a
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School
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Ressource
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High
School
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Ressource
Out High
4
3
2
1
Outside the high school program
/ Surfaces
18 exercises (including 16 corrected)
a
Usual surface area
(2 exercices)
Intersection of a surface and a plane
(8 exercices)
Volume
(1 exercice)
Annales - Study of surfaces
(1 exercice)
Annales - Surfaces and arithmetic
(5 exercices)
Annales - Volume
(2 exercices)
A
B
C
D
E
F
G
H
−
i
−
j
−
k
y
-6
-4
-2
A
2
4
6
8
z
-4
-2
2
4
6
8
10
12
14
j
k
1.
Usual
surfac
e
ar
e
a
E.4120
In
space,
consider
the
AB
C
D
E
F
GH
cub
e
sho
wn
b
elo
w
with
cen
ter
O
.
The
plane
is
giv
en
the
orthonormal
reference
frame
C
;
−
−
→
C
B
;
−
−
→
C
D
;
−
−
→
C
G
Determine
the
equations
of
the
follo
wing
planes
:
a
(
B
C
D
)
b
(
GH
D
)
c
(
F
E
H
)
d
(
AB
F
)
e
(
AB
G
)
f
(
B
D
G
)
E.4121
The
space
is
giv
en
a
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
Determine
the
Cartesian
equations
of
the
follo
wing
surfaces
:
1
The
cylinder
of
axis
O
;
−
→
j
passing
through
the
p
oin
t
A
0
;
0
;
3
.
2
The
cylinder
of
axis
O
;
−
→
k
passing
through
the
p
oin
t
B
3
;
3
;
0
.
3
The
cone
with
v
ertex
O
and
axis
of
rev
olution
O
;
−
→
i
passing
through
the
p
oin
t
C
3
;
2
;
2
.
4
The
cylinder
of
axis
O
;
−
→
i
and
whose
section
with
the
plane
of
equation
x
=
−
3
is
a
circle
of
radius
4
.
5
The
cone
with
v
ertex
O
and
axis
of
rev
olution
O
;
−
→
k
whose
section
with
the
plane
of
equation
z
=
2
is
a
circle
of
radius
2
.
2.
Interse
ction
of
a
surfac
e
and
a
plane
E.4098
Space
is
pro
vided
with
an
or-
thonormal
reference
frame
O
;
−
→
i
;
−
→
j
.
Consider
the
surfaces
S
1
and
S
2
of
resp
ectiv
e
equations
:
z
=
x
2
+
y
2
;
z
=
x
·
y
+
2
x
Note
P
the
plane
of
equation
x
=
2
,
E
1
the
in
tersection
of
surface
S
1
and
plane
P
and
E
2
the
in
tersection
of
surface
S
2
and
plane
P
.
In
app
endix
,
the
P
plane
is
represen
ted
fitted
with
the
A
;
−
→
j
;
−
→
k
where
A
is
the
p
oin
t
with
co
ordinates
2
;
0
;
0
.
1
a
Determine
the
nature
of
the
set
E
1
.
b
Determine
the
nature
of
the
set
E
2
.
2
a
Represen
t
the
sets
E
1
and
E
2
on
the
attac
hed
sheet.
b
In
the
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
giv
e
the
co
ordi-
nates
of
the
in
tersection
p
oin
ts
B
and
C
of
the
sets
E
1
and
E
2
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4120
sacados/4120
A
B
C
D
E
F
G
H
−
i
−
j
−
k
chapExoCorrec/4121
sacados/4121
chapExoCorrec/4098
sacados/4098
Extrait d'Antilles-Guyane Septembre 2009
y
-6
-4
-2
A
2
4
6
8
z
-4
-2
2
4
6
8
10
12
14
j
k
Figure n
o
1
Figure n
o
2
Figure n
o
3
Figure n
o
4
O
−
i
−
j
−
k
E.4114
In
the
space
equipp
ed
with
an
orthonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
,
w
e
consider
the
p
oin
ts
:
A
1
;
3
;
2
;
B
4
;
6
;
−
4
and
the
cone
(Γ)
with
axis
O
;
−
→
k
,
with
v
ertex
O
and
con-
taining
the
p
oin
t
A
.
