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MATHEMATICS MOCK PATENT – 24/01/2017

The subject consists of seven independent exercises. They can be treated in any order.

The calculator is allowed. The clarity of the writing and the care of the copy will be taken into account.

**Exercise 1 (3 points)**

Two numbers are twin primes if they are prime and if their difference is equal to 2.

Here are some pairs of twin primes: (3; 5), (5; 7), (11; 13). (17; 19) and (29; 31)

1) What is the next pair of twin primes?

2) Is the pair (429; 431) a pair of twin primes? Justify

**Exercise 2 (6 points)**

Two observation posts are placed on the coast (noted A and B on the map). They are 1800 m apart. At these stations, observers follow the course of a V sailboat.

At station A, we measure = 35

At station B, we measure = 55

Calculate the distances AV and BV, rounded to the nearest meter, that separate the sailboat from each observation post.

**Exercise 3 (7 points)**

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

The auditorium has the following shape:

The seats are arranged in four zones: two quarter discs and two trapezoids, separated by aisles two meters wide.

An average of 1.8 seats per square meter can be placed in the seating area.

Calculate the number of seats available in this theater.

**Reminders:**

**Exercise 4 (6 points)**

Jean-Baptiste, a ninth grader, is walking on the island of Manhattan in New York. He was asked to verify that 14th and 42nd Streets are indeed parallel, and that 6th Avenue is perpendicular to these two streets.

Jean-Baptiste leaves point C, goes up 6th Avenue to Bryant Park, turns left to Times Square, then down Broadway to Union Square Park.

John the Baptist measured the following lengths:

CE = 1400 m, EB = 560 m, BT = 192 m,

TE = 592 m and EU = 1480 m

1a) Show that the lines (BT) and (CU) are parallel.

1b) Deduce the distance from Jean-Baptiste’s starting point C to Union Square Park.

2) Show that 42nd Street and 6th Avenue form a right angle.

**Exercise 5 ( 3 points)**

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

Pat the pirate found a crate full of gold coins.

When he groups the pieces by 2, one remains.

When he groups the pieces by 3, 2 remain.

When he groups the pieces by 4, 3 remain.

When he groups the pieces by 5, 4 remain.

The case contains less than 100 pieces. How many gold coins did Pat the pirate find?

**Exercise 6 (6 points)**

For each of the following statements, several suggested answers are given. Only one is accurate. Give the correct answer on the subject. No justification is expected.

**Exercise 7 (5 points)**

1) How do you run program 1? What about program 2?

2) Tom thinks that these two programs give the same result whatever the number entered at the beginning.

Is he right? Justify your answer.

3) Consider the function f which associates the result obtained after the execution of program 2 to each number entered.

a) Give the algebraic expression of this function f.

b) What is the image of 5 by f?

c) Give an antecedent of 104.

d) Anatole plans to draw the graph of f.

Paul tells him “the point with coordinates (3; 13) belongs to this curve”. Anatole disagrees. Who is right?

View the answers for the 2017 maths brevet blanc

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