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Grade 12
/ Space annals
23 exercises (including 21 corrected)
a
Lines and planes
(7 exercices)
Distance, volume in space
(4 exercices)
Projected orthogonal
(5 exercices)
Qcm, affirmations...
(5 exercices)
A
B
C
D
E
F
G
H
I
J
K
L
−
n
O
A
B
C
Vue en perspective cavalière de la réπexion
d’un rayon lumineux sur le plan
(
OAB
1.
Lines
and
planes
E.3260
The
space
is
referenced
to
an
orthonormal
co
ordinate
system
O
;
−
→
i
;
−
→
j
;
−
→
k
.
Let
t
b
e
a
real
n
um
b
er.
Giv
en
the
p
oin
ts
A
8
;
0
;
8
,
B
10
;
3
;
10
,
and
the
line
D
with
parametric
equations
:
x
=
−
5
+
3
·
t
y
=
1
+
2
·
t
z
=
−
2
·
t
where
t
∈
R
1
a
Giv
e
a
system
of
parametric
equations
for
the
line
Δ
defined
b
y
A
and
B
.
b
Pro
v
e
that
D
and
Δ
are
not
coplanar.
2
a
The
plane
P
is
parallel
to
D
and
con
tains
Δ
.
Sho
w
that
the
v
ector
−
→
n
2
;
−
2
;
1
is
a
normal
v
ector
to
P
.
Determine
a
Cartesian
equation
of
P
.
b
Sho
w
that
the
distance
from
an
y
p
oin
t
M
on
D
to
P
is
indep
enden
t
of
M
.
(not
part
of
the
2012
curriculum)
c
Giv
e
a
system
of
parametric
equations
for
the
line
de-
fined
b
y
the
in
tersection
of
P
with
the
plane
xO
y
)
.
3
The
sphere
S
is
tangen
t
to
P
at
the
p
oin
t
C
10
;
1
;
6
.
The
cen
ter
Ω
of
S
is
at
a
distance
d
=
6
from
P
,
on
the
same
side
as
O
.
Giv
e
the
Cartesian
equation
of
S
.
E.6427
AB
C
D
E
F
GH
is
a
cub
e.
I
is
the
middle
of
segmen
t
[
AB
]
,
J
is
the
middle
of
segmen
t
[
E
H
]
,
K
is
the
middle
of
seg-
men
t
[
B
C
]
and
L
is
the
middle
of
segmen
t
[
C
G
]
.
Let’s
equip
the
space
with
the
orthonormal
reference
frame
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
.
1
a
Sho
w
that
the
straigh
t
line
(
F
D
)
is
orthogonal
to
the
plane
(
I
J
K
)
.
b
Deduce
a
standard
form
of
the
plane
(
I
J
K
)
.
2
Determine
a
parametric
represen
tation
of
the
line
(
F
D
)
.
3
Let
M
b
e
the
p
oin
t
of
in
tersection
of
the
line
(
F
D
)
and
the
plane
(
I
J
K
)
.
Determine
the
co
ordinates
of
the
p
oin
t
M
.
4
Determine
the
nature
of
the
triangle
I
J
K
and
calculate
its
area.
5
Calculate
the
v
olume
of
the
tetrahedron
F
I
J
K
.
6
Do
the
straigh
t
lines
(
I
J
)
and
(
K
L
)
in
tersect?
E.6881
A
retro-reflector
is
an
optical
de-
vice
formed
b
y
three
mirrors
in
the
shap
e
of
a
ˇ
cub
e
ı
corner,
with
the
reflecting
faces
facing
in
w
ards.
They
are
found
in
the
reflectors
of
some
v
ehicles,
as
w
ell
as
in
surv
eying
equipmen
t.
The
p
oin
ts
O
,
A
,
B
and
C
are
v
ertices
of
a
cub
e,
so
that
the
reference
frame
O
;
−
→
O
A
;
−
−
→
O
B
;
−
−
→
O
C
is
an
orthonormal
refer-
ence
frame.
This
co
ordinate
system
will
b
e
used
throughout
the
exercise.
The
three
mirrors
of
the
retro-reflector
are
represen
ted
b
y
the
planes
(
O
AB
)
,
(
O
B
C
)
and
(
O
AC
)
.
Ligh
t
ra
ys
are
mo
deled
b
y
straigh
t
lines.