1
Sho
w
that
an
equation
of
(Γ)
is
:
x
2
+
y
2
=
5
2
·
z
2
2
Let
(
P
)
b
e
the
plane
parallel
to
the
plane
(
xO
y
)
and
con
taining
the
p
oin
t
B
.
a
Determine
an
equation
of
(
P
)
.
b
Sp
ecify
the
nature
of
the
in
tersection
(
C
1
)
of
(
P
)
and
(Γ)
.
E.4117
In
space
pro
vided
with
the
orthonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
,
note
S
the
surface
of
equa-
tion
:
z
=
x
2
+
2
·
x
+
y
2
+
1
Sa
y
if,
y
es
or
no,
ˇ
the
section
of
S
with
the
plane
of
equa-
tion
z
=
5
is
a
circle
of
cen
ter
A
of
co
ordinates
−
1
;
0
;
5
and
radius
5
ı.
E.4138
Space
is
pro
vided
with
an
or-
thonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
Surface
study
S
equation
z
=
−
1
4
·
x
·
y
1
W
e
cut
S
through
the
plane
(
xO
y
)
.
Determine
the
sec-
tion
obtained.
2
W
e
cut
S
b
y
a
plane
P
parallel
to
the
plane
(
xO
y
)
.
What
is
the
nature
of
the
section
obtained?
3
In
this
question,
an
y
trace
of
researc
h,
ho
w
ev
er
incom-
plete,
or
initiativ
e,
ho
w
ev
er
unsuccessful,
will
b
e
tak
en
in
to
accoun
t
in
the
assessmen
t.
W
e
cut
S
b
y
the
plane
with
equation
x
+
y
=
0
.
What
is
the
nature
of
the
resulting
cross-section?
E.4135
In
the
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
,
S
is
the
surface
of
equation
:
4
·
z
=
x
·
y
.
The
follo
wing
figures
sho
w
the
in
tersections
of
S
with
certain
planes
of
space.
1
S
1
denotes
the
section
of
the
surface
S
through
the
plane
(
xO
y
)
.
One
of
the
figures
giv
en
represen
ts
S
1
whic
h
one?
2
S
2
designates
the
section
of
S
b
y
the
plane
R
of
equation
z
=
1
.
One
of
the
giv
en
figures
represen
ts
S
2
,
whic
h
one?
3
S
3
denotes
the
section
of
S
through
the
plane
of
equation
y
=
8
.
One
of
the
figures
giv
en
represen
ts
S
3
,
whic
h
one?
E.4136
F
or
eac
h
of
the
follo
wing
questions,
indicate
whether
the
prop
osed
statemen
t
is
true
or
false
and
justify
the
answ
er.
Space
is
pro
vided
with
an
orthonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
1
E
is
the
set
of
p
oin
ts
M
space
whose
co
ordinates
(
x
;
y
;
z
)
v
erify
the
equation
:
z
=
x
2
+
y
2
.
Note
S
the
section
of
E
b
y
the
plane
of
equation
y
=
3
.
Assertion
:
S
is
a
circle.
2
P
is
the
surface
of
equation
x
2
+
y
2
=
3
·
z
2
Assertion
:
O
is
the
only
p
oin
t
of
in
tersection
of
P
with
the
plane
(
y
O
z
)
with
in
teger
co
ordinates.
E.4311
Space
is
pro
vided
with
an
orthonor-
mal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
Consider
the
surface
S
,
one
equation
of
whic
h
is
:
z
=
4
·
x
·
y
Sho
w
that
the
section
of
the
surface
S
through
the
plane
of
equation
z
=
0
is
the
union
of
t
w
o
orthogonal
straigh
t
lines.
3.
V
olume
E.4140
F
or
eac
h
of
the
follo
wing
t
w
o
prop
ositions,
indicate
whether
it
is
true
or
false
and
giv
e
a
demonstration
of
the
c
hosen
answ
er.
An
unpro
v
en
answ
er
scores
no
p
oin
ts.
Space
is
referred
to
an
orthonormal
refer-
ence
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
The
surface
Σ
opp
osite
has
the
equation
:
z
=
x
2
+
y
2
1
The
section
of
the
surface
Σ
and
the
plane
of
equation
x
=
–
,
where
–
is
a
real,
is
a
h
yp
erb
ola.
2
The
plane
of
equation
z
=
9
·
2
2
di-
vides
the
solid
b
ounded
b
y
Σ
and
the
plane
of
equation
z
=
9
in
to
t
w
o
solids
of
equal
v
olume.
