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4
3
2
1
Outside the high school program
/ Complex geometry
24 exercises (including 19 corrected)
a
Plan configuration
(4 exercices)
Plane transformations
(8 exercices)
Annals on plane configurations
(4 exercices)
Annals on plane transformations
(2 exercices)
Autes annales
(1 exercice)
1.
Plan
c
onfigur
ation
E.3811
The
complex
plane
is
pro-
vided
with
a
direct
orthonormal
reference
frame
O
;
−
→
u
;
−
→
v
;
(gr
aphic
unit
:
2
cm
)
.
1
Sho
w
that
the
p
oin
ts
A
of
affix
1+i
·
3
and
B
of
affix
1
−
i
·
3
are
on
the
same
circle
with
cen
ter
O
whose
radius
w
e’ll
sp
ecify
.
Dra
w
this
circle
and
then
construct
the
p
oin
ts
A
and
B
.
2
Let
O
b
e
the
image
of
the
p
oin
t
O
b
y
the
rotation
r
1
of
cen
ter
A
and
angle
−
.
ı
2
and
B
the
image
of
the
p
oin
t
B
b
y
the
rotation
r
2
with
cen
ter
A
and
angle
+
ı
2
.
Calculate
the
affixes
of
the
p
oin
ts
O
and
B
and
con-
struct
these
p
oin
ts.
E.3840
The
complex
plane
is
referred
to
a
direct
orthonormal
reference
frame
O
;
−
→
u
;
−
→
v
of
graphic
unit
1
cm
.
Mak
e
a
figure
to
b
e
completed
as
the
questions
are
ask
ed.
1
Place
the
p
oin
ts
A
,
B
and
C
with
affixes
:
z
A
=
−
11
+
4
·
i
;
z
B
=
−
3
−
4
·
i
;
z
C
=
5
+
4
·
i
2
Calculate
the
mo
dulus
and
an
argumen
t
of
the
quotien
t
z
A
−
z
B
z
C
−
z
B
and
deduce
the
nature
of
the
triangle
AB
C
.
3
Let
E
b
e
the
image
of
p
oin
t
C
b
y
the
rotation
R
of
cen
ter
B
and
angle
ı
4
.
Sho
w
that
the
affix
of
E
v
erifies
:
z
E
=
−
3
+
8
2
−
4
·
i
.
Place
the
p
oin
t
E
.
4
Let
D
b
e
the
image
of
the
p
oin
t
E
b
y
the
homothet
y
H
of
cen
ter
B
and
ratio
2
2
.
Sho
w
that
D
is
the
cen
ter
of
the
circumscrib
ed
circle
of
the
triangle
AB
C
.
Place
the
p
oin
t
D
.
E.3842
The
complex
plane
is
equipp
ed
with
a
direct
orthonormal
reference
frame
O
;
−
→
u
;
−
→
v
of
graphic
units
2
cm
.
Consider
the
p
oin
ts
A
and
B
with
re-
sp
ectiv
e
affixes
:
z
A
=
1
+
i
·
√
3
;
z
B
=
2
·
i
1
a
W
rite
z
A
and
z
B
in
exp
onen
tial
form.
b
Place
p
oin
ts
A
and
B
on
a
figure
that
will
b
e
com-
pleted
during
the
exercise.
c
Determine
the
nature
of
triangle
O
AB
.
2
W
e
denote
r
the
rotation
with
cen
ter
O
that
transforms
A
in
to
B
.
F
or
an
y
p
oin
t
M
of
affix
z
,
w
e
denote
M
the
image
of
M
b
y
r
and
z
the
affix
of
the
p
oin
t
M
.
a
Calculate
an
argumen
t
of
the
quotien
t
z
B
z
A
.
In
terpret
this
result
geometrically
.
b
Deduce
the
complex
rotation
notation
r
.
3
Let
Γ
b
e
the
circle
with
cen
ter
A
passing
through
O
and
Γ
the
circle
with
cen
ter
B
passing
through
O
.
Let
C
b
e
the
second
p
oin
t
of
in
tersection
of
Γ
and
Γ
(other
than
O
)
.
Let
z
C
b
e
its
affix.
a
Justify
that
the
circle
Γ
is
the
image
of
the
circle
Γ
b
y
rotation
r
.
b
Calculate
the
affix
z
I
of
the
cen
ter
I
of
[
AB
]
.
c
Determine
the
nature
of
the
quadrilateral
O
AC
B
.
d
Deduce
that
I
is
the
midp
oin
t
of
[
O
C
]
then
sho
w
that
the
affix
of
C
is
:
z
C
=
1
+
2
+
√
3
·
i
4
Let
D
b
e
the
affix
p
oin
t
z
D
=
2
·
i
·
√
3
.
a
Justify
that
the
p
oin
t
D
b
elongs
to
the
circle
Γ
.
Place
D
on
the
figure.
b
Place
of
image
D
b
y
rotating
r
as
defined
in
question
2
.
W
e
note
z
D
the
affix
of
D
.
Sho
w
that
:
z
D
=
−
3
+
3
·
i
.
5
Sho
w
that
the
v
ectors
−
−
→
D
C
and
−
−
→
D
D
are
collinear.
What
can
w
e
deduce
from
this?
E.3819
The
complex
plane
is
referred
to
a
direct
orthonormal
reference
frame
O
;
−
→
u
;
−
→
v
;
i
denotes
the
complex
n
um
b
er
of
mo
dulus
1
and
argumen
t
ı
2
Let
the
p
oin
ts
A
,
B
and
C
ha
v
e
affixes
:
i
;
1
+
i
;
−
1
+
i
,
resp
ectiv
ely
.
