Download Exercise 1. Direct Similitude - ISA-EMT

ENTRANCE EXAMINATION AT THE SCHOOL OF PETROLEUM
ENGINEERS OF ISA - EMT
Paper: MATHEMATICS
Duration : 3 Hours
/20mks
Exercise 1. Direct Similitude
The plan is provided with an orthonormal reference frame (𝑂, ⃗⃗𝑖 , 𝑗 ). Let’s consider the equation
(E): 𝑧 3 − 7𝑖𝑧 2 − 15𝑧 + 25𝑖 = 0
1) a) Show that 5𝑖 is solution of (E)
b) Solve the equation (E) in C (complex numbers set)
2) A, B and C are the points of respective affix 2 + 𝑖 ; 5𝑖 𝑎𝑛𝑑 − 2 + 𝑖
The line (D) of equation y = 2 cut the line (AB) in K and the line (OA) in L.
() and (') are the circles which delimit the triangles OAB and ALK.
(S) is the direct similitude such as S(B) = O and S(K) = L; is the center of (S)
a) Show that () and  (') and that A.
b) Give the complex expression of S, and then deduce the co-ordinates of 
Exercise 2. Differential Equation and function
I-
Let’s consider the following differential equations :
1
1
(E) : 𝑦 ′ + 𝑦 = 0 et (E’) : 𝑦 ′ + 𝑦 = − 𝑒 −2𝑥 − 2
2
1) Show that there is a function ℎ defined by ℎ(𝑥) = 𝑝𝑒
1
2
− 𝑥
+ 𝑞 which is solution of (E ') where
p and q are real numbers to be determine.
2) Show that a function 𝑓 = 𝑔 + ℎ is a solution of (E ') if and only if 𝑔 is a solution of (E).
3) Solve equation (E), and then deduce the solutions of the equation (E ')
BP : 1215 Douala
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𝑥
II-
−
Let’s consider the numerical function defined by: 𝑓(𝑥) = 𝑒 −𝑥 − 𝑒 2 − 2
(𝐶𝑓 ) its representative curve in an orthonormal reference frame (𝑂, 𝑖⃗⃗ , 𝑗 ). The unit on the axes is
equal to 1 cm.
1) Show that the function 𝑓 satisfy the differential equation (E ') mentioned above.
2) Study the variations of 𝑓, then draw its variation table.
3) a) Study the infinite branches of (𝐶𝑓 )
b) Plot the curve (𝐶𝑓 )
𝛼
4) a) Calculate the real 𝐴(𝛼) = ∫𝑙𝑛4[−2 − 𝑓(𝑥)]𝑑𝑥 where 𝛼 is a real number and 𝛼 > 4.
b) Calculate lim 𝐴(𝛼); then give a geometrical interpretation of the result obtained.
𝛼→+∞
Exercise 3. Probability
A dice numbered from 1 to 6 is thrown three times consecutively.
 Let’s consider a the first number thrown
 b the second number thrown
 c the third number thrown
Determine the probability of the following events:
A. The numbers a, b and c are following an arithmetical progression
B. The numbers a, b and c are following a geometrical progression
C. Differential equation 𝑦 ′′ + 𝑎𝑦 ′ + 𝑏𝑦 = 0 have solutions defined by 𝑦 = (𝐴𝑥 + 𝐵)𝑒 𝑟𝑥
Exercise 4. Numerical progression
1
Let’s consider the progression (𝐼𝑛 ) defined by: 𝐼𝑛 = ∫0 𝑥 𝑛 𝑒 𝑥 𝑑𝑥 (𝑛 is a natural integer different
from zero)
1) Calculate 𝐼𝑛
2) Show that, 𝐼𝑛+1 = 𝑒 − (𝑛 + 1)𝐼𝑛
3) Then deduce the values of 𝐼2 and 𝐼3
The Direction of Exams
September 2014
BP : 1215 Douala
Situé à Makèpe à 500 m de l’Hôpital Général
Tél : 33 78 85 87 / 74 18 94 40
Site web : www.a-eim.com
E-mail : [email protected]