Reminder.
Let
V
b
e
the
v
olume
of
the
solid
b
ounded
b
y
Σ
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4114
sacados/4114
chapExoCorrec/4117
sacados/4117
chapExoCorrec/4138
sacados/4138
Extrait d'Ameriques du Sud Novembre 2008
chapExoCorrec/4135
sacados/4135
Extrait Centres Etrangers Juin 2009
Figure n
o
1
Figure n
o
2
Figure n
o
3
Figure n
o
4
chapExoCorrec/4136
sacados/4136
Extrait d'Antilles-Guyanes Juin 2009
sacados/4311
chapExoCorrec/4140
sacados/4140
Extrait de Liban Mai 2008
O
−
i
−
j
−
k
Figure n
o
1
Figure n
o
2
Figure n
o
3
and
the
planes
of
equations
z
=
a
and
z
=
b
where
0
a
b
9
.
V
is
giv
en
b
y
the
form
ula
V
=
b
a
S
(
k
)
d
k
where
S
(
k
)
is
the
cross-sectional
area
of
the
solid
through
the
plane
of
equation
z
=
k
where
k
∈
a
;
b
.
4.
A
nnales
-
Study
of
surfac
es
E.4143
The
space
(
E
)
is
equipp
ed
with
an
orthonormal
basis
O
;
−
→
i
;
−
→
j
;
−
→
k
.
Consider
the
p
oin
ts
:
A
0
;
5
;
5
;
B
0
;
0
;
10
1
In
this
question,
w
e
are
w
orking
in
the
plane
P
0
with
equation
x
=
0
relativ
e
to
the
co
ordinate
system
O
;
−
→
i
;
−
→
j
.
Let
C
b
e
the
circle
with
cen
ter
B
and
passing
through
A
.
Pro
v
e
that
the
line
(
O
A
)
is
tangen
t
to
the
circle
C
.
2
W
e
call
S
the
sphere
generated
b
y
the
rotation
of
the
circle
C
around
the
axis
(
O
z
)
and
Γ
the
cone
generated
b
y
the
rotation
of
the
line
(
O
A
)
around
the
axis
(
O
z
)
.
a
Pro
v
e
that
the
cone
Γ
has
the
equation
:
x
2
+
y
2
=
z
2
b
Determine
the
in
tersection
of
the
cone
Γ
and
the
sphere
S
.
Sp
ecify
the
nature
of
this
in
tersection
and
its
c
harac-
teristic
elemen
ts.
c
Illustrate
these
ob
jects
with
a
diagram
in
space.
3
W
e
cut
the
cone
Γ
b
y
the
plane
P
1
with
equation
x
=
1
.
In
P
1
,
one
of
the
three
figures
b
elo
w
represen
ts
this
in-
tersection.
Iden
tify
this
figure
and
pro
vide
the
necessary
justifica-
tion.
4
Let
M
(
x
;
y
;
z
)
b
e
a
p
oin
t
on
the
cone
Γ
whose
co
ordi-
nates
are
non-zero
relativ
e
in
tegers.
Pro
v
e
that
x
and
y
cannot
b
oth
b
e
o
dd.
5.
A
nnales
-
Surfac
es
and
arithmetic
E.4087
P
art
A
Consider,
in
a
co
ordinate
system
O
;
−
→
i
;
−
→
j
;
−
→
k
in
space,
the
surface
S
with
equation
:
z
=
x
−
y
)
2
1
W
e
denote
E
1
the
in
tersection
of
S
with
the
plane
P
1
of
equation
z
=
0
.
Determine
the
nature
of
E
1
.
2
W
e
note
E
2
the
in
tersection
of
S
with
the
plane
P
2
of
equation
x
=
1
.
Determine
the
nature
of
E
2
.
P
art
B
Consider,
in
a
co
ordinate
system
O
;
−
→
i
;
−
→
j
;
−
→
k
in
space,
the
surface
S
with
equation
:
z
=
x
·
y
1
W
e
note
E
3
the
in
tersection
of
S
with
the
plane
P
1
of
equation
z
=
0
.
Determine
the
nature
of
E
3
.
2
W
e
note
E
4
the
in
tersection
of
S
with
the
plane
P
3
of
equation
z
=
1
.
Determine
the
nature
of
E
4
.