Rules
for
reflecting
a
ligh
t
ra
y
admitted
:
when
a
ligh
t
ra
y
with
director
v
ector
−
→
v
a
;
b
;
c
is
re-
flected
b
y
the
plane
(
O
AB
)
,
a
director
v
ector
of
the
re-
flected
ra
y
is
−
→
v
a
;
b
;
−
c
;
when
a
ligh
t
ra
y
with
director
v
ector
−
→
v
a
;
b
;
c
is
re-
flected
b
y
the
plane
(
O
B
C
)
,
a
direction
v
ector
of
the
reflected
ra
y
is
−
→
v
−
a
;
b
;
c
;
when
a
ligh
t
ra
y
of
direction
v
ector
−
→
v
a
;
b
;
c
is
reflected
b
y
the
plane
(
O
AC
)
,
a
direction
v
ector
of
the
reflected
ra
y
is
−
→
v
a
;
−
b
;
c
;
1
Prop
ert
y
of
retroreflectors
Using
the
previous
rules,
demonstrate
that
if
a
ligh
t
ra
y
of
direction
v
ector
−
→
v
a
;
b
;
c
is
reflected
successiv
ely
b
y
the
planes
(
O
AB
)
,
(
O
B
C
)
and
(
O
AC
)
,
the
final
radius
is
parallel
to
the
initial
radius.
T
o
con
tin
ue,
w
e
consider
a
ligh
t
ra
y
mo
deled
b
y
a
straigh
t
line
d
1
with
director
v
ector
−
→
v
1
−
2
;
−
1
;
−
1
whic
h
strik
es
the
plane
O
AB
at
p
oin
t
I
1
2
;
3
;
0
.
The
reflected
ra
y
is
mo
delled
b
y
the
straigh
t
line
d
2
with
director
v
ector
−
→
v
2
−
2
;
−
1
;
1
and
passing
through
the
p
oin
t
I
1
.
2
Reflection
of
d
2
on
the
plane
(
O
B
C
)
a
Giv
e
a
parametric
repr±en
tation
of
the
line
d
2
.
b
Giv
e,
without
justification,
a
normal
v
ector
to
the
(
O
B
C
)
plane
and
a
standard
form
of
this
plane.
c
Let
I
2
b
e
the
p
oin
t
with
co
ordinates
0
;
2
;
1
.
Chec
k
that
the
(
O
B
C
)
plane
and
the
d
2
line
are
secan
t
at
I
2
.
Note
d
3
the
straigh
t
line
represen
ting
the
ligh
t
ra
y
after
re-
flection
on
the
(
O
B
C
)
plane.
d
3
is
therefore
the
straigh
t
line
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3260
sacados/3260
chapExoCorrec/6427
sacados/6427
Liban Mai 2015
A
B
C
D
E
F
G
H
I
J
K
L
chapExoCorrec/6881
sacados/6881
Asie Juin 2016
−
n
O
A
B
C
Vue en perspective cavalière de la réπexion
d’un rayon lumineux sur le plan
(
OAB
A
B
C
D
O
I
S
A
B
C
D
E
F
G
H
with
director
v
ector
−
→
v
3
2
;
−
1
;
1
passing
through
the
p
oin
t
I
2
0
;
2
;
1
.
3
Reflection
of
d
3
on
the
plane
(
O
AC
)
Calculate
the
co
ordinates
of
the
p
oin
t
of
in
tersection
I
3
of
the
line
d
3
with
the
plane
(
O
AC
)
.
Note
d
4
the
straigh
t
line
represen
ting
the
ligh
t
ra
y
after
re-
flection
on
the
(
O
AC
)
plane.
It
is
therefore
parallel
to
the
straigh
t
line
d
1
.
4
Ligh
t
path
study
W
e
giv
e
the
v
ector
−
→
u
1
;
−
2
;
0
,
and
note
P
the
plane
defined
b
y
the
straigh
t
lines
d
1
and
d
2
.
a
Demonstrate
that
the
v
ector
−
→
u
is
a
normal
v
ector
to
the
plane
P
.
b
Do
the
straigh
t
lines
d
1
,
d
2
and
d
3
lie
in
the
same
plane?
c
Do
the
straigh
t
lines
d
1
,
d
2
and
d
4
lie
in
the
same
plane?
E.6883
Consider
the
regular
p
yramid
S
AB
C
D
with
v
ertex
S
consisting
of
square
base
AB
C
D
and
equilateral
triangles
sho
wn
b
elo
w.
The
p
oin
t
O
is
the
cen
ter
of
the
base
AB
C
D
with
O
B
=
1
.
