**bac S of maths**Report an error / Note?

G E N E R A L B A C C A L A U R E A T

SESSION 2019

MATHEMATICS

Series : S

TEST TIME: 4 hours. – COEFFICIENT : 7

**Exercise 1: (6 points)**

**Common to all candidates**

**Part A**

Let a and b be real numbers. Consider a function defined on [0 ; +∞[ by

The curve Cf representing the function in an orthogonal reference frame is given below.

The curve Cf passes through the point A(0; 0.5).

The tangent to the curve Cf at point A passes through point B(10; 1).

1. Justify that a = 1.

2. We admit that the function f is derivable on [0 ; +∞[ and we note f ‘ its function

derived.

3. Using the data in the statement, determine b.

**Part B**

The proportion of individuals who own a certain type of equipment in a

population is modeled by the function p defined on [0 ; +∞[ by

The actual represents the time elapsed, in years, since January 1, 2000.

The number p(x) models the proportion of individuals equipped after x years.

Thus, for this model, p(0) is the proportion of individuals equipped on January 1, 2000

and p(3.5) is the proportion of individuals equipped in mid-2003.

1. What is, for this model, the proportion of individuals equipped on January 1st

2010 ? We will give a value rounded to the hundredth.

2.

a. Determine the direction of variation of the function p on [0 ; +∞[.

b. Compute the limit of the function p in +∞.

c. Interpret this limit in the context of the exercise.

3. It is considered that when the proportion of individuals equipped exceeds 95%, the

market is saturated.

Determine, by explaining the process, the year in which this happens.

product.

4. The average proportion of individuals equipped between 2008 and 2010 is defined by

a.

b. Deduce a primitive of the function p on [0; +∞[.

c. Determine the exact value of m and its rounding to the hundredth.

**Exercise 2: (5 points)**

**Common to all candidates**

*The two parts of this exercise are independent.*

Alex and Elisa, two drone pilots, are training on a field consisting of a flat area that is bordered by an obstacle.

We consider an orthonormal reference frame , one unit corresponding to ten meters. To model the relief of the area, we define six points O, P, Q, T, U and V by

their coordinates in this frame:

O(0; 0; 0), P(0; 10; 0), Q(0; 11; 1), T(10; 11; 1), U(10; 10; 0) and V(10; 0; 0)

The flat part is delimited by the OPUV rectangle and the obstacle by the

rectangle PQTU.

The two drones are assimilated to two points and are assumed to follow rectilinear trajectories:

– Alex’s drone follows the trajectory carried by the line (AB) with A(2 ; 4 ; 0,25)

and B(2; 6; 0.75);

– the drone of Elisa follows the trajectory carried by the line (CD) with C(4; 6; 0.25)

and D(2; 6; 0.25).

**Part A: Study of Alex’s drone trajectory**

1. Determine a parametric representation of the line (AB).

2.

a. Justify that the vector is a normal vector to the plane (PQU).

b. Deduce a Cartesian equation of the plane (PQU).

3. Prove that the line (AB) and the plane (PQU) are secant at the point I of coordinates .

4. Explain why, by following this trajectory, Alex’s drone does not encounter the obstacle.

**Part B: Minimum distance between the two paths**

To avoid a collision between their two aircraft, Alex and Elisa impose a

minimum distance of 4 meters between the trajectories of their drones.

The objective of this part is to verify that this instruction is respected.

For this, consider a point M on the line (AB) and a point N on the line (CD).

Then there are two real numbers and such that and .

We are therefore interested in the distance MN.

1. Show that the coordinates of the vector are (2 – 2b; 2 – 2a; -0.5a).

2. We admit that the lines (AB) and (CD) are not coplanar. We also admit that the distance MN is minimal when the line (MN) is perpendicular to both the line (AB) and the line (CD).

Then show that the distance MN is minimal when and b = 1.

3. Deduce the minimum value of the distance MN then conclude.

**Exercise 3: (4 points)**

**Common to all candidates**

For each of the following four statements, indicate whether it is true or false, justifying the answer.

*One point is awarded for each correct answer that is correctly justified. A no answer*

*justified does not earn any points. A lack of response is not penalized.*

The complex plane has a direct orthonormal reference point .

Consider the complex number and the points S and T with affixes and respectively.

1.** Assertion 1:**

The number can be written .

2. **Assertion 2:**

For any natural number , is a real number.

3. **Assertion 3:**

The points O, S and T are aligned.

4. **Assertion 4:**

For any non-zero natural number , .

**Exercise 4: (5 points)**

**Candidates who have not followed the speciality course**

*The three parts of this exercise are independent.*

**Part A**

One evening, a television station broadcast a game. This chain has

then proposed a program of analysis of this match.

The following information is available:

– 56% of viewers watched the game;

– a quarter of the viewers who watched the game also watched the show;

– 16.2% of viewers watched the program.

A viewer is randomly interviewed. We note the events:

– M: “the viewer watched the game”;

– E: “the viewer watched the program”.

We note the probability that a viewer watched the program knowing that he did not watch the game.

1. Construct a weighted tree illustrating the situation.

2. Determine the probability of .

3. a. Check that .

b. Deduce the value of .

4. The viewer interviewed did not watch the program.

What is the probability, rounded to , that he watched the game?

**Part B**

To determine the audience of television channels, a polling institute collects information from thousands of French households by means of individual boxes.

This institute decides to model the time spent, in hours, by a viewer in front of the television on the evening of the match, by a random variable T following the normal distribution of expectation and standard deviation .

1. What is the probability, rounded to , that a viewer spent between one and two hours in front of his or her television on the night of the game?

2. Determine the rounding to of the real such that .

Interpret the result.

**Part C**

The life of an individual box, expressed in years, is modeled by a random variable noted S which follows an exponential law of parameter strictly positive.

Recall that the probability density of S is the function f defined on [0; +∞[ by

The survey institute found that a quarter of the cases have a lifespan of between one and two years.

The factory that manufactures the cases claims that their average life span is over three years.

Is the plant’s statement correct? The answer must be justified.

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