P
art
C
Let
E
5
b
e
the
in
tersection
of
S
and
S
.
In
this
section,
w
e
w
an
t
to
sho
w
that
the
only
p
oin
t
b
elong-
ing
to
E
5
whose
co
ordinates
are
natural
n
um
b
ers
is
the
p
oin
t
O
0
;
0
;
0
.
W
e
assume
that
there
is
a
p
oin
t
M
b
elonging
to
E
5
and
whose
co
ordinates
x
,
y
and
z
are
natural
in
tegers.
1
Sho
w
that
if
x
=
0
,
then
the
p
oin
t
M
is
the
p
oin
t
O
2
W
e
no
w
assume
that
the
in
teger
x
,
y
and
z
v
érifien
t
:
x
2
−
3
·
x
·
y
+
y
2
=
0
.
Deduce
that
there
then
exist
natural
n
um
b
ers
x
and
y
premiers
b
et
w
een
them
suc
h
that
:
x
2
−
3
·
x
·
y
+
y
2
=
0
3
Sho
w
that
x
divides
y
2
,
then,
that
x
divise
y
.
4
Establish
that
y
v
erify
the
relationship
:
1
−
3
·
y
+
y
2
=
0
5
Conclude.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4143
sacados/4143
Pondichery Avril 2004
Figure n
o
1
Figure n
o
2
Figure n
o
3
chapExoCorrec/4087
sacados/4087
Pondichery Avril 2011 5 points
E.4132
The
space
is
equipp
ed
with
an
orthonor-
mal
basis
O
;
−
→
i
;
−
→
j
;
−
→
j
.
W
e
consider
the
surface
S
1
of
equation
z
=
x
2
+
y
2
and
the
surface
S
2
of
equation
z
=
x
·
y
+2
·
x
P
art
A
Let
P
b
e
the
plane
of
equation
x
=
2
,
E
1
the
in
tersection
of
the
surface
S
1
and
the
plane
P
and
E
2
the
in
tersection
of
the
surface
S
2
and
the
plane
P
.
In
the
app
endix,
the
plane
P
is
mark
ed
with
the
reference
A
;
−
→
i
;
−
→
j
where
A
is
the
p
oin
t
with
co
ordinates
2
;
0
;
0
.
1
a
Determine
the
nature
of
the
set
E
1
.
b
Determine
the
nature
of
the
set
E
2
.
2
a
Represen
t
the
sets
E
1
and
E
2
on
the
sheet
ap-
p
endix
.
b
In
the
reference
p
oin
t
O
;
−
→
i
;
−
→
j
;
−
→
k
giv
e
the
co
ordi-
nates
of
the
p
oin
ts
of
in
tersection
B
and
C
of
the
sets
E
1
and
E
2
.
P
art
B
The
follo
wing
prop
ert
y
can
b
e
used
without
pro
of
:
ˇLet
a
,
b
and
c
b
e
in
tegers
with
a
prime.
If
a
divides
b
·
c
then
a
divides
b
or
a
divides
c
.
ı
The
ob
jectiv
e
of
this
section
is
to
determine
the
p
oin
ts
of
in-
tersection
M
(
x
;
y
;
z
)
of
the
surfaces
S
1
and
S
2
where
y
and
z
are
relativ
e
in
tegers
and
x
is
a
prime
in
teger.
W
e
consider
suc
h
a
p
oin
t
M
(
x
;
y
;
z
)
.
1
a
Sho
w
that
:
y
(
y
−
x
)
=
x
(2
−
x
)
.
b
Deduce
that
the
prime
n
um
b
er
x
divides
y
.
2
W
e
set
y
=
k
·
x
with
k
∈
Z
.
a
Sho
w
that
x
divides
2
,
then
that
x
=
2
.
b
Deduce
the
p
ossible
v
alues
of
k
.
3
Determine
the
p
ossible
co
ordinates
of
M
and
compare
the
results
with
those
in
part
A
question
2
b
.
E.4137
The
space
is
equipp
ed
with
an
orthonormal
basis
O
;
−
→
i
;
−
→
j
;
−
→
k
.
1
Let
F
b
e
the
p
oin
t
with
co
ordinates
0
;
0
;
1
4
and
P
b
e
the
plane
with
equation
z
=
−
1
4
.
Let
d
(
M
;P
)
b
e
the
distance
from
a
p
oin
t
M
to
the
plane
P
.
