Rešenja matematičkog prijemnog ispita: Pripremni ciklus

>@
>@
>@
DUJ 
DUJ 
itr
tre t
ca r
ba
ca
ba
AC rAB
AB AC
T
TS
TS
TS
°
®{
°
¯
°
°
®
§·
°
{
¨¸
°
©¹
¯
°
®{
°
¯
JJJG JJJG
45pSRQVH'
A B
C
VRQWDOLJQpVVLHWVHXOHPHQWVL
ca
ba
\
VLHWVHXOHPHQWVL
t\
45pSRQVH$
/HWULDQJOH
ABC
HVWUHFWDQJOHHQ
A
VLHWVHXOHPHQWVL
>@
AB AC
SS
{
JJJG JJJG
VLHWVHXOHPHQWVL
ti
\
([HUFLFH
45pSRQVH%
/HQRPEUHGHSDUWLHVGH
E
HVW
card E n
45pSRQVH&
8QHSDUWLHVGH
E
TXLFRQWLHQQHQWXQHWXQVHXOpOpPHQWGH
A
est la réunion d’un singleton de
A
HW
d’une partie de
?
EA
2QD
p
IDoRQSRXUFKRLVLUXQpOpPHQWGH
A
HW 
np
SRXUFKRLVLUXQHSDUWLHGH
?
EA
FDU
?
card E A n p
DLQVLOHQRPEUH GHSDUWLHVGH
E
TXLFRQWLHQQHQWXQHWXQVHXOpOpPHQWGH
A
HVW
np
p
45pSRQVH$
2QSHXWYpULILHUTXHSRXUOHFDVSDUWLFXOLHU
p
HW
q
OHQRPEUHGHFKHPLQSRVVLEOHHVW
Correction du Concours d’entrée en 1qUHDQQpHGXF\FOHSUpSDUDWRLUH
(FROH1DWLRQDOH'HV6FLHQFHV$SSOLTXpHV
Correction mathématique
4UpSRQVH&
RQD

2U
     CCCet
u
DORUVODUpSRQVHHVW$
45pSRQVH&
2QD
xfxx
x
\
DORUV
ff
§·
¨¸
©¹
RU
z
DORUV
z
DORUV
f
n’est pas
injective. ( de même pour n’importHUpHOQRQQXO
D
RQD
ff
DD
§·
¨¸
©¹
6XUMHFWLYLWpGH
f
SRXUWRXVUpHOV
x
HW
y
RQD
fx y yx x y   
Le discriminant de l’équation
 xyxxy \
HVW
 y' 
DLQVL
f
n’est pas surjective
LOVXIILWGHSUHQGUH
@>@>
 yf  f
/DUpSRQVHHVWGRQF&
45pSRQVH$
2QD
uuuuuuuuuuuuu
uuuuu
'RQFOHQRPEUHGHGLYLVHXUVGH HVW
      uuuuu
45pSRQVH$
3RXUFKDTXHTXHVWLRQRQFKRLVLHXQHUpSRQVHXQLTXHGRQFOHQRPEUHGHJULOOHVSRVVLEOHVHVW
C
45pSRQVH$
2QD
D
t
FDU
>@ >@
   tt  
d’autre part, pour
x
HW
y
TXHOFRQTXH
xy
$LQVL
D
45pSRQVH$
2QD

kk
kk k
kk
CC
¦¦
45pSRQVH'
2QD
ij ij
ij j j
i
iii
ij i ijj
iij j
ii
ii
dd d d dd d d
dd
dd
dd dd dd
§·
¨¸
©¹
uuu
§·
¨¸
©¹
uu u
u  u u
¦¦ ¦¦
¦¦ ¦ ¦
¦
¦¦¦
2QUDSSHOTXH
n
k
nn
k
¦
HW
n
k
nn n
k
¦
RQDDXVVL
n
k
nn
k
¦
45pSRQVH%
7RXWHIRQFWLRQGLVFRQWLQXHHVWQRQGpULYDEOH
45pSRQVH$
2QD
VLQFRVxfxx
xx
§· §·
c
¨¸ ¨¸
©¹ ©¹
\
3XLVTXH
VLQxx x
x
§·
 d
¨¸
©¹
\
DORUV
VLQ 
x
limx x
o
§·
¨¸
©¹
2UODIRQFWLRQ
FRVxx
§·
¨¸
©¹
6
n’admet pas de limite en 0DORUVVL
fc
n’aGPHWGHOLPLWHHQ
'RQFODIRQFWLRQ
fc
n’est pas continuHHQ
VLQ
xx
fx f
f lim limx
xx
oo
§·
c
¨¸
©¹
FDU
VLQxx x
x
§·
 d
¨¸
©¹
\

45pSRQVH%
2QD
OQ 
xx
xx
xx
x
lim lim e
x
§·
¨¸
©¹
of of
§·
¨¸
©¹
(W
OQ
OQ  
xx
x
xx
x
lim x lim x
xx
x
of of
§·
¨¸
§· ©¹
 u
u 
¨¸ 
©¹ 
FDU
OQ OQ
xt
xt
x
lim lim
xt
x
of o
§·
¨¸
©¹
$LQVL
x
x
x
lim e
x
of
§·
¨¸
©¹
45pSRQVH$
2QD
@>
FRV VLQ  
  xx xxx xx
§· §·
¨¸ ¨¸
©¹ ©¹
 f t
2U
x
lim xx
o f
DORUV
FRV VLQ 
x
xx
lim xx
o
§· §·
¨¸ ¨¸
©¹ ©¹ f
45pSRQVH&
2QSRVH
Px x x x x    
'RQF
Px xr xr xr xr    
DLQVL
i
irP
uuu

45pSRQVH%
2QD
OQ OQ
OQ OQ
OQ
OQ OQ OQ
OQ
ee
ee
e
e
e
e
xx
dx dx
xx xx
xdx
xx
xx
§·
¨¸
¨¸
¨¸
©¹
ªº
¬¼
³³
³
2XELHQ
OQ
OQ
OQ OQ
OQ OQ
OQ
ee
ee
e
e
xx
xdx dx
xx xx
xx
c
ª º
¬¼
³³
45pSRQVH$
3DULQWpJUDWLRQSDUSDUWLHVRQREWLHQW
VLQ FRV FRV
FRV VLQ VLQ
FRV VLQ FRV
x x dx x x x x dx
xx xx xdx
xx xx x
SSS
SS
SSS
SS S
SSS
SS S
S
S
ªº
«»
¬¼
ªº
ªº
¬¼
«»
¬¼
ªº
ªºªº
¬¼¬¼
«»
¬¼
³³
³
45pSRQVH&
2QD
FRV VLQ
VLQ FRV 
FRV VLQ OQ VLQ FRV 
VLQ FRV
xx
J I dx dx
xx
xx
JI dx x x
xx
SS
SS
S
°
°
®
° ª  º
¬¼
°
¯
³³
³
'RQF
IJ
S
.