Bac-D 2015
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ASSOCIATION OF MATHEMATICS TEACHERS OF BENIN
Registeredundernumber96-59MISAT/DC/DAI/SAAP-Assoc.ofMay13,1996
May 23, 2015
MATH TEST, 2015 Edition
BAC D 4 hours
Context: Construction of a hotel
Pierre, a great economic operator, decides to build a five-star hotel whose
The creation of the model is entrusted to engineer Gérard. He equips the space.
of a direct orthonormal frame On his drawing, is a point of
ceiling plan belonging to the set points checking:
Gérard measures the height h of the building by the distance from point H to the plane (P) of the ground.
defined by the points and Moreover, the reception (person
physical) will be positioned at C at a distance of d Gérard also has the
mission to submit a development project for one of the access roads to the hotel,
within which the construction of a large VIP nightclub is planned.
Bob, son of Gérard, a senior in class D, is made aware of these projects.
the execution is entrusted to his father. He seeks to know more.
Task: You are invited to help Bob find answers to his concerns.
solving the following three problems.
Problem 1
1- Determine a Cartesian equation of the plane (P).
2- a) Verify that H is a point of .
b) Justify that is a line for which you will specify a reference point.
3- a) Determine the coordinates of point D, the orthogonal projection of H onto (P).
b) Show that point D belongs to the line .
c) Deduce the position of in relation to (P).
d) Determine the height h of the building.
4- Calculate the distance d.
1
Problem 2
The VIP nightclub will be illuminated by a large luminous object of peaks.
and triangular in shape. In the complex plane, the respective affixes and
points and are the solutions in the equation (E):
, where .
Furthermore, the point is the antecedent of the point by the transformation S that
complex writing is: .
5 - Solve the equation (E) and specify the coordinate of each point and.
6- Demonstrate that : .
7- Determine the nature and characteristic elements of S.
8- a) Linearize the expressions: and .
b) Deduce from this that: .
c) Determine the affix of the point.
d) Deduce the exact shape of the luminous object with vertices .
Problem 3
In the plane equipped with an orthonormal system the path to be developed is
similar to a portion of the curve (C) of the numeric function defined by:
9- a) Study the variation of the function g defined by:
b) Determine the extrema of g and deduce the sign of g based on the values of a.
real variable.
10- a) Determine the domain of definition of.
b) Show that it is continuous at 0 and -1.
c) (i) - Assuming that the limited development of order 3 of the function
the natural exponential near 0 is such that
2
with lim(x) 0 determine the third order Taylor expansion of the function
x 0
inthevicinityof0.
(ii) - Study the differentiability at 0 and give a geometric interpretation of it.
result.
d) (i) Study the differentiability at -1
(ii) Provide a geometric interpretation of the results.
11- a) Determine the set of differentiability E of.
b) Demonstrate that: .
c) Study the sign of .
d) Complete the variations def.
We put and the function:
a) Determine .
b) Show that it admits a reciprocal bijection .
c) Determine the set of differentiability E’ of .
d) Study the differentiability of in 0 and specify the possible half-tangent at ,
representative curve of , at the last point of abscissa 0.
e) Prove that the point F(1 ; 0) belongs to and determines an equation of the
tangent to in F.
13- a) Demonstrate that on a : .
b) Deduce the infinite branches of the curve (C).
14- Build the curves (C) and in the same reference frame.
- FIN -