Sho
w
that
the
set
(
S
)
of
p
oin
ts
M
with
co
ordinates
(
x
;
y
;
z
)
that
satisfy
d
(
M
;
P
)
=
M
F
a
has
the
equation
:
x
2
+
y
2
=
z
.
2
a
What
is
the
nature
of
the
in
tersection
of
the
set
(
S
)
with
the
plane
of
equation
z
=
2
?
b
What
is
the
nature
of
the
in
tersection
of
the
set
(
S
)
with
the
plane
of
equation
x
=
0
?
Represen
t
this
in
tersection
in
the
co
ordinate
system
O
;
−
→
j
;
−
→
k
3
In
this
question,
x
and
y
denote
natural
n
um
b
ers.
a
What
are
the
p
ossible
remainders
of
the
Euclidean
di-
vision
of
x
2
b
y
7
?
b
Pro
v
e
that
7
divides
x
2
+
y
2
if,
and
only
if,
7
divides
x
and
7
divides
y
.
4
In
this
question,
an
y
attempt
at
researc
h,
ev
en
if
incom-
plete,
or
an
y
initiativ
e,
ev
en
if
unsuccessful,
will
b
e
tak
en
in
to
accoun
t
in
the
assessmen
t.
Are
there
an
y
p
oin
ts
that
b
elong
to
the
in
tersection
of
the
set
(
S
)
and
the
plane
of
equation
z
=
98
and
whose
co
ordinates
are
all
natural
n
um
b
ers?
If
so,
determine
them.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4132
sacados/4132
chapExoCorrec/4137
sacados/4137
La R&union Juin 2009
E.4141
In
the
space
equipp
ed
with
an
orthonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
j
,
w
e
consider
the
p
oin
ts
A
1
;
3
;
2
and
B
4
;
6
;
−
4
and
the
cone
(Γ)
with
axis
O
;
−
→
k
,
with
v
ertex
O
and
con
taining
the
p
oin
t
A
.
P
art
A
1
Sho
w
that
an
equation
of
(Γ)
is
:
x
2
+
y
2
=
5
2
·
z
2
2
Let
(
P
)
b
e
the
plane
parallel
to
the
plane
(
xO
y
)
and
con
taining
the
p
oin
t
B
,
a
Determine
an
equation
of
(
P
)
.
b
Sp
ecify
the
nature
of
the
in
tersection
(
C
1
)
of
(
P
)
and
(Γ)
.
3
Let
(
Q
)
b
e
the
equation
plane
y
=
3
.
W
e
note
(
C
2
)
the
in
tersection
of
(Γ)
and
(
Q
)
.
Without
justification,
iden-
tify
the
nature
of
(
C
2
)
from
among
the
follo
wing
prop
o-
sitions
:
a
t
w
o
parallel
lines
;
b
t
w
o
in
tersecting
lines
;
c
a
parab
ola
;
d
a
h
yp
erb
ola
;
e
a
circle.
P
art
B
Let
x
,
y
and
z
b
e
three
relativ
e
in
tegers
and
M
the
p
oin
t
with
co
ordinates
(
x
;
y
;
z
)
.
The
sets
(
C
1
)
and
(
C
2
)
are
the
sections
defined
in
part
A
.
1
Consider
the
equation
(
E
)
:
x
2
+
y
2
=
40
where
x
and
y
are
relativ
e
in
tegers.
a
Solv
e
the
equation
(
E
)
.
b
Deduce
all
p
oin
ts
of
(
C
1
)
whose
co
ordinates
are
rela-
tiv
e
in
tegers.
2
a
Pro
v
e
that
if
the
p
oin
t
M
with
co
ordinates
(
x
;
y
;
z
)
where
x
,
y
and
z
denote
relativ
e
in
tegers,
is
a
p
oin
t
of
(Γ)
then
z
is
divisible
b
y
2
and
x
2
+
y
2
is
divisible
b
y
10
.
b
Sho
w
that
if
M
is
a
p
oin
t
of
(
C
2
)
,
in
tersection
of
(Γ)
and
(
Q
)
,
then
:
x
2
≡
1
(
mo
d.
10)
c
Determine
a
p
oin
t
(
C
2
)
,
distinct
from
A
,
whose
co
or-
dinates
are
relativ
e
in
tegers.