Let
f
b
e
the
application
whic
h,
to
an
y
p
oin
t
M
of
the
plane
other
than
A
,
of
affix
z
,
asso
ciates
the
p
oin
t
M
of
the
plane
of
affix
z
suc
h
that
:
z
=
i
·
z
+
2
z
−
i
1
a
Determine
images
from
B
and
C
via
the
f
applica-
tion.
b
Sho
w
that,
for
an
y
complex
n
um
b
er
z
other
than
i
,
w
e
ha
v
e
the
relationship
:
(
z
−
i)(
z
−
i)
=
1
c
Let
D
b
e
the
p
oin
t
with
affix
1+2
·
i
.
Place
the
p
oin
ts
A
,
B
,
C
and
D
on
a
figure
(gr
aphic
unit
4
cm
)
.
Deduce
from
the
previous
question
a
construction
of
the
p
oin
t
D
image
of
the
p
oin
t
D
b
y
the
application
f
.
2
Let
R
b
e
a
strictly
p
ositiv
e
real
n
um
b
er.
What
is
the
image
b
y
the
application
f
of
the
circle
with
cen
ter
A
and
radius
R
?
2.
Plane
tr
ansformations
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3811
sacados/3811
Extrait Pondichery Avril 2004
chapExoCorrec/3840
sacados/3840
Extrait Antilles-Guyane Septembre 2009
chapExoCorrec/3842
sacados/3842
chapExoCorrec/3819
sacados/3819
Extrait La Reunion Juin 2004
E.3820
The
complex
plane
is
referred
to
a
O
;
−
→
u
;
−
→
v
orthonormal
reference
frame.
1
Determine
the
exp
onen
tial
form
of
the
complex
n
um
b
er
:
2
4
·
−
1
+
i
2
Consider
the
transformation
f
of
the
plane
whic
h,
to
an
y
p
oin
t
M
of
affix
z
,
asso
ciates
the
p
oin
t
M
of
affix
z
suc
h
that
:
z
=
2
4
·
−
1
+
i
·
z
Determine
the
nature
and
c
haracteristic
elemen
ts
of
the
transformation
f
.
E.3821
The
complex
plane
is
referenced
to
O
;
−
→
u
;
−
→
v
orthonormal
direct.
Consider
the
p
oin
t
I
of
affix
i
and
the
p
oin
t
A
of
affix
:
z
=
3
+
2
·
i
1
Sho
w
that
the
p
oin
t
A
b
elongs
to
the
circle
Γ
with
cen
ter
p
oin
t
I
and
radius
2
.
On
a
figure
(gr
aphic
unit
1
cm
)
to
b
e
completed
as
the
exercise
progresses,
place
the
p
oin
t
I
,
dra
w
the
circle
Γ
,
then
construct
the
p
oin
t
A
.
2
Consider
the
rotation
r
with
cen
ter
p
oin
t
I
and
angle
ı
2
.
Sho
w
that
p
oin
t
B
,
image
of
p
oin
t
A
b
y
rotation
r
,
has
affix
:
z
B
=
−
1
+
i
·
3
+
1
Justify
that
the
p
oin
t
B
b
elongs
to
the
circle
Γ
.
3
Calculate
the
affix
of
the
p
oin
t
C
image
of
the
p
oin
t
A
b
y
symmetry
of
cen
ter
I
.
4
What
is
the
nature
of
the
triangle
AB
C
?
Justify
.
E.3132
P
arts
A
and
B
are
indep
en-
den
t
Consider
the
equation
(
E
)
:
z
3
−
(4
+
i)
z
2
+
(7
+
i)
z
−
4
=
0
where
z
denotes
a
complex
n
um
b
er.
P
art
A
1
a
Sho
w
that
(
E
)
has
a
real
solution,
denoted
b
y
z
1
.
b
Determine
the
t
w
o
complex
n
um
b
ers
a
and
b
suc
h
that,
for
an
y
complex
n
um
b
er
z
,
w
e
ha
v
e
:
z
3
−
(4
+
i)
·
z
2
+
(7
+
i)
·
z
−
4
=
z
−
z
1
·
z
−
2
−
2i
·
az
+
b
2
Solv
e
(
E
)
.
P
art
B
In
the
plane
equipp
ed
with
a
direct
orthonormal
co
ordinate
system
O
;
−
→
u
;
−
→
v
,
consider
the
three
p
oin
ts
A
,
B
and
C
with
resp
ectiv
e
affixes
1
,
2+2i
and
1
−
i
.
1
Represen
t
A
,
B
and
C
.
2
Determine
the
mo
dulus
and
an
argumen
t
of
2+2
i
1
−
i
.
De-
duce
the
nature
of
triangle
O
B
C
.
3
What
do
es
line
(
O
A
)
represen
t
for
triangle
O
B
C
?
Jus-
tify
y
our
answ
er.
4
Let
D
b
e
the
image
of
O
under
the
rotation
b
y
angle
−
ı
2
with
cen
ter
C
.
Determine
the
affix
of
D
.
5
What
is
the
nature
of
O
C
D
B
?
h
ttps:/
/c
hingmath.fr
sacados/3820
sacados/3821
Extrait de France Septembre 2010
chapExoCorrec/3132
sacados/3132
E.3120
The
complex
plane
is
referred
to
a
direct
orthonormal
reference
frame
O
;
−
→
u
;
−
→
v
.
The
p
oin
ts
in
tro
duced
in
the
text
(graphic
unit
:
2
cm
)
will
b
e
placed
on
the
same
figure,
whic
h
will
b
e
completed
as
w
e
go
along
1
a
Solv
e
equation
:
(
E
)
:
z
2
−
2
3
·
z
+4
=
0
b
Consider
the
complex
n
um
b
ers
:
z
1
=
3
+
i
;
z
2
=
3
−
i
And
w
e
denote
b
y
M
and
N
the
p
oin
ts
of
affixes
z
1
and
z
2
resp
ectiv
ely
.
