QCM Analyse numérique licence master
Q
u
e
s
t
i
o
n
1
On
désigne
par
x
la
partie
entière
de
x
.
lim
x
→
1
x
2
−
1
x
2
−
1
=
□
2
□
0
□
1
□
n
'
e
x
i
s
t
e
pas
Question
2
On
considére
la
suite
x
n
pour
tout
n
∊
ℕ
définie
par
;
x
n
=
x
n
−
1
+
x
n
−
2
+
....
+
x
0
,
∀
n
∊
ℕ
Quelle
est
la
nature
de
x
n
?
□
x
n
est
une
suite
géomtrique
□
x
n
est
une
suite
arithmétique
□
x
n
est
une
suite
constante
□
On
ne
peut
pas
en
conclure
Question
3
L
'
e
q
u
a
t
i
o
n
x
2
−
2
2
−
5
2
=
1
admet
dans
ℝ
:
□
4
solutions
□
5
solutions
□
6
solutions
□
7
solutions
□
8
solutions
Question
4
Soit
f
d
é
rivable
en
a
,
lim
h
→
0
f
2
a
+
3
h
−
f
2
a
−
h
h
=
□
8
f
a
f
′
a
□
8
f
2
a
f
′
a
□
4
f
a
f
′
a
□
2
f
a
f
′
a
Question
5
On
pose
f
x
=
1
x
2
∫
3
x
2
u
−
3
f
′
u
du
.
f
′
3
=
□
1
/
2
□
3
/
2
□
1
/
4
□
−
1
/
3
Question
6
Siut
u
n
la
suite
réelle
définit
par
1
−
u
n
u
n
+
1
>
1
/
4
et
0
≤
u
n
≤
1
∀
n
∊
ℕ
,
lim
u
n
=
n
→
+
∞
□
0
□
1
/
2
□
1
□
+
∞
□
Aucune
de
ces
réponses
Question
7
Pour
quelle
valeur
de
α
la
série
dont
le
terme
générale
est
−
1
n
−
1
n
+
n
α
converge
pour
n
≥
2
?
□
α
≥
3
/
2
□
α
≤
2
/
3
□
α
≤
1
/
2
□
α
>
2
/
3
□
α
>
3
/
2
Question
8
Soit
l
'
a
p
p
l
i
c
a
t
i
o
n
f
dans
ℂ
définie
par
f
(
z
)
=
z
+
i
z
−
i
pour
tout
z
different
de
i
et
on
considére
E
=
z
∊
ℂ
,
∣
z
∣
=
1
et
i
ℝ
=
z
∊
ℂ
,
Rel
z
=
0
□
f
réalise
une
bijection
de
ℂ
∖
i
dans
ℂ
.
□
f
i
ℝ
=
ℝ
□
f
i
ℝ
=
E
□
f
E
=
i
ℝ
□
Aucune
réponses
n
'
e
s
t
c
o
r
r
e
c
t
e
.
Question
9
On
désigne
P
E
l
'
e
n
s
e
m
b
l
e
des
parties
de
E
.
Soit
E
une
ensemble
de
cardinal
fini
n
.
le
nombre
de
couples
A
,
B
∊
P
E
×
P
E
tel
que
A
⊂
B
v
a
u
t
:
□
2
n
□
3
n
−
1
□
3
n
□
2
n
2
□
n
2
n
−
1
Question
10
Une
urne
contient
2
boules
blanches
et
n
−
2
boules
r
o
u
g
e
s
.
On
effectue
des
tirages
successifs
sans
remise
d
'
u
n
e
boule
et
on
note
X
le
rang
pour
lequel
la
boule
blanche
apparait
pour
la
premiere
f
o
i
s
.
E
X
=
□
n
+
1
2
□
2
n
+
1
2
□
3
n
+
1
2
□
n
+
1
3
Question
11
Soit
S
l
'
ensemble
des
entiers
x
tel
que
x
2
−
x
+
1
est
divisible
par
7
.
S
est
alors
:
□
S
=
7
k
+
2
,
7
k
+
5
,
k
∊
ℤ
□
S
=
7
k
+
1
,
7
k
+
5
,
k
∊
ℤ
□
S
=
7
k
+
3
,
7
k
+
5
,
k
∊
ℤ
□
S
=
7
k
+
2
,
7
k
+
3
,
k
∊
ℤ
Question
12
lim
n
→
+
∞
∑
k
=
0
n
sin
1
n
+
k
=
□
0
□
sin
1
□
ln
2
□
1
□
+
∞
Question
13
Soit
une
fonction
f
d
é
rivable
en
x
0
=
0
d
é
finie
de
ℝ
→
ℝ
tel
que
f
0
=
0
et
f
'(
0
)
=
1
∀
n
∊
ℕ
*
lim
x
→
0
f
x
×
f
2
x
×
f
3
x
×
.....
×
f
n
−
1
x
x
n
−
1
=
□
n
n
−
1
/
2
□
n
!
□
n
n
+
1
/
2
□
n
−
1
!
□
+
∞
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Question
14
Soit
n
un
entier
n
a
t
u
r
e
l
.
On
d
é
f
i
n
i
t
l
'
e
n
t
i
e
r
A
n
=
1
+
5
n
+
5
2
n
+
5
3
n
.
A
n
≡
0
13
si
est
seulement
si
:
□
n
est
un
multiple
de
5
□
n
est
un
multiple
de
4
□
n
n
'
e
s
t
pas
un
multiple
de
5
□
n
n
'
e
s
t
pas
un
multiple
de
4
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