E.4139
The
space
is
referenced
to
the
orthonormal
co
ordinate
system
O
;
−
→
i
;
−
→
j
;
−
→
k
.
W
e
call
(
S
)
the
surface
with
equation
:
x
2
+
y
2
−
z
2
=
1
1
Sho
w
that
the
surface
(
S
)
is
symmetrical
with
resp
ect
to
the
plane
xO
y
.
2
W
e
call
A
and
B
the
p
oin
ts
with
the
resp
ectiv
e
co
ordi-
nates
:
3
;
1
;
−
3
;
−
1
;
1
;
1
a
Determine
a
parametric
represen
tation
of
the
line
(
D
)
passing
through
the
p
oin
ts
A
and
B
.
b
Pro
v
e
that
the
line
(
D
)
is
included
in
the
surface
(
S
)
.
3
Determine
the
nature
of
the
section
of
the
surface
(
S
)
b
y
a
plane
parallel
to
the
plane
(
xO
y
)
.
4
a
Consider
the
curv
e
(
C
)
,
the
in
tersection
of
the
sur-
face
(
S
)
and
the
plane
with
equation
z
=
68
.
Sp
ecify
the
c
haracteristic
elemen
ts
of
this
curv
e.
b
M
b
eing
a
p
oin
t
of
(
C
)
,
w
e
designate
a
as
its
abscissa
and
b
as
its
ordinate.
W
e
prop
ose
to
sho
w
that
there
is
only
one
p
oin
t
M
of
(
C
)
suc
h
that
a
and
b
are
natural
in
tegers
satisfying
:
a
<
b
;
ppcm
(
a
;
b
)
=
440
.
That
is,
suc
h
that
(
a
;
b
)
is
a
solution
to
the
system
:
(1)
:
a
<
b
a
2
+
b
2
=
4625
ppcm
(
a
;
b
)
=
440
Sho
w
that
if
(
a
;
b
)
is
a
solution
of
(1)
then
pg
cd
(
a
;
b
)
is
equal
to
1
or
to
5
.
Conclude.
In
this
question
an
y
trace
of
researc
h,
ev
en
if
incom-
plete,
or
initiativ
e,
ev
en
if
unsuccessful,
will
b
e
tak
en
in
to
accoun
t
in
the
assessmen
t.
6.
A
nnales
-
V
olume
E.4199
The
space
(
E
)
is
equipp
ed
with
an
orthonormal
basis
O
;
−
→
i
;
−
→
j
.
W
e
consider
the
surface
T
with
equation
:
x
2
·
y
=
z
with
−
1
x
1
and
−
1
y
1
.
The
figure
opp
osite
is
a
represen
tation
of
the
surface
T
in
the
cub
e
with
cen
ter
O
and
side
length
2
.
1
Elemen
ts
of
symmetry
of
the
surface
T
.
a
Sho
w
that
if
p
oin
t
M
(
x
;
y
;
z
)
b
elongs
to
T
,
then
p
oin
t
M
(
−
x
;
y
;
z
)
also
b
elongs
to
T
.
Deduce
a
plane
of
symmetry
of
T
.
b
Sho
w
that
the
origin
O
of
the
co
ordinate
system
is
the
cen
ter
of
symmetry
of
T
.
2
In
tersections
of
the
surface
T
with
planes
parallel
to
the
axes.
a
Determine
the
nature
of
the
in
tersection
curv
es
of
T
with
planes
parallel
to
the
plane
(
xO
z
)
.
b
Determine
the
nature
of
the
in
tersection
curv
es
of
T
with
planes
parallel
to
the
plane
(
y
O
z
)
.
3
In
tersections
of
the
surface
T
with
planes
parallel
to
the
plane
(
xO
y
)
of
equations
z
=
k
,
a
v
ec
k
∈
0
;
1
.
a
Determine
the
in
tersection
of
the
surface
T
and
the
plane
of
equation
z
=
0
.
b
F
or
k
>
0
,
w
e
note
K
the
p
oin
t
with
co
ordinates
0
;
0
;
k
.
Determine,
in
the
co
ordinate
system
K
;
−
→
i
;
−
→
j
,
the
equation
of
the
in
tersection
curv
e
of
T
and
the
plane
with
equation
z
=
k
.
c
Dra
w
the
shap
e
of
this
curv
e
in
the
co
ordinate
system
K
;
−
→
i
;
−
→
j
.