Determine
the
mo
dulus
and
argumen
t
of
z
1
and
z
2
;
place
M
and
N
on
the
figure.
c
Determine
the
affixes
of
the
p
oin
ts
Q
and
P
resp
ec-
tiv
e
images
of
M
and
N
b
y
the
v
ector
translation
−
→
w
=
−
2
·
−
→
u
.
Place
P
and
Q
on
the
figure.
Sho
w
that
M
N
P
Q
is
a
square.
2
Let
R
b
e
the
symmetrical
of
P
with
resp
ect
to
O
,
E
the
image
of
P
b
y
the
rotation
of
cen
ter
O
and
angle
ı
2
,
S
the
image
of
E
b
y
the
homothet
y
of
cen
ter
O
and
ratio
3
.
Place
these
p
oin
ts
on
the
figure.
Calculate
the
affixes
of
R
and
S
.
Sho
w
that
S
b
elongs
to
segmen
t
[
M
N
]
.
3
W
e
p
ose
:
a
=
2
−
3
:
a
Sho
w
that
:
1+
a
2
=
4
a
;
1
−
a
2
=
2
a
3
b
Express
the
affixes
Z
of
−
→
P
R
and
Z
of
−
→
P
S
in
terms
of
a
.
c
Sho
w
that
:
⏐
⏐
Z
⏐
⏐
=
⏐
⏐
Z
⏐
⏐
;
Z
Z
=
e
i
π
3
d
Deduce
from
the
previous
questions
the
nature
of
the
triangle
P
R
S
E.3136
1
The
complex
plane
is
referred
to
a
direct
orthonormal
reference
frame
O
;
−
→
u
;
−
→
v
.
W
e
p
ose
:
a
=
3
;
b
=
5
−
2
·
i
;
c
=
5+2
·
i
W
e
denote
b
y
A
,
B
and
C
the
p
oin
ts
with
affixes
a
,
b
and
c
resp
ectiv
ely
.
Let
M
b
e
a
p
oin
t
of
affix
z
in
the
plane,
distinct
from
p
oin
ts
A
and
B
.
a
Sho
w
that
AB
C
is
an
isosceles
righ
t-angled
triangle.
b
Giv
e
a
geometric
in
terpretation
of
the
argumen
t
of
the
complex
n
um
b
er
z
−
3
z
−
5+2
·
i
.
c
Then
determine
the
set
of
p
oin
ts
M
of
affix
z
suc
h
that
z
−
3
z
−
5+2
·
i
is
a
strictly
negativ
e
real
n
um
b
er.
2
Let
Γ
b
e
the
circumscrib
ed
circle
of
the
triangle
AB
C
and
Ω
the
p
oin
t
with
affix
2
−
i
.
a
Giv
e
the
complex
writing
of
the
rotation
r
of
cen
ter
Ω
and
angle
−
ı
2
.
b
Determine
the
image
Γ
of
Γ
b
y
the
rotation
r
.
Deter-
mine
a
parametric
equation
of
Γ
.
E.3182
1
In
the
complex
plane
referred
to
an
orthonormal
refer-
ence
frame
O
;
−
→
u
;
−
→
v
,
consider
the
p
oin
ts
:
A
of
affix
a
where
a
∈
R
;
B
affix
b
+i
where
b
∈
R
;
C
image
of
B
b
y
the
rotation
of
cen
ter
A
and
angle
ı
3
a
Determine
a
relationship
b
et
w
een
a
and
b
so
that
the
p
oin
t
C
b
elongs
to
the
axis
O
;
−
→
v
.
b
Then
express
the
affix
of
the
p
oin
t
C
as
a
function
of
a
.
2
In
this
question,
w
e
p
ose
a
=
3
and
b
=
0
.
Consider
the
p
oin
ts
C
of
affix
c
=
−
i
and
D
of
affix
d
=
2+
3
−
2i
√
3
a
What
is
the
nature
of
the
triangle
AB
C
?
b
Calculate
the
quotien
t
d
−
a
c
−
a
;
what
can
b
e
deduced
for
the
triangle
AC
D
?
c
Determine
the
affix
of
the
p
oin
t
E
image
of
D
in
the
rotation
of
cen
ter
A
and
angle
ı
3
.
d
Determine
the
affix
of
the
p
oin
t
F
image
of
D
in
the
translation
of
v
ector
−
→
AC
.
e
Determine
the
nature
of
the
triangle
B
E
F
.
E.3202
The
complex
plane
is
referenced
to
a
direct
orthonormal
O
;;
−
→
u
;
−
→
v
.
The
graphic
unit
is
2
cm
.
Designate
b
y
i
the
complex
n
um
b
er
of
mo
dulus
1
and
argu-
men
t
+
ı
2
.
W
e’ll
create
a
figure
and
complete
it
as
w
e
go
along.
1
Solv
e
in
the
set
C
of
complex
n
um
b
ers
the
equation
z
−
4
z
=
i
.
W
rite
the
solution
in
algebraic
form.
2
Solv
e
in
C
the
equation
z
2
−
2
z
+4
=
0
.
W
rite
the
solutions
in
exp
onen
tial
form.
3
Let
A
,
B
,
A
and
D
b
e
p
oin
ts
in
the
complex
plane
with
affixes
:
a
=
2
;
b
=
4
;
a
=
2i
;
d
=
2
+
2i
What
is
the
nature
of
the
triangle
O
D
B
?
4
Let
E
and
F
b
e
the
p
oin
ts
with
resp
ectiv
e
affixes
:
e
=
1
−
i
3
;
f
=
1
+
i
3
What
is
the
nature
of
the
quadrilateral
O
E
AF
?
5
Let
C
b
e
the
circle
of
cen
ter
A
and
radius
2
.
Let
C
‘
b
e
the
circle
with
cen
ter
A
and
radius
2.