Recall
that
segmen
t
[
S
O
]
is
the
heigh
t
of
the
p
yramid
and
that
all
edges
ha
v
e
the
same
length.
1
Justify
that
the
reference
frame
O
;
−
−
→
O
B
;
−
−
→
O
C
;
−
→
O
S
is
or-
thonormal.
In
the
rest
of
the
exercise,
w
e
place
ourselv
es
in
the
reference
frame
O
;
−
−
→
O
B
;
−
−
→
O
C
;
−
→
O
S
.
2
W
e
define
the
p
oin
t
K
b
y
the
relation
−
−
→
S
K
=
1
3
·
−
→
S
D
and
note
I
the
midp
oin
t
of
the
segmen
t
[
S
O
]
.
a
Determine
the
co
ordinates
of
p
oin
t
K
.
b
Deduce
that
the
p
oin
ts
B
,
I
and
K
are
aligned.
c
Note
L
the
p
oin
t
of
in
tersection
of
edge
[
S
A
]
with
plane
(
B
C
I
)
.
Justify
that
the
straigh
t
lines
(
AD
)
and
(
K
L
)
are
par-
allel.
d
Determine
the
co
ordinates
of
p
oin
t
L
.
3
Consider
the
v
ector
−
→
n
1
;
1
;
2
in
the
reference
frame
O
;
−
−
→
O
B
;
−
−
→
O
C
;
−
→
O
S
.
a
Sho
w
that
−
→
n
is
a
normal
v
ector
to
the
(
B
C
I
)
plane.
b
Sho
w
that
the
v
ectors
−
→
n
,
−
→
AS
and
−
→
D
S
are
coplanar.
c
What
is
the
relativ
e
p
osition
of
the
planes
(
B
C
I
)
and
(
S
AD
)
?
E.5452
Consider
a
cub
e
AB
C
D
E
F
GH
of
edge
length
1
.
W
e
place
ourselv
es
in
the
orthonormal
reference
frame
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
.
Consider
the
p
oin
ts
:
I
1
;
1
3
;
0
;
J
0
;
2
3
;
1
;
K
3
4
;
0
;
1
;
L
a
;
1
;
0
with
a
a
real
n
um
b
er
b
elonging
to
the
in
terv
al
0
;
1
.
P
arts
A
and
B
are
indep
enden
t.
P
art
A
1
Determine
a
parametric
represen
tation
of
the
line
(
I
J
)
.
2
Sho
w
that
the
straigh
t
line
(
K
L
)
has
parametric
repre-
sen
tation
:
x
=
3
4
+
t
·
a
−
3
4
y
=
t
z
=
1
−
t
où
t
∈
R
.
3
Demonstrate
that
the
straigh
t
lines
(
I
J
)
and
(
K
L
)
are
secan
t
if,
and
only
if,
a
=
1
4
.
P
art
B
Throughout
the
rest
of
the
exercise,
w
e
p
ose
a
=
1
4
.
The
p
oin
t
L
therefore
has
co
ordinates
1
4
;
1
;
0
.
1
Demonstrate
that
the
quadrilateral
I
K
J
L
is
a
parallelo-
gram.
2
The
figure
b
elo
w
sho
ws
the
in
tersection
of
the
plane
(
I
J
K
)
with
the
faces
of
the
cub
e
AB
C
D
D
E
F
GH
as
obtained
using
dynamic
geometry
soft
w
are.
W
e
denote
b
y
M
the
p
oin
t
of
in
tersection
of
the
plane
(
I
J
K
)
and
the
line
(
B
F
)
and
b
y
N
the
p
oin
t
of
in
tersec-
tion
of
the
plane
(
I
J
K
)
and
the
line
(
D
H
)
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6883
sacados/6883
A
B
C
D
O
I
S
chapExoCorrec/5452
sacados/5452
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
I
J
K
L
M
N
A
B
C
D
E
F
G
H
M
P
N
A
B
C
D
E
F
G
H
I
J
K
L
M
N
The
aim
of
this
question
is
to
determine
the
co
ordinates
of
the
p
oin
ts
M
and
N
.
a
Pro
v
e
that
the
v
ector
−
→
n
of
co
ordinates
8
;
9
;
5
is
a
normal
v
ector
to
the
plane
(
I
J
K
)
.
b
Deduce
that
the
plane
(
I
J
K
)
has
equation
:
8
x
+
9
y
+
5
z
−
11
=
0
c
Deduce
the
co
ordinates
of
p
oin
ts
M
and
N
.
