Bac Maths niveau 2 2014

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lim
!→!𝑓𝑥𝑑𝑥
!
!!
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%&'()$*)+,-.*)./012)30+4$)563)+$*)$3641/,4($
+789:;<7=$7<$>=78?@7?$$-7AA@BC$DE!#$
Mathématiques,niveau,II,
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Exercice'1''''''FGB@HA$IJ'
$KJ$/LAB8H?7$=MLN8K<@BC$H@OOL?7C<@7==7$$𝑦′+𝑦=8𝑥+4𝑒!!$P$
-B@<$=K$OBC9<@BC$$𝑓(𝑥)=4𝑥!+4𝑥+𝑚𝑒!!Q$KR79$$𝑚∈ℝ$P$
$SJ$ TB8?$N87==7FAJ$RK=78?FAJ$H7$𝑚$=K$OBC9<@BC$𝑓$7A<.7==7$GK?<B8<$HL9?B@AAKC<7$U$
0C$GBA7$$𝑚=0Q$K@CA@$$𝑓(𝑥)=4𝑥!+4𝑥𝑒!!$$7<$BC$HBCC7$$𝑓!! 𝑥=4𝑥!−12𝑥𝑒!!P$
$9J$ (<8H@7?$=K$OBC9<@BC$𝑓$FVL?BFAJQ$9BWGB?<7W7C<$KAXWG<B<@N87Q$GB@C<FAJ$Y$<KCZ7C<7$
:B?@VBC<K=7$7<$GB@C<FAJ$H[@CO=7\@BCQ$Z?KG:7JP$
$HJ$ *L<7?W@C7?$=[LN8K<@BC$H7$$𝑝(𝑥)Q$=7$GB=XC]W7$H7$4KX=B?$H7$H7Z?L$I$K8<B8?$H7$0$H7$=K$
OBC9<@BC$𝑓P$
)C$<7CKC<$9BWG<7$H7A$VL?BA$7<$H7A$GB@C<A$Y$<KCZ7C<7$:B?@VBC<K=7$H8$GB=XC]W7$$𝑝Q$
H7AA@C7?$ABC$Z?KG:7$HKCA$=7$W^W7$AXA<_W7$H[K\7$N87$97=8@$H7$𝑓P$
$7J$ /7G?LA7C<7?$A8?$=7$H7AA@C$=K$A8?OK97$𝑆$HL=@W@<L7$GK?$=7$Z?KG:7$H7$𝑓Q$=MK\7$H7A$𝑥Q$7<$=7A$
H?B@<7A$H[LN8K<@BC$$𝑥=−1$$7<$$𝑥=1P$
*L<7?W@C7?$GK?$9K=98=$=[K@?7$H7$𝑆P$
$OJ$ *L<7?W@C7?Q$A@$7==7$7\@A<7Q$=K$RK=78?$H7$=K$=@W@<7$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$P$
$ZJ$ 0C$CBWW7$𝑡!$=K$H?B@<7$<KCZ7C<7$K8$Z?KG:7$H7$𝑓$K8$GB@C<$H[KSA9@AA7$$𝑥=𝑘P$
-KCA$9K=98=AQ$HBCC7?$=[7CA7WS=7$H7A$RK=78?A$𝑘$<7==7A$N87$$𝑡!$7<$=7$Z?KG:7$H7$$𝑓$K@7C<$
7\K9<7W7C<$H78\$GB@C<A$9BWW8CAP$
$ $

D"#$
%&'()$*)+,-.*)./012)30+4$)563)+$*)$3641/,4($
+789:;<7=$7<$>=78?@7?$$-7AA@BC$DE!#$
Mathématiques,niveau,II,
$
Exercice'2''''''FGB@HA$IJ'
*KCA$𝑉
!$W8C@$H[8C7$SKA7$B?<:BCB?WL7$(𝑢!;𝑢!;𝑢!)Q$BC$9BCA@H_?7$=[KGG=@9K<@BC$=@CLK@?7$𝑓$N8@$
HL9?@<$8C7$?B<K<@BC$H7$90°$K8<B8?$H7$𝑢!Q$A8@R@7$H[8C7$AXWL<?@7$GK?$?KGGB?<$K8$G=KC$OB?WL$
GK?$𝑢!$7<$𝑢!$F=7$AB=JP$
$KJ$ *BCC7?$=K$WK<?@97$𝐴$H7$fQ$ABC$HL<7?W@CKC<Q$AK$RK=78?$G?BG?7$7<$8C$R79<78?$G?BG?7$
9B??7AGBCHKC<P$
0C$9BCA@H_?7$=[KGG=@9K<@BC$=@CLK@?7$𝑔$HL9?@RKC<$8C7$?B<K<@BC$H7$90°$K8<B8?$H7$=[K\7$GK?K==_=7$
K8$R79<78?$$𝑐=!
