Baccalauréat Blanc 2006 Mathématiques - lycée Descartes

P(O;→
u,→ v)
z2−4z+ 8 = 0
z1z2z1
PA B z1z2
f M z M0
z0=1
z
z z
A0B0A B f
M O O, M M0
−−→
OM .−−−→ OM0= 1
|z−2|= 2 ⇐⇒ 
1
2−z0
=|z0|
CI
MCO
M0d
Cd
MCO, A B
M0M f

•a b
cos(a−b) = cos acos b+ sin asin b; sin(a−b) = sin acos b−cos asin b
•hRλ h
h(x) = λ x +µ µ
(E)
y00 + 4y0+ 4y= 5 sin x
(E0)y00 + 4y0+ 4y= 0
θi0 ; π
2htan θ=4 3
cos θ=3
5sin θ
ϕRϕ(x) = sin(x−θ)
ϕ(E)
f(E)f−ϕ
(E0)
g(E0)
x h(x) = g(x) e2x
hR
h0(x)h(x)x
(E0)
(E)

p a,
ap−1≡1 mod p
29
x∈Nn n ≡1 mod 28.
xn≡xmod 29.
(E) : 17x−28y= 1 (x, y)∈Z2.
(E)
A={x∈N, x < 29}={0,1,2,· · · ,28}.
x∈A, f (x)x17 29 g(x)
x529.
f(x)∈A x17 ≡f(x) mod 29.
g(x)∈A x5≡g(x) mod 29.
x∈A, g (f(x)) = x.
f
f717 ≡24 mod 29 117 ≡1 mod 29
f

fRf(x) = ln 1+e−xC
f(O;→
i,
→
j)
lim
x→+∞ln x= +∞
f−∞ +∞
x f(x) = −x+ ln (1 + ex)
C −∞ d y =−x
dC
f
d∆C
g h [ 0 ; +∞[
g(t) = ln(1 + t)−tet h(t) = ln(1 + t)−t+t2
2
g h
t t −t2
2≤ln(1 + t)≤t
xe−x−e−2x
2≤f(x)≤e−x
a n
An=1
a+1 a2+· · · +1 an
Ann(An)n∈N∗
1
a−1
(An)n∈N∗
(Sn)n∈N∗(Tn)n∈N∗
Sn=1
e+1 e2+· · · +1 enet Tn=1 e2+1 e4+· · · +1 e2n
(Sn)n∈N∗(Tn)n∈N∗

(un)n∈N∗
u1= 1 + 1
en un+1 =1 + 1 en+1 un
n un>0
(vn)n∈N∗vn= ln(un)n
n
vn=f(1) + f(2) + · · · +f(n)
(vn)
n
Sn−1
2Tn≤vn≤Sn
(vn)n∈N∗
`(vn)n∈N∗
2e+ 1
2(e2−1) ≤`≤1 e−1
(un)n∈N∗