June 2018
Written test - Mathematics
2 hours
The use of documents and calculators is forbidden.
The quality of writing is an important element for the evaluation.
Exercise 1
Q. 1.1Justify the existence of
I=
!
R
e−x2dx.
Q. 1.2Compute the value of the integral I.
Exercise 2
Let P be a polynomial with real coefficients, P∈R[X]. Assume all its roots are real. Letα∈R. We want to prove that all the roots of P′+αP are real too.
Q. 2.1Assume at first that P has only simple roots. Show that P′+αP, where P′the derivative polynomial of P, has only real simple roots, as well.
Q. 2.2We do not assume that the roots of P are all simple anylonger. Show that the roots of P′+αP are all real.
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Exercise 3
Q. 3.1Determine the convergence radius R of the series"nn n!zn.
Q. 3.2Study the convergence of the series on the circle of center 0 and radius R.
Exercise 4
Q. 4.1Study the linear independence of the following families:
1. fn:x$→(sin(x))n,n∈N 2. fa:x$→|x−a|,a∈R 3. fα:x$→exp(αx),α∈C
4. (x$→(sin(x))n),(x$→(cos(x))n),n∈N.
Q. 4.2Letn≥1. Let fi:C→C,i∈{1,...,n} be a family of functions. Set A :Cn−→M\(C)
(x1,...,xn)$−→(fi(xj))1≤i,j≤n
Find a necessary and sufficient condition on the function det(A) so that the family (fi)1≤i≤nis linearly independent.
Exercise 5
Let X be a real random variable following Poisson’s law with parameterλ. Let Y be defined as
Y=
⎧⎪
⎪⎨
⎪⎪
⎩ X
2 if X is even 0 if X is odd Q. 5.1Determine the law of Y.
Q. 5.2Compute the expectation of Y.
Q. 5.3Compute the variance of Y.
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