!
𝑢!+!
!
𝑢!Q$A8@R@7$H[8C7$AXWL<?@7$GK?$?KGGB?<$K8$G=KC$G7?G7CH@98=K@?7$Y$97<$
K\7P$
$SJ$*BCC7?$=7$R79<78?$𝑏$H7$AB?<7$N87$$(𝑢1;𝑏;𝑐)$$AB@<$8C7$SKA7$B?<:BCB?WL7$GK?$?KGGB?<$Y$
=KN87==7$=K$WK<?@97$𝐵′$H7$𝑔$7A<$=K$W^W7$N87$=K$WK<?@97$𝐴P$
$9J$ *BCC7?$=K$WK<?@97$𝐵$H7$𝑔$HKCA$=K$SKA7$B?<:BCB?WL7$$(𝑢!;𝑢!;𝑢!)P$
0C$9BCA@H_?7$=K$WK<?@97$$𝐶=
−15 16 12
0−15 20
20 12 9
P$$
$HJ$3BC<?7?$N87$𝐶$HL9?@<$8C7$:BWB<:L<@7$A8@R@7$H[8C7$?B<K<@BCQ$G8@A$G?L9@A7?$=7$OK9<78?$
H[:BWB<:L<@7P$
$7J$`L?@O@7?$N87$=[K\7$H7$?B<K<@BC$7A<$GK?K==_=7$K8$R79<78?$$$$$$$$$$$$$$$$$$Q$G8@A$HBCC7?$=[KCZ=7$H7$=K$
?B<K<@BCP$
*KCA$=M7AGK97Q$BC$9BCA@H_?7$=MKOO@C@<L$ZLCL?K=7$𝑙$H7$WK<?@97$𝐶$N8@$=K@AA7$O@\7$=[B?@Z@C7P$
$OJ$ *L9?@?7$ZLBWL<?@N87W7C<$=[@WKZ7$GK?$𝑙$H7$=K$H?B@<7$𝑑:
𝑥=1+𝜆
𝑦=𝜆
𝑧=2𝜆
$$P$
$'
𝑣=
1
1
2
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I"#$
%&'()$*)+,-.*)./012)30+4$)563)+$*)$3641/,4($
+789:;<7=$7<$>=78?@7?$$-7AA@BC$DE!#$
Mathématiques,niveau,II,
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Exercice'3''''''FGB@HA$DJ'
0C$9BCA@H_?7$=K$OBC9<@BC$H7$ℂ$HKCA$ℂ$HLO@C@7$GK?$$𝑓𝑧=!
!
𝑖𝑧+!
!
−𝑖!P$
$KJ$'K=98=7?$$𝑓2+𝑖P$
$SJ$*KCA$=7$G=KC$H7$2K8AAQ$BC$9BCA@H_?7$=7A$GB@C<A$$𝐴(2;1)$7<$$𝐵(−1;−2)$K@CA@$N87$=K$H?B@<7$
𝑑$N8@$GKAA7$GK?$97A$D$GB@C<AP$
$!J$ *BCC7?Q$AB8A$=K$OB?W7$$𝑦=𝑚𝑥 +ℎQ$=[LN8K<@BC$H7$=K$H?B@<7$$𝑑$$K@CA@$N87$=[LN8K<@BC$
H7$=K$H?B@<7$𝑑′Q$@WKZ7$GK?$$𝑓$H7$$𝑑P$
$DJ$a87$G78<.BC$H@?7$H7$$𝑑$$7<$$𝑑′$U$
$9J$ *L<7?W@C7?$=7A$GB@C<A$O@\7A$H7$$𝑓$$FGBA7?$$𝑧=𝑥+𝑦𝑖JP$
$HJ$3BC<?7?$N87$=K$OBC9<@BC$$𝑓$$7A<$=K$9BWGBA@<@BC$$𝑓
!∘𝑓 !$$H7A$OBC9<@BCA$$𝑓 !𝑧=!