Sp
ecify
in
particular
the
co
ordinates
of
the
endp
oin
ts
of
the
arc.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4141
sacados/4141
Polynesie Juin 2007
chapExoCorrec/4139
sacados/4139
Ameriques du Nord Mai 2008
chapExoCorrec/4199
sacados/4199
4
W
e
denote
(
D
)
the
domain
formed
b
y
the
p
oin
ts
of
the
unit
cub
e
lo
cated
b
elo
w
the
surface
T
:
(
D
)
=
M
(
x
;
y
;
z
)
∈
(
E
)
⏐
⏐
⏐
0
x
1
;
0
y
1
;
0
z
x
2
·
y
a
F
or
0
<
k
1
,
the
plane
of
equation
z
=
k
coup
e
the
do-
main
(
D
)
selon
a
surface
that
can
b
e
visualized
on
the
graph
in
question
3
c
.
This
is
the
set
of
p
oin
ts
M
of
the
unit
cub
e
with
co
or-
dinates
(
x
;
y
;
z
)
suc
h
as
:
y
k
x
2
et
z
=
k
.
Calculate
as
a
function
of
k
the
area
S
(
k
)
expressed
in
units
of
area
of
this
surface.
b
W
e
p
ose
S
(0)
=
1
;
calculate
in
units
of
v
olume,
the
v
olume
V
du
domain
(
D
)
.
Recall
that
:
V
=
1
0
S
(
k
)
dk
.
E.4319
P
art
A
:
Organized
kno
wledge
retriev
al
1
Prerequisite
:
an
y
in
teger
n
strictly
greater
than
1
has
at
least
one
prime
divisor.
Pro
v
e
that
ev
ery
in
teger
n
strictly
greater
than
1
is
prime
or
can
b
e
decomp
osed
in
to
a
pro
duct
of
prime
factors
(we
do
not
ask
you
to
pr
ove
the
uniqueness
of
this
de
c
omp
o-
sition)
.
2
Giv
e
the
decomp
osition
in
to
prime
factors
of
629
.
P
art
B
In
an
orthonormal
co
ordinate
system
O
;
−
→
i
;
−
→
j
;
−
→
k
,
w
e
consider
the
surfaces
Γ
and
C
with
resp
ectiv
e
equations
:
Γ
:
z
=
x
·
y
;
C
:
x
2
+
z
2
=
1
1
Giv
e
the
nature
of
the
surface
C
and
determine
its
c
har-
acteristic
elemen
ts.
2
P
oin
ts
of
in
tersection
with
in
teger
co
ordinates
of
surfaces
Γ
and
C
.
a
Sho
w
that
the
co
ordinates
(
x
;
y
;
z
)
of
the
in
tersection
p
oin
ts
of
Γ
and
C
are
suc
h
that
:
x
2
·
1
+
y
2
=
1
b
Deduce
that
Γ
and
C
ha
v
e
t
w
o
p
oin
ts
of
in
tersection
whose
co
ordinates
are
relativ
e
in
tegers.
3
P
oin
ts
of
in
tersection
with
in
teger
co
ordinates
of
Γ
and
a
plane.
F
or
an
y
non-zero
natural
n
um
b
er
n
,
w
e
denote
b
y
P
n
the
plane
of
equation
z
=
n
4
+4
a
Determine
the
set
of
in
tersection
p
oin
ts
of
Γ
and
the
plane
P
1
whose
co
ordinates
are
relativ
e
in
tegers.
for
the
rest
of
the
exercise,
w
e
assume
n
2
.
b
Chec
k
that
:
n
2
−
2
·
n
+
2
n
2
+
2
·
n
+
2
=
n
4
+
4
c
Pro
v
e
that,
regardless
of
the
natural
n
um
b
er
n
2
,
the
in
teger
n
4
+4
is
not
prime.
d
Deduce
that
the
n
um
b
er
of
p
oin
ts
of
in
tersection
of
Γ
and
the
plane
P
n
whose
co
ordinates
are
relativ
e
in
te-
gers
is
greater
than
or
equal
to
8
.
e
Determine
the
p
oin
ts
of
in
tersection
of
Γ
and
the
plane
P
5
whose
co
ordinates
are
relativ
e
in
tegers.
h
ttps:/
/c
hingmath.fr
sacados/4319
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