Groupes, Permutations, Anneaux, Arithmétique dans Z, Corps commutatif, Les polynômes formels à une indéterminée à coefficients dans un corps K, Fonctions polynomiales, racines, Espaces vectoriels, K-algèbres, Espaces vectoriels de type fini, Matrices, Déterminants, Fractions rationnelles, Produit scalaire sur un R-ev, Espace vectoriel euclidien, R-ev euclidien orienté de dimension 2, R-ev euclidien orienté de dimension 3, Espaces affines, Géométrie dans un espace affine euclidien

1. = !" # $ × !
! %
& ' ⊕ ' (
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⊕
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! ⋅
→×
λλ !
"
, - !⋅⊕ . + + !
-
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!!
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33 =⋅
! 33 =⋅∈∀ λλ
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33 =⋅λ "
4! 333 ==⇔=⋅∈∀∈∀ λλλ
5 ⇐ ! !"
& - 3=⋅λ - 3≠λ "
# - ' 3= "
, −
λ - 3≠λ !"
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2. =⋅=⋅×=⋅⋅ −−
!! λλλλ '
33! =⋅=⋅⋅ −−
λλλ ' "
3=
6! !!! −⋅=⋅−=⋅−∈∀∈∀ λλλλ
( ) ( ) 3!!! =⋅−+=⋅−⊕⋅ λλλλ " !! ⋅−=⋅− λλ
( ) 33!!! =⋅=−⊕⋅=−⋅⊕⋅ λλλλ " !! ⋅−=−⋅ λλ "
7! ⋅=∈∀∈∀ "
' !⊕ 8) ( !
3≥ 6! 3≤ "
%
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9
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3. 4
& ∈ !"""= !"""= "
⊕=
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( ' ⊕ "
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,
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µλ
µλ
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µλµλµλ
µλµλµλ
µλµλ
!"""!"""
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=
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λλλ
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4. 6
! 9
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5 A B ) C '- "
%%
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& 3≠β ⋅= −
β
α
& 3=β 3=⋅α " , 3≠α ( )33! ≠βα " 3= "
!⋅+ ; + "
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5. 7
! %
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+ { }3
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+ "
> '@ ' + (
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& { }3F∈ { }∈⋅ λλ +
"
• 5 + 4
(
+ { }43
+ { }43F4
∈ { }∈⋅ λλ "
+ { }43F4
∈ { }∈⋅+⋅ µλµλ
!
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= ' % @ +⋅ ∈ "
, B /0 /0 < ∈ !
! """!3
5' % D
1! > +
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E + + "
touscours.net

6. J
& ∈! % + "
& { }∈∈∀∈==
∈
∈3 ∈∈∀ 3
$
& ∈ " ∈∈∈∀ ∈+∈∀ " ∈+
⋅
& ∈∈ λ " ∈∈∀ ∈⋅∈∀ λ ∈⋅λ "
! % -
& ⊂ "
∈
⇔
∈⋅+∈∀∈∀
∈
⇔
∈⋅+⋅∈∀∈∀
∈
⇔
∈⋅∈∀∈∀
∈+∈∀
∈
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!
!
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λλ
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4 ! ∈⋅∈∀∈∀∈∀
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& !3!4 - !4 3= ! 4 !
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7. K
L %
5' ε + '
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+
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• & + ⊂ ⊂!9
%% ε∈ { }ε∈
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& """ "
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% !""" A C
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==
=⋅ !λλλλ !
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+
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9
• & + % % !
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/0 ' @ % " %% - ' %
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touscours.net

8. ∈!"""
• [ ]∈ ≠
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• [ ]∈ { }3F∈
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• [ ]∈ ∈λ
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4 !
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µ
λλλλ
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λ
µ
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4
4
4
6 4
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∈ !"""9!"""9!"""9 =⇔∈
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touscours.net

