MT 27 - Autumn 2015
Final Exam - January, the 11th 2016
Exercise 1 Find the determinant for then×nmatrix where n∈N, n≥2,
1 n n · · · n n 2 n · · · n n n 3 · · · n ... ... ... . .. ...
n n n · · · n .
Exercise 2 Let the matrixA=
0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
.
1. Check that A is invertible.
2. Calculate A2 and deduceA−1 and A2015. 3. What are the eigenvalues of A ?
4. Explain why Ais diagonalizable.
5. Find the similarity transformation matrix P and its inverse to diagonalize A.
Exercise 3 We consider the curve parametrized by
{ x(t) = cos3t y(t) = sin3t.
1. Determine the optimal domain of study.
2. Find the stationnary points in the domain of study and their characterization.
3. Study the variations of the curve and plot it together with the tangent lines at some points.
4. Calculate the length of the curve.
5. Calculate the area of the plan part inner to the curve.
Exercise 4 Letf the 2π-periodic function defined by
∀x∈]−π, π], f(x) =|x|. 1. Calculate the Fourier coefficients of f.
2. Prove that the Fourier series is convergent and find its limit for anyx∈R. 3. Deduce the sum ∑∞
p=1 1 (2p+1)2.
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Exercise 5 ForE =R3[X], and the mapping φ:
{ E×E −→ R (P, Q) 7−→ φ(P, Q) =∫+∞
0 P(t)Q(t)e−t dt.
We admit that ∀k∈N, ∫+∞
0 tke−tdt=k!.
1. Verify that the mappingφ defines an inner product onE.
2. Show that the minimum
m= min
(a,b,c)∈R3
∫ +∞
0
(t3−a t2−b t−c)2
e−tdt
can be expressed as a distance between the polynomialt3 and a subspace F of E.
3. Find an orthonormal basis ofF. 4. Calculate this minimum.
Exercise 6 Apply the d’Alembert test to the series (a) ∑ n
2n (b) ∑sinn
n! (c) ∑ 1 n2 and find in which ones it is conclusive.
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