Let
r
b
e
the
rotation
of
cen
ter
O
and
angle
+
ı
2
.
a
Let
E
b
e
the
image
of
p
oin
t
E
under
rotation
r
.
Cal-
culate
the
affix
e
of
p
oin
t
E
.
b
Pro
v
e
that
p
oin
t
E
is
a
p
oin
t
on
circle
C
.
c
V
erify
that
:
e
−
d
=
3
+2
e
−
d
.
Deduce
that
p
oin
ts
E
,
E
and
D
are
collinear.
6
Let
D
b
e
the
image
of
p
oin
t
D
under
rotation
r
.
Pro
v
e
that
triangle
E
E
D
is
a
righ
t
triangle.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3120
sacados/3120
Antilles-Guyane Septembre 1998 5 points
chapExoCorrec/3136
sacados/3136
chapExoCorrec/3182
sacados/3182
Antilles-Guyane Juin 2006 5 points
chapExoCorrec/3202
sacados/3202
E.3855
F
or
eac
h
of
the
follo
wing
six
state-
men
ts,
indicate
whether
it
is
true
or
false
and
pro
vide
a
justi-
fication
for
y
our
answ
er.
An
answ
er
without
justification
will
not
earn
an
y
p
oin
ts.
The
complex
plane
is
related
to
an
orthonormal
direct
refer-
ence
frame
O
;
−
→
u
;
−
→
v
.
1
Let
z
b
e
a
complex
n
um
b
er
with
argumen
t
ı
3
.
Prop
osition
1
:
ˇ
z
100
is
a
real
n
um
b
erı.
2
Let
(
E
)
b
e
the
set
of
p
oin
ts
M
with
affix
z
differen
t
from
1
in
the
plane
suc
h
that
:
⏐
⏐
⏐
z
1
−
z
⏐
⏐
⏐
=
1
.
Prop
osition
2
:
ˇthe
set
(
E
)
is
a
line
parallel
to
the
real
axisı.
3
Let
r
b
e
the
rotation
b
y
angle
−
ı
2
and
whose
cen
ter
K
has
affix
1+i
·
3
.
Prop
osition
3
:
ˇthe
image
of
the
p
oin
t
O
b
y
the
rota-
tion
r
has
affix
1
−
3
+i
·
1+
3
ı
4
Consider
the
follo
wing
equation
(
E
)
:
z
2
+
2
·
ı
5
·
z
+
1
=
0
Prop
osition
4
:
ˇThe
equation
(
E
)
has
t
w
o
complex
solutions
with
mo
duli
equal
to
1
.ı
3.
A
nnals
on
plane
c
onfigur
ations
E.4074
The
complex
plane
is
referenced
to
the
direct
orthonormal
co
ordinate
system
O
;
−
→
u
;
−
→
v
;
the
graphic
unit
is
1
cm
.
1
Solv
e
the
follo
wing
equation
in
the
set
of
complex
n
um-
b
ers
:
z
2
+
4
·
z
+
8
=
0
Giv
e
the
solutions
in
algebraic
form,
then
in
trigonomet-
ric
form.
2
Let
A
and
B
b
e
the
p
oin
ts
on
the
plane
with
resp
ectiv
e
affixes
:
a
=
2
−
2
·
i
;
b
=
−
a
Place
these
p
oin
ts
on
a
graph
that
will
b
e
completed
as
the
exercise
progresses.
a
Determine
the
affix
c
of
p
oin
t
C
,
image
of
p
oin
t
B
b
y
rotation
ab
out
cen
ter
O
and
angle
ı
2
.
b
W
e
note
D
the
image
of
C
b
y
rotation
ab
out
cen
ter
A
and
angle
ı
2
;
pro
v
e
that
the
affix
d
of
p
oin
t
D
is
:
d
=
2
−
6
·
i
c
Place
the
p
oin
ts
C
and
D
on
the
graph.
What
is
the
nature
of
the
quadrilateral
AB
C
D
?
3
¸
b
eing
a
non-zero
real
n
um
b
er,
w
e
denote
b
y
G
α
,
the
barycen
ter
of
the
system
:
(
A
;
1
)
;
(
B
;
−
1
)
;
(
C
;
¸
)
a
Express
the
v
ector
−
−
−
→
C
G
α
en
as
a
function
of
the
v
ector
−
−
→
B
A
.
b
Determine
the
set
of
p
oin
ts
G
α
when
¸
décrit
the
set
of
non-zero
reals.
Construct
this
set.
c
F
or
what
v
alue
of
¸
will
:
G
α
=
D
?
4
It
is
assumed
in
this
question
that
¸
=
2
.
In
this
question,
an
y
trace
of
researc
h,
ho
w
ev
er
incom-
plete,
or
of
unsuccessful
initiativ
e,
will
b
e
tak
en
in
to
ac-
coun
t
in
the
assessmen
t.
Determine
and
construct
the
set
of
p
oin
ts
M
of
the
plane
suc
h
that
:
−
−
→
M
A
−
−
−
→
M
B
+
2
·
−
−
→
M
C
=
4
·
2
E.3903
The
complex
plane
is
pro
vided
with
a
direct
orthonormal
reference
frame
O
;
−
→
u
;
−
→
v
d
graphic
unit
2
cm
.
Consider
the
p
oin
ts
A
,
B
and
C
of
affixes
:
z
A
=
−
2
·
i
;
z
B
=
−
3
+
i
;
z
C
=
3
+
i
1
Écrire
z
A
,
z
B
et
z
C
sous
exp
onen
tial
form.
2
Determine
the
cen
ter
and
radius
of
the
circle
Γ
passing
through
the
p
oin
ts
A
,
B
et
C
.