E.6268
Consider
a
AB
C
D
D
E
F
GH
cub
e
giv
en
b
elo
w
:
Note
M
the
middle
of
segmen
t
[
E
H
]
,
N
that
of
[
F
C
]
and
P
the
p
oin
t
suc
h
that
:
−
−
→
H
P
=
1
4
·
−
−
→
H
G
.
P
art
A
:
Section
of
the
cub
e
b
y
the
(
M
N
P
)
plane
1
Justify
that
the
straigh
t
lines
(
M
P
)
and
(
F
G
)
in
tersect
at
a
p
oin
t
L
.
Construct
the
p
oin
t
L
.
2
A
dmit
that
the
straigh
t
lines
(
LN
)
and
(
C
G
)
are
secan
t
and
note
T
their
p
oin
t
of
in
tersection.
W
e
admit
that
the
straigh
t
lines
(
LN
)
and
(
B
F
)
are
se-
can
t
and
w
e
note
Q
their
p
oin
t
of
in
tersection.
a
Construct
the
p
oin
ts
T
and
Q
,
lea
ving
the
construction
lines
visible.
b
Construct
the
in
tersection
of
planes
(
M
N
P
)
and
(
AB
F
)
.
3
Deduce
a
construction
of
the
cub
e’s
section
through
the
(
M
N
P
)
plane.
P
art
B
Space
is
referred
to
the
reference
frame
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
.
1
Giv
e
the
co
ordinates
of
the
p
oin
ts
M
,
N
and
P
in
this
frame.
2
Determine
the
co
ordinates
of
p
oin
t
L
.
3
Assume
that
the
p
oin
t
T
has
co
ordinates
1
;
1
;
5
8
.
Is
the
triangle
T
P
N
righ
t-angled
at
T
?
E.3238
1
Let
[
K
L
]
b
e
a
segmen
t
of
space
;
note
I
its
midp
oin
t.
The
p
erp
endicular
plane
at
I
to
the
line
(
K
L
)
.
is
called
the
mediator
plane
of
[
K
L
]
Sho
w
that
the
media-
tor
plane
of
[
K
L
]
is
the
set
of
p
oin
ts
in
space
equidistan
t
from
K
and
L
.
2
Here,
space
is
pro
vided
with
an
orthonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
;
consider
the
p
oin
ts
:
A
4
;
0
;
−
3
;
B
2
;
2
;
2
;
C
3
;
−
3
;
−
1
;
D
0
;
0
;
−
3
a
Sho
w
that
the
median
plane
of
[
AB
]
has
equation
:
4
x
−
4
y
−
10
z
−
13
=
0
F
or
the
follo
wing,
w
e
admit
that
the
median
planes
of
[
B
C
]
and
[
C
D
]
ha
v
e
resp
ectiv
ely
for
equation
:
2
x
−
10
y
−
6
z
−
7
=
0
;
3
x
−
3
y
+
2
z
−
5
=
0
b
Demonstrate,
b
y
solving
a
system
of
linear
equa-
tions,
that
these
three
planes
ha
v
e
a
single
common
p
oin
t
E
whose
co
ordinates
will
b
e
giv
en.
c
Using
the
question
1
sho
w
that
the
p
oin
ts
A
,
B
,
C
and
D
are
on
a
sphere
with
cen
ter
E
.
What
is
the
radius
of
this
sphere?
2.
Distanc
e,
volume
in
sp
ac
e
E.6884
AB
C
D
E
F
GH
is
a
cub
e
with
edge
equal
to
1
.
Space
is
pro
vided
with
the
orthonormal
reference
frame
D
;
−
−
→
D
C
;
−
−
→
D
A
;
−
−
→
D
H
.
h
ttps:/
/c
hingmath.fr
A
B
C
D
E
F
G
H
I
J
K
L
M
N
chapExoCorrec/6268
sacados/6268
A
B
C
D
E
F
G
H
M
P
N
chapExoCorrec/3238
sacados/3238
Antilles-Guyane Juin 2005 6 points
chapExoCorrec/6884
sacados/6884
Antilles-Guyane Juin 2016
A
B
C
D
E
F
G
H
I
J
K
L
M
N
A
B
C
D
E
F
G
H
In
this
frame
of
reference,
w
e
ha
v
e
:
D
0
;
0
;
0
C
1
;
0
;
0
A
0
;
1
;
0
H
0
;
0
;
1
E
0
;
1
;
1
Let
I
b
e
the
middle
of
[
AB
]
.