9. O
%
& ( + "
5
{ } { }+=×∈∃∈=∈∈+=+ !
%
+ + ' B !9 ∪ "
%%
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( % + !
+ ' 3+ < ∈3 ! "
> ∪ "
% + ∪ +
$
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3"
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!!
3 !! ε+++=∈∀
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+ 6
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=−=−∈==
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& %
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%
5
!****
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(
5 ( "
, '-
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4! { }3=∩
! !****!**! ==+=+×∈∀×∈∀ !
touscours.net

10. 3
+ , ;) - !! ' !33!** = !
+ # - !4! " & !
& ∩∈
, 3! =−+ " , ∈ ∈− ∈ ⋅ !
' ! 3= 3=− ! '< ' "
+ # - !!4 " & 4!"
& ×∈×∈ !**! " & - **+=+ "
−=− ** ∈−∈− ** ∩∈−∩∈− ** "
3*=− 3* =− * +)+ *= *= "
'< - "
N
& ⊕ "
%
& ( + "
, - -
{ }=∩
=+
3
- ⊕= "
( +
' - + < ∈ ∈ "
+ "
! %
& →ϕ "
, - ϕ . + + -
!!
!*!!**
ϕλλϕλ
ϕϕϕ
⋅=⋅∈∀∈∀
+=+∈∀
& ϕ ϕ
!+ !+ "
9
• 5' !
• ? '
! M ! "
• ? ' % " !
:" 5' % ' "
touscours.net

11. -
& →ϕ "
!4!*"!!*"!*
!!*"!"!*""!*!!!
ϕλϕλϕλ
ϕβϕαβαϕβαϕ
+=+∈∀∈∀⇔
+=+∈∀∈∀⇔∈
%%
!4!! "
# - !!4 "
, - 4! =λ " !*!!*!* ϕϕϕ +=+∈∀
!33!* = 3!3 =ϕ "
!!3!!3!3 ϕλϕλϕλϕλϕλ ⋅=⋅+=⋅+=⋅+∈∀∈∀
(
• 5' "
• 5' "
• 5 (
% ⋅ < ∈
o ;) % ⋅
!*"!!*"!*"!*"* λλλλλ +=+=+=+∈∀∈∀
o > !∈ "
∈ !"!"! ==
!= "! =∈∀ "
• 5'
*
!! → "
• 5'
!
!! → % !!
!! ' !
• 5'
!
!
π
ψ →
!∈" !!!!!!! """" ψψπππψ +=+=+=+
!∈ ∈λ !"!"!!"!" ψλπλπλλψ === "
• 5'
!
→ ' "
# ) % ( → % ( !
, -
!
→ "
1! N @
& !∈ϕ "
5 @ ϕ ' @
{ }3!Q =∈= ϕϕ "
ϕϕϕ Q*!*!* ∈−⇔=∈∀ "
ϕ ; { }3Q =⇔ ϕ "
touscours.net

12. %%
+ & ϕ ;
& ϕQ∈ " !33! ϕϕ == " 3= "
'< ' ' "
+ & - { }3Q =ϕ "
& !*! ϕϕ = ϕQ*∈− 3*=− " *= "
ϕ ; "
ϕQ + "
;) ⊂ϕQ ϕQ3 ∈ "
& ∈∈ λ* " ,
33"3!*"!!*" =+=+=+ λϕλϕλϕ
5' ϕ { } { }=∈∃∈=∈== !!!> ϕϕϕϕ "
ϕ ; =ϕ> "
ϕ> + "
;) ⊂ϕ> ϕ>3 ∈ 3!3 =ϕ "
ϕ> $ ⋅
& ∈∈ λϕ>* "
> ( ∈* - !**! ϕϕ == "
!*"!*"!*" λϕϕλϕλ +=+=+ " ϕλ >*" ∈+ "
! > - ' +
& !∈ϕ "
5' ϕ ' + +
"
5' - ϕ ' + +
"
!ϕ + ' ϕ> !
{ }!3−
ϕ + ' ϕQ !
,
!
touscours.net