3
Mak
e
a
figure
and
place
the
p
oin
t
A
,
dra
w
the
circle
Γ
then
place
the
p
oin
ts
B
et
C
.
4
a
W
rite
the
quotien
t
z
B
−
z
A
z
C
−
z
A
sous
algebraic
form
and
then
in
exp
onen
tial
form.
b
Determine
the
nature
of
the
triangle
AB
C
.
5
W
e
note
r
the
rotation
of
cen
ter
A
and
angle
measuring
ı
3
radians.
a
Sho
w
that
the
p
oin
t
O
,
image
of
O
par
r
,
has
affix
−
3
−
i
.
b
Demonstrate
that
the
p
oin
ts
C
and
O
son
t
diametri-
cally
opp
osed
on
the
circle
Γ
.
c
T
race
the
image
Γ
of
the
circle
Γ
par
the
rotation
r
.
d
Justify
that
the
circles
Γ
and
Γ
se
in
tersect
at
A
et
B
.
6
a
Determine
the
set
(
E
)
des
p
oin
ts
M
d’affix
z
tels
that
:
|
z
|
=
⏐
⏐
z
+
3
+
i
|
b
Sho
w
that
p
oin
ts
A
and
B
b
elong
to
(
E
)
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3855
sacados/3855
Extrait Liban Juin 2008
chapExoCorrec/4074
sacados/4074
chapExoCorrec/3903
sacados/3903
A
B
C
A’
B’
C’
P
Q
R
E.3217
In
the
orien
ted
plane
equipp
ed
with
a
direct
orthonormal
reference
frame,
consider
AB
C
a
direct
triangle
on
whic
h
three
equilateral
triangles
are
con-
structed
externally
:
B
C
A
,
AC
B
,
and
AB
C
.
W
e
consider
the
p
oin
ts
P
,
Q
and
R
to
b
e
the
resp
ectiv
e
cen
ters
of
gra
vit
y
of
the
triangles
B
C
A
,
AC
B
and
AB
C
,
resp
ectiv
ely
:
W
e
note
a
,
b
,
c
,
a
,
b
,
c
,
p
,
q
and
r
the
resp
ectiv
e
affixes
of
p
oin
ts
A
,
B
,
C
,
A
,
B
,
C
,
P
,
Q
and
R
.
1
a
T
ranslate,
with
the
affixes
of
the
p
oin
ts
concerned,
that
C
is
the
image
of
A
b
y
a
rotation
whose
angle
and
cen
ter
will
b
e
sp
ecified.
b
Sho
w
that
:
a
+
b
+
c
=
a
+
b
+
c
.
2
Deduce
that
:
p
+
q
+
r
=
a
+
b
+
c
.
3
Deduce
that
the
triangles
AB
C
,
A
B
C
and
P
QR
ha
v
e
the
same
cen
ter
of
gra
vit
y
.
4
Sho
w
that
:
3(
q
−
p
)
=
(
b
−
c
)
+
(
c
−
a
)
+
(
a
−
b
)
.
W
e
will
admit
that,
similarly
:
3(
r
−
p
)
=
(
a
−
c
)
+
(
b
−
a
)
+
(
c
−
b
)
.
5
Justify
the
follo
wing
equalities
:
a
−
c
=
e
i
π
3
(
b
−
c
)
;
b
−
a
=
e
i
π
3
(
c
−
a
)
;
c
−
b
=
e
i
π
3
(
a
−
b
)
6
Deduce
from
questions
4
and
5
that
the
triangle
P
QR
is
equilateral.
E.3822
The
complex
plane
is
equipp
ed
with
an
orthonormal
direct
reference
frame
O
;
−
→
u
;
−
→
v
(unit
1
cm
)
.
W
e
will
dra
w
a
figure
that
w
e
will
complete
as
w
e
go
along
with
the
questions.
W
e
consider
the
p
oin
ts
A
,
B
,
S
,
and
Ω
with
resp
ectiv
e
affixes
:
a
=
−
2
+
4i
;
b
=
−
4
+
2i
s
=
−
5
+
5i
;
w
=
−
2
+
2i
Let
h
b
e
the
homothet
y
with
cen
ter
S
and
ratio
3
.
Let
C
b
e
the
image
of
p
oin
t
A
under
h
and
D
b
e
the
image
of
p
oin
t
B
under
h
.
1
a
Determine
the
complex
form
of
h
.
b
Sho
w
that
p
oin
t
C
has
affix
c
=
4+2
i
and
that
p
oin
t
D
has
affix
d
=
−
2
−
4
i
.
2
Sho
w
that
p
oin
ts
A
,
B
,
C
,
and
D
lie
on
the
same
circle,
whose
cen
ter
and
radius
will
b
e
sp
ecified.
3
Demonstrate
that
line
(
S
Ω)
is
the
p
erp
endicular
bisector
of
segmen
t
[
AB
]
.
4
Let
P
b
e
the
midp
oin
t
of
segmen
t
[
AC
]
.
a
Determine
the
affix
p
of
p
oin
t
P
.
b
Sho
w
that
!
−
p
d
−
b
=
−
1
2
·
i
.
Use
this
to
figure
out
the
mea-
sure
of
angle
−
−
→
B
D
;
−
→
P
Ω
.
5
Let
Q
b
e
the
midp
oin
t
of
segmen
t
[
B
D
]
.
What
do
es
p
oin
t
Ω
represen
t
for
triangle
P
QS
?
4.
A
nnals
on
plane
tr
ansformations
E.3175
In
the
complex
plane
referred
to
a
direct
orthonormal
reference
frame
O
;
−
→
u
;
−
→
v
(gr
aphic
unit
2
cm
)
,
consider
the
p
oin
ts
A
,
B
and
C
with
affixes
:
z
A
=
2
;
z
B
=
1
+
i
3
;
z
C
=
1
−
i
3
P
art
A
1
a
Giv
e
the
exp
onen
tial
form
of
z
B
and
then
of
z
C
.
b
Place
p
oin
ts
A
,
B
and
C
.
