Exercice I. Exercice II. 1. a) Soit M = Exercice III. Exercice IV

F= (x;y;z;t)∈R4/x +y−2z=t et t −x+y= 0
G1=V ect((0; 1; 1; 0),(−1; 4; 0; 1)) G2=V ect((0; −1; 1; 0),(−1; 4; 0; 1))
F⊕G1=R4F⊕G2=R4
M=
2−2 1
2−3 2
−1 2 0
(M−I3)(M+ 3I3)
M M−1
(yk) (vk)∀k∈NMk=ukI3+vkM
ukvkMkk
x y
A=
1x1y
1y1x
x1y1
y1x1
A=
1 0 0
0−2−1
0 1 2
A
B=
1 0 0
0 1 0
0 0 1
An
m
mx +y+z= 1
x+my +z= 1
x+y+mz =m
In=R1
0
xn
1+xndx
In=ln 2
n−1 nR1
0ln(1 + xn)dx
0≤In≤1
n+1

In
∀n∈Nfn
∀x∈Rfn(x) = Z1
0
(1 −t2)ncos(tx)dt
In=fn(0) = R1
0(1 −t2)ndt
n fn
n x fn+2(x)fn+1(x)
fn(x)
In+1 In
In
F]0; +∞[
F:x7→ Zx
1
ln(1 + t)
tdt
F C1F(x) + F(1
x) = (ln x)2 2
Zx
1
ln(x+t)
tdt =1 2(ln x)2+F(x)
X
X
X E(X)
xiieme
25
X
i=1
xi= 49,5kg et
25
X
i=1
x2
i= 98,3kg2
5%