kWh by 2% and increase the price of subscriptions by 3% for the year 2018 compared to 2017.

1. The supplier uses the spreadsheet below to calculate the subscription and kWh prices for the year 2018.

a. To calculate the subscription amounts for 2018, what formula can he enter in cell D2 before copying it to column D?

b. To calculate the price per kWh for 2018, what formula can he enter in cell E2 before copying it to column E?

2. Calculate the range as well as the average value of the annual subscription including tax in 2017.

3. Paul has a 6 kVA meter and consumed 5,361 kWh in 2017.

a. What is the amount, rounded to the penny, of his electric bill for 2017?

b. If her consumption remains the same, what would be the amount, rounded to the penny, of her electric bill for the year 2018?

**Exercise 8 (7 points):**

Questions 1-6 are multiple choice.

For each, one or more answers are correct.

Circle the answer(s) without justification.

The answer to question 7 must be justified in the box provided.

Consult the online answer key

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