!
𝑧−!
!
𝑖$$7<$$
𝑓
!𝑧=𝑖𝑧+1−𝑖P$
$7J$ %K$OBC9<@BC$𝑓$HL9?@<Q$HKCA$=7$G=KC$9BWG=7\7Q$=K$9BWGBA@<@BC$H[8C7$:BWB<:L<@7$7<$H[8C7$
AXWL<?@7$K\@K=7P$*BCC7?$=[LN8K<@BC$H7$=[K\7$H7$AXWL<?@7$K@CA@$N87$=7$97C<?7$7<$=7$OK9<78?$
H7$=[:BWB<:L<@7P$
$OJ$ 0C$GBA7$𝑧!=1$$7<$$𝑧!!!=𝑓𝑧!$$FGB8?$𝑛∈ℕ∗JP$
!J$'K=98=7?$𝑧!=𝑓𝑧!$7<$𝑧!=𝑓𝑧!P$
DJ$*LWBC<?7?$GK?$?L98??7C97$N87$$𝑧!=!
!!−1−! !!𝑖P$
IJ$'K=98=7?$$$$$$$$$$$$$$$$P$
$ $
lim
!→!
𝑧!$

#"#$
%&'()$*)+,-.*)./012)30+4$)563)+$*)$3641/,4($
+789:;<7=$7<$>=78?@7?$$-7AA@BC$DE!#$
Mathématiques,niveau,II,
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Exercice'4''''''FGB@HA$DJ'
bKW7A$GBAA_H7$8C7$SBc<7$N8@$9BC<@7C<$!E$G@_97A$H7$d$O?KC9AP$
,=$X$K$I$AB?<7A$H7$G@_97A$e$d$G@_97A$CB?WK=7AQ$I$G@_97A$<?8N8L7A$N8@$G?LA7C<7C<$OK97$H7A$H78\$
9]<LA$7<$D$G@_97A$<?8N8L7A$N8@$G?LA7C<7C<$G@=7$H7A$H78\$9]<LAP$
$KJ$ bKW7A$G?7CHQ$K8$:KAK?HQ$8C7$G@_97$H7$AK$SBc<7$7<$=K$=KC97P$
3BC<?7?$N87$=K$G?BSKS@=@<L$N87$OK97$KGGK?K@AA7$RK8<$$!!
!"$P$
$SJ$ bKW7A$G?7CHQ$K8$:KAK?HQ$8C7$G@_97$H7$AK$SBc<7$7<$=K$=KC97P$%K$G@_97$G?LA7C<7$=7$9]<L$G@=7P$
a87==7$7A<$=K$G?BSKS@=@<L$N87$=K$G@_97$AB@<$<?8N8L7$U$$
bKW7A$ALGK?7$A7A$G@_97A$7C$H78\$Z?B8G7A$e$=7A$G@_97A$CB?WK=7A$7<$=7A$G@_97A$<?8N8L7AP$
$9J$,=$G?7CH$=7A$d$G@_97A$CB?WK=7AQ$=7A$=KC97Q$G8@A$=7A$K=@ZC7$H7RKC<$=8@P$
a87==7$7A<$=K$G?BSKS@=@<L$N87$=7A$d$G@_97A$G?LA7C<7C<$G@=7$U$
$HJ$,=$G?7CH$=7A$d$G@_97A$<?8N8L7AQ$=7A$=KC97Q$G8@A$=7A$K=@ZC7$H7RKC<$=8@P$
a87==7$7A<$=K$G?BSKS@=@<L$N87$=7A$9]<LA$G@=7$C7$AB@7C<$GKA$RB@A@CA$U$
bKW7A$?7W7<$<B8<7A$=7A$G@_97A$HKCA$AK$SBc<7$7<$<@?7$K8$:KAK?H$D$G@_97A$7<$=7A$=KC97P$
$7J$a87==7$7A<$=K$G?BSKS@=@<L$N87$=7A$D$G@_97A$AB@7C<$H7$=K$W^W7$AB?<7$U$
bKW7A$?7W7<$7C9B?7$8C7$OB@A$<B8<7A$=7A$G@_97A$HKCA$AK$SBc<7P$,=$<@?7$K8$:KAK?H$!$G@_97$N8M@=$
=KC97$G8@A$@=$=K$?7W7<$HKCA$=K$SBc<7P$,=$?LG_<7$=MBGL?K<@BC$f8AN8MY$97$N8M@=$BS<@7CC7$OK97P$
6GG7=BCA$𝑋$=7$CBWS?7$H7$=KC97?AP$+B8A$AKRBCA$HLfY$N87$!𝑃(𝑋=1)=!!