13. 4
! & '
! &
& !∈ψϕ ∈λ "
, %
!! ψϕ
ψϕ
+
→+
!"
"
ϕλ
ϕλ →
!" ∈+ ϕλψϕ "
, !! ⋅+ !! ⋅+ +
B + !! ⋅+ !"
;) % - !! ⋅+ + D
! ! 3 $ ⋅
& !∈ψϕ ∈λ "
, ∈* ∈µ
!*!"!!
!*"!!*"!
!*"!*"!*"!
ψϕµψϕ
ψµψϕµϕ
µψµϕµψϕ
+++=
+++=
+++=++
!!*!""!!"
!!*""!!"
!!*"!"
!*""!*"!"
ϕλµϕλ
ϕµλϕλ
ϕµϕλ
µϕλµϕλ
+=
+=
+=
+=+
!" ∈+ ϕλψϕ ! +
!! ⋅+ + "
!
5 - % (
"
& !∈ϕ !∈ψ " ϕψ %
"
∈* ∈µ
!*!"!!
!!*"!!
!!*"!
!!*"!*"!
ϕψµϕψ
ϕψµϕψ
ϕµϕψ
µϕψµϕψ
+=
+=
+=
+=+
!*∈ϕϕ !*∈ψψ ∈λ
!*!*
!*!*
ϕψϕψϕψψ
ϕψϕψϕϕψ
+=+
+=+
touscours.net

14. 6
!6!"!"
!4!"!"
ϕψλϕψλ
ϕψλϕλψ
=
=
;) % "
∈
!/*0
!!*!!!!*!!
!/*!0!/!*0!!/*0
ϕψϕψ
ϕψϕψϕψϕψ
ϕϕψϕϕψϕϕψ
+=
+=+=
+=+=+•
'< !"
!/*0!!*!!
!!*!!!!!*!/!*0
ϕψϕψϕψϕψ
ϕψϕψϕψψϕψψ
+=+=
+=+=+•
'< ! ' D !
!!/"0
!!"!!"!/"0!/!"0!!/"0
ϕψλ
ϕψλϕψλϕλψϕλψϕλψ
=
====•
'< 4!
!!/"0
!!"!!"!!!"!/!"0
ϕψλ
ϕψλϕψλϕψλϕψλ
=
===•
'< 6! ' !
-
% ! !! +
!! + % !! ⋅+ + !"
! ! - $ -
! !"
% @ ) > "
' ' % "
(
!!
→
!3!
"
−
→ "
!∈
& ∈∈ λ* !***! == "
!*"!
!**"!!*"*"
!*"*"!!**"!!*"
λ
λλλ
λλλλ
+=
+=++=
++=+=+
!" ∈
& ∈∈ λ* !***! == "
!*"!
!3**"!3!3!*"*"
!*"*"!!**"!!*"
""
"""
λ
λλλ
λλλλ
+=
−+−=+−+=
++=+=+
,
!!
"
−−
→
!33!
" →
- "
touscours.net

15. 7
4! > !
& !∈ϕ " & ϕ ; !∈−
ϕ " , -
ϕ "
( - ' (
' ' "
& ∈* ∈λ "
, - !*"!!*" −−−
+=+ ϕλϕλϕ * +)+ -
*"λ+ !*"! −−
+ ϕλϕ ϕ -
*"!!*"!!!!*"! λϕϕλϕϕϕλϕϕ +=+=+ −−−−
9
R ; "
R "
R ; % "
5' ! "
! !!
!" ' ' !! + "
' % "
(
&
!! −+
→
!!
" →
" ; "
" ; >= =−
!
=
=
≠
+−
−+
!3!
!3!
!!
!!
"
"
""
"
"
6!
& % !"
& ∈3 "
'
3!"
→φ "
%%
& ∈∈ λ "
!"!!""!"!""!" 333 φλφλλλφ +=+=+=+
(
5'
!
4
→
4
!****! ∈== ∈λ
!*"!*"!!*"*"*"!*" λλλλλλ +=+=+++=+
A ; - 4
C
touscours.net

