Lycée Jean Perrin Classe de TSI Révisions d'oral (1) Sujet 1

TSI2
X
n>2
2xn
n2−1
a b 2
n2−1=a n−1+b n+ 1
x∈]−R, R[
A∈ Mn(R)AT=A
∀x∈Rx2−2x+ 8 >7
P D A =P DP −1
A2−2A+ 8In=P(D2−2D+ 8In)P−1tr(A2−2A+ 8In)>7n
A=
2 1 1
1 2 −1
0 0 3
Ann∈N∗
A Ann=−1
a > 0Z+∞
0
e−a√ttZ+∞
0
2ue−au u
u=√t
Z+∞
0
2ue−au u
a∈R∗
+k∈N∗Zk+1
k
e−a√tt6e−a√k6Zk
k−1
e−a√tt
X
k>1
e−a√k
+∞
X
k=1
e−a√k62
a2
A=
0−1 1 0
1001
000−1
0010
A−1
f0(x) = sin x f1(x) = cos x f2(x) = xsin x f3(x) = xcos x
B= (f0, f1, f2, f3)
EBφ f ∈E f0
φ E B
x(sin x+ cos x)
25%

TSI2
P∈R[X]φ(P) = P(X)−P(X−1)
ψ=φ|R2[X]
P∈R2[X]φ3(P) = 0
Q∈R[X]D(Q) = 
Q(1) Q(2) Q(3) Q(4)
Q(2) Q(3) Q(4) Q(5)
Q(3) Q(4) Q(5) Q(6)
Q(4) Q(5) Q(6) Q(7)
D(Q) = 
Q(1) φ(Q)(2) φ(Q)(3) φ(Q)(4)
Q(2) φ(Q)(3) φ(Q)(4) φ(Q)(5)
Q(3) φ(Q)(4) φ(Q)(5) φ(Q)(6)
Q(4) φ(Q)(5) φ(Q)(6) φ(Q)(7)
∀Q∈R2[X], D(Q)=0
En
Pn=P(En)Qn=P(En)
P1=a n
1/10 n4/10
Pn+1 =1
10Pn+4 10QnPn+1 Pn
P(X) = X3−7X+ 6 C[X]
Q(X) = X9−7X3+ 6 C
Q
Qn(X) = X3n−7Xn+ 6
QR[X]
Φ(x, y) = 3x+y
4,x−y 4
∀(u, v)∈R2Φ(x, y) = (u, v)
ΦC1ϕ−1
gC1R2∂g(u, v)
∂v = 7(u+v)
∂f(x, y)
∂x −3∂f(x, y) ∂y = 7x
f(x, y)