!"$P$
$OJ$ 3BC<?7?$N87$!𝑃(𝑋=2)=!!
!""$Q$9K=98=7?$!𝑃(𝑋=3)$G8@A$HBCC7?$!𝑃(𝑋=𝑛).$
$ZJ$ a87==7$7A<$=K$G?BSKS@=@<L$N8M@=$HB@R7$=KC97?$WB@CA$H7$g$OB@A$U$
$:J$ a87==7$7A<$=K$G?BSKS@=@<L$N87$𝑋$AB@<$8C$W8=<@G=7$H7$I$U$

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%**/$-0%14,0+$*)$%[)563)+$*)$3641/,4($DE!#$$3K<:A$,,$
Exercice'1'
KJ$𝑦′+𝑦=8𝑥+4𝑒!!$P$
0C$GBA7$=𝑢∙𝑣!⟹𝑢!+𝑢=0!⟹𝑣=𝑒!!
𝑣!∙𝑒!!=8𝑥+4𝑒!!⟹𝑣!=8𝑥+4!⟹!𝑣=4𝑥!+4𝑥+𝑐
!!$
𝑦=4𝑥!+4𝑥+𝑐𝑒!!P$
SJ$ 𝑦′=−4𝑥!+4𝑥+4−𝑚𝑒!!$
−4𝑥!+4𝑥+4−𝑚<0!⟹4!+16 4−𝑚<0!⟹𝑚>5P$
9J$hL?BA$e$𝑂(0;0)$7<$𝐼(−1;0)P$
𝑂𝑥$7A<$KAXWG<B<7$:B?@VBC<K=7$à,droiteP$
!𝑀𝑖𝑛(−0,62;−1,75)Q$𝑀𝑎𝑥(1,62;3,36)P$
!Points!d'inflexion!:$𝑂(0;0)$$7<$$𝐽(3;2,39)P$
2?KG:7$9@.9BC<?7P $
HJ$𝑓𝑥=4𝑥!+4𝑥𝑒!!Q$$
𝑓′𝑥=−4𝑥!+4𝑥+4𝑒!!Q$$
𝑓!! 𝑥=4𝑥!−12𝑥𝑒!!$$
𝑓!!! 𝑥=−4𝑥!+20𝑥−12 𝑒!!$$
𝑝𝑥=𝑝0+𝑝!0∙𝑥+!
!𝑝!0∙𝑥!+! !𝑝!!! 0∙𝑥!$$
$$$$$$$$$$=4𝑥−2𝑥!$P$
$hL?BA$e$𝑂(0;0)Q$$𝐼!(−2;0)Q$7<$𝐼(2;0)Q$
𝑝!(𝑥)=4−6𝑥!$Q$ABWW7<A$e$𝑀𝑖𝑛(−0,82;−2,18)Q$𝑀𝑎𝑥(0,82;2,18)P$
2?KG:7$9@.9BC<?7P$
7J$ 𝑆=−𝑓𝑥𝑑𝑥 +𝑓𝑥𝑑𝑥
!
!
!
!!$
$$$$=−−4𝑥!−12𝑥−12 𝑒!!
!!
!!+[−4𝑥!−12𝑥−12 𝑒!!]! !$
$$$$=12 −4𝑒−!"
!
+12 =24 −!"!!!!
!
≅2,826$
OJ$ lim!→!𝑓(𝑥)
!
!!𝑑𝑥 =−12 +4𝑒+lim!→!𝑓𝑥
!
!𝑑𝑥$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$=−12 +4𝑒+lim!→![−4𝑥!−12𝑥−12 𝑒!!]!
!$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$=−12 +4𝑒+12 =4𝑒P$
ZJ$ 𝑘∈]−0,62;0[∪]0;1,62]∪{3}P$
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