2
Determine
the
nature
of
the
quadrilateral
O
B
AC
.
3
Determine
and
construct
the
set
D
of
p
oin
ts
M
of
the
plane
suc
h
that
:
|
z
|
=
|
z
−
2
|
P
art
B
T
o
an
y
p
oin
t
M
of
affix
z
suc
h
that
z
=
z
A
,
w
e
asso
ciate
the
p
oin
t
M
of
affix
z
defined
b
y
:
z
=
−
4
z
−
2
1
a
Solv
e
in
C
the
equation
:
z
=
−
4
z
−
2
.
b
Deduce
the
p
oin
ts
asso
ciated
with
B
and
C
.
c
Determine
and
place
the
p
oin
t
G
asso
ciated
with
the
cen
ter
of
gra
vit
y
G
of
the
triangle
O
AB
.
2
a
Course
question
:
Prerequisite
:
the
mo
dulus
of
an
y
complex
n
um
b
er
z
,
denoted
|
z
|
,
v
erifies
|
z
|
2
=
z
·
z
where
z
is
the
conjugate
of
z
.
Sho
w
that
:
for
all
complex
n
um
b
ers
z
1
and
z
C
O
P
Y
01
2
:
⏐
⏐
z
1
×
z
2
⏐
⏐
=
⏐
⏐
z
1
⏐
⏐
×
⏐
⏐
z
2
⏐
⏐
.
for
an
y
non-zero
complex
n
um
b
er
z
:
⏐
⏐
⏐
1
z
⏐
⏐
⏐
=
1
|
z
|
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3217
sacados/3217
Antilles-Guyane Septembre 2004 5 points
A
B
C
A’
B’
C’
P
Q
R
sacados/3822
chapExoCorrec/3175
sacados/3175
b
Sho
w
that
for
an
y
complex
n
um
b
er
z
distinct
from
2
:
⏐
⏐
z
−
2
⏐
⏐
=
2
|
z
|
|
z
−
2
|
c
It
is
assumed
in
this
question
that
M
is
an
y
p
oin
t
of
D
,
where
D
is
the
set
defined
in
question
3
from
part
A
.
Sho
w
that
the
p
oin
t
M
asso
ciated
with
M
b
elongs
to
a
circle
Γ
whose
cen
ter
and
radius
will
b
e
sp
ecified.
T
race
Γ
.
E.3156
The
plane
is
referred
to
a
direct
orthonormal
co
ordinate
system
O
;
−
→
u
;
−
→
v
(gr
aphic
unit
:
1
cm
)
.
P
art
A
In
the
reference
frame
O
;
−
→
u
;
−
→
v
,
consider
the
curv
e
H
of
equation
:
y
2
−
x
2
=
16
.
1
Sho
w
that
H
is
the
union
of
t
w
o
curv
es
C
and
C
where
C
is
the
represen
tativ
e
curv
e
of
the
function
f
defined
on
R
b
y
f
(
x
)
=
x
2
+16
and
where
C
is
the
image
of
C
b
y
a
simple
transformation
to
b
e
sp
ecified.
2
a
Study
the
function
f
(limits
at
the
b
ounds
of
the
defining
set
and
dir
e
ction
of
variation)
.
b
Sho
w
that
the
straigh
t
line
of
equation
y
=
x
is
an
asymptote
of
C
.
c
Dra
w
H
in
the
reference
frame
O
;
−
→
u
;
−
→
v
.
W
e
name
A
and
B
the
p
oin
ts
on
the
curv
e
with
abscis-
sas
−
3
and
3
resp
ectiv
ely
.
Consider
the
domain
D
of
the
plane
made
up
of
the
p
oin
ts
M
(
x
;
y
)
v
erifying
:
−
3
x
3
;
x
2
+
16
y
5
Hatc
h
the
domain
D
and
express
the
area
of
D
using
an
in
tegral
that
w
e
w
on’t
attempt
to
calculate.
P
art
B
W
e
call
r
the
rotation
of
cen
ter
O
and
angle
−
ı
4
.
1
a
Giv
e
the
complex
writing
of
r
.
b
Denote
b
y
x
and
y
the
co
ordinates
of
the
p
oin
t
M
,
image
of
the
p
oin
t
M
(
x
;
y
)
in
the
plane.
Chec
k
that
:
x
=
1
2
·
(
x
+
y
)
y
=
1
2
·
(
−
x
+
y
)
Determine
the
co
ordinates
of
p
oin
ts
A
and
B
,
resp
ec-
tiv
e
images
of
A
and
B
b
y
rotation
r
.
Place
the
p
oin
ts
A
and
B
in
the
reference
frame
O
;
−
→
u
;
−
→
v
.
2
Let
H
b
e
the
h
yp
erb
ola
of
equation
:
x
·
y
=
8
.
a
T
race
H
in
the
reference
frame
O
;
−
→
u
;
−
→
v
.
b
Sho
w
that
H
is
the
image
of
H
b
y
rotation
r
.
3
Let
D
b
e
the
image
of
D
b
y
rotation
r
.
W
e
admit
that
D
is
the
set
of
p
oin
ts
M
(
x
;
y
)
of
the
plane
v
erifying
:
2
x
4
·
2
;
8
x
y
5
·
2
−
x
.
a
Chop
D
.
b
Calculate
the
area
of
D
,
expressed
in
cm
2
.
Deduce
an
appro
ximate
v
alue
to
the
nearest
10
−
3
of
the
area
of
D
.
5.
A
utes
annales
E.3901
The
plane
is
equipp
ed
with
a
di-
rect
orthonormal
reference
frame
O
;
−
→
u
;
−
→
v
.