16. J
B
!
4
→
!
4
4 → "
+ > ! - ∈#
#"""!
4
++
→
4#++= "
+ 6! - ∈#
!3"""!
4
#++
→
!3"=
!3!"
4
→
B
!*"*"*"3!
4
#++
→
'<
!*"*"*""""!
4
## ++++
→ "
, - 4
@ "
, ) !
; + "
! S !
%
? @
"α
→ < ∈α "
∈α '
"αα → α
" 3=α ; ' α.
1! ; !
%
& ( + " 5 ;
' → < ' - +=
∈∈ " % ' +
- ∈ ∈ !
, %
+=
→⊕= "
5' @ ' "
& ∈∈ λ* "
∈∈∈∈
+=+= *** "
∈∈
+++=+ *"*"*" λλλ !*"!*"!*" λλλ +=+=+ "
touscours.net

17. K
N @
& ∈
∈∈
+= " , -
∈⇔=⇔=⇔∈ 33!Q
>
, ;) - ⊂> " > >⊂
' ' ( + B "
%
& → " , - ; - ' ( ( +
- ; "
9
; "
∈ ! ; "
E
& !∈ "
; =⇔ "
& ; < =⊕
=
& ∈ "
∈∈
+= =! "
!!!!! ==== "
' = "
& !∈ - = "
>= Q= "
;) ( + " # - '
"
& ∈ " ∈!
!! −+=
∈
3!!!!! =−=− ∈− !
;) =+ "
# - { }3=∩
& ∩∈ "
∈ " !*= < ∈* "
∈ 3! = 3!!* = " = 3!* = "
3!* == '< ' '
"
=⊕
# - ; "
& ∈ "
∈∈
−+= !!! " !
%
∈∈
+
touscours.net

18. 2 -
& ;
{ }
!>Q
!
−=
=
=∈=
%% ∈⇔=⇔=⇔=
=−
3!
3!!>
%
& ; "
5 ; ) ; $ "
$ >=+ "
!! $+=
5 ; { }3 ' "
5 ; { }3 ' "
! &@ !
%
& ( + " 5 @
) '
−+=
→⊕= "
touscours.net

19. O
& @ - )
!∈• "
%% - - $ >−=−= < ;
$ ; ) !
• ; B "
>= => ; ! { }3Q =
; !
{ }
{ } !>Q!
!>Q!
+=−=∈=
−==∈=•
E
& !∈ "
@ >=⇔
⇔ !
⇔ ' ≤ ! !
;) "
⇐ - >= "
!>Q −= !>Q += "
( + @ (
' "
{ }3=∩ ∩∈ =! −=!
3" = 3= 3≠ !
'
∈∈
+= !!!! −++= "
, ∈+ ! !!!!!!! +=+=+=+
∈− ! !!!!!!!! +−=−=−=−
% @ ) " %%
&
∈∈
+= !!+= !!−= "
!=− "
! " #
; + "
! %
& !"""= % "
T
%
⇔ - 3
%% "
[ ] =∈∀=∈∀⇔
=
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touscours.net

20. 3
9
!"""• !"""
%
⇔ ' "
• 5 - !""" @ 3=
=
λ < λ
' "
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"
(
+ % "
+ ' % "
5 % ! 3≠⇔
+ ' %
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1!
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& 3= 3"
3
=
≠
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• & !""" !""" !"""9∈ "
%%
> ( µλλλ """ -
3"""" =+++ µλλλ "
3≠µ ' λ
≤≤ "
=
−
= λµ
• !""" '
"
$ " #
%
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⇔ % "
⇔ ' -
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=
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touscours.net

21. - !""" "
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=
= "
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=
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[ ] *=∈∀ "
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⇐ & - ' - D
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3=
=
λ [ ] 3=∈∀ λ (
%% 3 )
=
λ
=
"3 "
(
!/33!33!330 4
' - 4
"
!/!!0 −−− " 5 '
! 4 +++
"
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=+
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%
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,
−
−
−=
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−
=
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6
6
6
6
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π
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#
#
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touscours.net