Let
A
,
B
,
and
P
b
e
the
resp
ectiv
e
affix
p
oin
ts
:
a
=
5
+
5
·
i
;
b
=
5
−
5
·
i
;
p
=
10
Consider
a
p
oin
t
M
,
distinct
from
O
,
with
affix
z
.
W
e
denote
U
the
p
oin
t
with
affix
u
,
image
of
the
p
oin
t
M
b
y
the
rotation
R
A
with
cen
ter
A
and
angle
of
measuremen
t
−
ı
2
.
W
e
denote
T
the
affix
p
oin
t
t
,
image
of
p
oin
t
M
b
y
rotation
R
B
with
cen
ter
B
and
angle
of
measuremen
t
ı
2
.
Let
D
b
e
the
image
of
p
oin
t
M
b
y
the
symmetry
of
cen
ter
O
.
1
Demonstrate
that
the
affix
of
p
oin
t
U
is
u
=
i(10
−
z
)
;
ex-
press
in
terms
of
z
the
affix
of
p
oin
t
T
then
justify
that
quadrilateral
M
U
D
T
is
a
parallelogram
with
cen
ter
O
.
2
Determine
the
set
Γ
of
p
oin
ts
M
with
affix
z
suc
h
that
:
z
·
z
−
5
·
z
−
5
·
z
=
0
Justify
that
the
quadrilateral
O
AP
B
is
inscrib
ed
in
Γ
.
3
W
e
assume
that
p
oin
t
M
is
distinct
from
O
,
A
and
P
.
P
oin
ts
O
,
M
,
and
U
are
therefore
distinct
from
eac
h
other
in
pairs.
a
Pro
v
e
that
p
oin
ts
O
,
M
and
U
are
collinear
if,
and
only
if
:
u
z
=
u
z
b
Pro
v
e
that
p
oin
ts
O
,
M
and
U
are
collinear
if,
and
only
if,
M
b
elongs
to
Γ
.
4
Determine
the
set
of
p
oin
ts
M
on
the
plane
suc
h
that
O
M
U
is
an
isosceles
triangle
at
O
.
What
is
the
nature
of
the
quadrilateral
M
U
D
T
in
this
case?
5
Determine
the
set
of
complex
n
um
b
ers
z
suc
h
that
u
z
is
purely
imaginary
.
Deduce
the
nature
of
the
quadrilateral
M
U
D
T
in
the
case
where
M
is
a
p
oin
t
on
the
line
(
O
P
)
dev
oid
of
O
and
P
.
Finally
,
pro
v
e
that
there
is
a
unique
p
osition
of
the
p
oin
t
M
suc
h
that
M
U
D
T
is
a
square.
6.
Unclassifie
d
financial
ye
ars
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3156
sacados/3156
chapExoCorrec/3901
sacados/3901
E.4236
The
complex
plane
is
referenced
to
O
;
−
→
u
;
−
→
v
orthonormal
direct.
Consider
the
p
oin
t
I
of
affix
i
and
the
p
oin
t
A
of
affix
z
A
=
√
3
+2
·
i
1
Sho
w
that
the
p
oin
t
A
b
elongs
to
the
circle
Γ
with
cen
ter
p
oin
t
I
and
radius
2
.
2
Consider
the
rotation
r
with
cen
ter
the
p
oin
t
I
and
angle
ı
2
.
Sho
w
that
the
p
oin
t
B
image
of
the
p
oin
t
A
b
y
the
rota-
tion
r
has
affix
:
z
B
=
−
1
+
i
·
3
+
1
Justify
that
the
p
oin
t
B
b
elongs
to
the
circle
Γ
.
3
Calculate
the
affix
of
the
p
oin
t
C
,
image
of
the
p
oin
t
A
b
y
cen
tral
symmetry
of
the
p
oin
t
I
.
4
What
is
the
nature
of
the
triangle
AB
C
?
Justify
.
E.4248
The
complex
plane
is
referred
to
a
reference
frame
O
;
−
→
u
;
−
→
v
orthonormal
direct
Consider
the
p
oin
ts
A
and
B
with
resp
ectiv
e
affixes
:
a
=
i
;
b
=
1
+
i
Note
:
r
A
the
rotation
with
cen
ter
A
and
angle
ı
2
;
r
B
the
rotation
of
cen
ter
B
and
angle
ı
2
;
r
O
the
rotation
of
cen
ter
O
and
angle
−
ı
2
.
Consider
:
the
p
oin
t
C
of
affix
c
=
3
·
i
;
the
p
oin
t
D
image
of
C
b
y
r
A
;
p
oin
t
G
image
of
D
b
y
r
B
;
p
oin
t
H
image
of
C
b
y
r
0
.
Let
d
,
g
and
h
b
e
the
resp
ectiv
e
affixes
of
the
p
oin
ts
D
,
G
and
H
.
1
Demonstrate
that
d
=
−
2+i
.
2
Determine
g
and
h
.
3
Demonstrate
that
the
quadrilateral
C
D
GH
is
a
rectan-
gle.
E.3859
The
complex
plane
is
referred
to
a
reference
frame
O
;
−
→
u
;
−
→
v
orthonormé.
Let
(
C
)
the
circle
with
cen
ter
O
et
and
radius
1
.
Consider
the
p
oin
t
A
of
(
C
)
d
of
affix
:
z
A
=
e
i
·
π
3
1
Determine
the
affix
z
B
du
p
oin
t
B
image
of
A
par
the
rotation
of
cen
ter
O
et
of
angle
2
ı
3
.
Determine
the
affix
z
C
of
p
oin
t
C
image
of
B
par
the
rotation
of
cen
ter
O
et
of
angle
2
ı
3
.
2
a
Justify
that
(
C
)
est
the
circumscrib
ed
circle
of
the
triangle
AB
C
.
Construct
the
p
oin
ts
A
,
B
and
C
on
the
sheet
of
graph
pap
er.
b
What
is
the
nature
of
the
triangle
AB
C
?
Justify
.
3
So
h
the
homothet
y
of
cen
ter
O
and
ratio
−
2
.
a
Complete
the
figure
b
y
placing
the
p
oin
ts
P
,
Q
et
R
images
resp
ectiv
e
of
the
p
oin
ts
A
,
B
et
C
par
h
.
b
What
is
the
nature
of
the
triangle
P
QR
?
Justify
.
4
In
this
question,
the
candidate
is
in
vited
to
write
on
his
cop
y
the
steps
of
his
approac
h
ev
en
if
it
do
es
not
succeed
.
a
Giv
e
the
complex
writing
of
h
.
b
Calculer
z
A
+
z
B
+
z
C
.
Deduce
that
A
is
the
middle
of
segmen
t
[
QR
]
.
c
What
can
b
e
said
of
the
straigh
t
line
(
QR
)
in
relation
to
the
circle
(
C
)
?
h
ttps:/
/c
hingmath.fr
sacados/4236
sacados/4248
Extrait Amerique du Nord Mai 2011
chapExoCorrec/3859
sacados/3859
E.3968
P
art
1
:
Organized
restitution
of
kno
wledge
The
complex
plane
is
referred
to
a
O
;
−
→
u
;
−
→
v
orthonormé
direct
reference
frame.
Prerequisites
:
Remem
b
er
that
the
complex
writing
of
a
direct
similitude
in
the
plane
is
of
the
form
z
=
¸
·
z
+
˛
,
where
¸
est
is
a
non-zero
complex
n
um
b
er
and
˛
est
is
a
complex
n
um
b
er.
Let
A
,
B
,
C
,
D
quatre
p
oin
ts
of
the
plane
;
on
the
one
hand,
assume
that
the
p
oin
ts
A
et
C
son
t
distinct
and,
on
the
other
hand,
that
the
p
oin
ts
B
et
D
son
t
distinct.
Sho
w
that
there
is
a
single
direct
similitude
suc
h
that
:
s
(
A
)
=
B
;
s
(
C
)
=
D
P
art
2
:
The
complex
plane
is
referred
to
the
reference
frame
A
;
−
−
→
AB
;
−
−
→
AD
orthonormal
direct
suc
h
that
:
−
−
→
AB
;
−
−
→
AD
=
ı
2
[2
·
ı
]
Consider
p
oin
t
C
suc
h
that
AB
C
D
is
a
square.
Let
E
b
e
the
midp
oin
t
of
segmen
t
[
AD
]
,
w
e
consider
the
square
E
D
GF
suc
h
that
:
−
−
→
E
D
;
−
−
→
E
F
=
ı
2
[2
·
ı
]
.
1
a
Dra
w
a
figure
b
y
placing
the
p
oin
ts
A
,
B
,
C
,
D
,
E
,
F
,
G
.
The
figure
will
b
e
completed
during
the
exercise.
b
Sp
ecify
the
complex
n
um
b
ers
a
,
b
,
c
,
d
,
e
,
f
,
g
,
resp
ec-
tiv
e
affixes
of
p
oin
ts
A
,
B
,
C
,
D
,
E
,
F
and
G
.
c
Sho
w
that
there
is
a
unique
direct
similarit
y
s
of
the
plane
suc
h
that
:
s
(
D
)
=
F
;
s
(
B
)
=
D
2
W
e
prop
ose
to
sp
ecify
the
c
haracteristic
elemen
ts
of
di-
rect
similarit
y
s
.
a
Determine
the
ratio
k
and
angle
„
of
the
direct
simi-
larit
y
s
.
b
Giv
e
the
complex
form
of
this
similarit
y
.
c
Determine
the
cen
ter
Ω
of
the
direct
similarit
y
s
.
E.3128
The
plane
is
referenced
to
O
;
−
→
u
;
−
→
v
direct
orthonormal
(gr
aphic
unit
2
cm
)
.
The
figure
will
b
e
completed
as
the
exercise
progresses.
Let
I
b
e
the
p
oin
t
with
affix
2i
.
W
e
call
f
the
transformation
whic
h,
to
an
y
p
oin
t
M
of
affix
z
asso
ciates
the
p
oin
t
M
of
affix
z
suc
h
that
z
=
i
·
z
.
1
a
Sp
ecify
the
nature
of
f
and
its
c
haracteristic
ele-
men
ts.
b
Determine
the
affix
of
the
p
oin
t
A
,
image
b
y
f
of
the
p
oin
t
A
of
affix
1+
2
+i
.
c
Sho
w
that
the
p
oin
ts
A
,
I
and
A
are
aligned.
2
a
Sho
w
that
the
set
(Γ)
of
p
oin
ts
M
of
the
plane
suc
h
that
M
,
I
and
M
are
aligned,
is
the
circle
with
cen
ter
Ω
of
affix
1+i
and
radius
2
.
b
V
erify
that
the
p
oin
t
A
b
elongs
to
(Γ)
.
c
Determine
the
set
(Γ
)
describ
ed
b
y
the
p
oin
t
M
when
the
p
oin
t
M
describ
es
(Γ)
.
3
Let
B
b
e
the
p
oin
t
with
affix
2+2i
and
B
b
e
the
image
of
B
b
y
f
.
a
Demonstrate
that
the
straigh
t
lines
(
AB
)
and
(
A
B
)
are
p
erp
endicular.
b
Let
C
b
e
the
p
oin
t
of
in
tersection
of
the
lines
(
AB
)
and
(
A
B
)
.
Determine
the
nature
of
the
quadrilateral
O
AC
A
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3968
sacados/3968
chapExoCorrec/3128
sacados/3128
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