Let
P
b
e
the
plane
parallel
to
plane
(
B
GE
)
and
passing
through
p
oin
t
I
.
W
e
admit
that
the
section
of
the
cub
e
b
y
the
plane
P
rep-
resen
ted
ab
o
v
e
is
a
hexagon
whose
v
ertices
I
,
J
,
K
,
L
,
M
and
N
b
elong
to
the
[
AB
]
,
[
B
C
]
,
[
C
G
]
,
[
GH
]
,
[
H
E
]
and
[
AE
]
edges
resp
ectiv
ely
.
1
a
Sho
w
that
the
v
ector
−
−
→
D
F
is
normal
to
the
plane
(
B
GE
)
.
b
Deduce
a
standard
form
of
the
plane
P
.
2
Sho
w
that
p
oin
t
N
is
the
midp
oin
t
of
segmen
t
[
AE
]
.
3
a
Determine
a
parametric
represen
tation
of
the
line
(
H
B
)
.
b
Deduce
that
the
straigh
t
line
(
H
B
)
and
the
plane
P
are
secan
t
at
a
p
oin
t
T
whose
co
ordinates
w
e
will
sp
ec-
ify
.
4
Calculate,
in
units
of
v
olume,
the
v
olume
of
the
tetrahe-
dron
F
B
GE
.
E.6265
In
space
pro
vided
with
an
or-
thonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
,
consider
the
tetra-
hedron
AB
C
D
whose
v
ertices
ha
v
e
co
ordinates
:
A
1
;
−
3
;
0
;
B
1
;
3
;
0
;
C
−
2
;
0
;
0
;
D
0
;
0
;
2
2
1
Sho
w
that
the
plane
(
AB
D
)
has
standard
form
4
x
+
z
2
=
4
02
2
Note
D
the
line
whose
parametric
represen
tation
is
:
x
=
t
y
=
0
z
=
t
2
où
t
∈
R
a
Demonstrate
that
D
is
the
line
that
is
parallel
to
(
C
D
)
and
passes
through
O
.
b
Determine
the
co
ordinates
of
p
oin
t
G
,
in
tersection
of
line
D
and
plane
(
AB
D
)
.
3
a
Note
L
the
midp
oin
t
of
segmen
t
[
AC
]
.
Sho
w
that
the
line
(
B
L
)
passes
through
the
p
oin
t
O
and
is
orthogonal
to
the
line
(
AC
)
.
b
Pro
v
e
that
the
triangle
AB
C
is
equilateral
and
deter-
mine
the
cen
ter
of
its
circumscrib
ed
cen
ter.
4
Demonstrate
that
the
tetrahedron
AB
C
D
is
regular,
i.e.
a
tetrahedron
whose
six
edges
ha
v
e
the
same
length.
E.6406
Let
b
e
a
cub
e
AB
C
D
E
F
GH
of
edge
1
.
Dans
le
rep
ère
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
,
consider
the
p
oin
ts
M
,
N
and
P
with
co
ordinates
:
M
1
;
1
;
3
4
;
N
0
;
1
2
;
1
;
P
1
;
0
;
−
5
4
.
1
Place
M
,
N
and
P
on
the
figure
b
elo
w.
2
Determine
the
co
ordinates
of
the
v
ectors
−
−
→
M
N
and
−
−
→
M
P
.
Deduce
that
the
p
oin
ts
M
,
N
and
P
are
not
aligned.
3
Consider
the
function
f
from
the
algorithm
1
b
elo
w
:
Fonction
f(
x
M
,
y
M
,
z
M
,
x
N
,
y
N
,
z
N
,
x
P
,
y
P
,
z
P
)
d
←
x
N
−
x
M
e
←
y
N
−
y
M
f
←
z
N
−
z
M
g
←
x
p
−
x
M
h
←
y
p
−
y
M
i
←
z
P
−
z
M
k
←
d
×
g+e
×
h+f
×
i
Resend
k
a
What
is
the
v
alue
returned
b
y
the
function
f
when
called
with
the
co
ordinates
of
the
p
oin
ts
M
,
N
and
P
giv
en
ab
o
v
e.
b
What
do
es
this
v
alue
corresp
ond
to?
What
can
w
e
deduce
for
the
triangle
M
N
P
?
4
Consider
the
function
g
of
an
algorithm
2
giv
en
b
elo
w.
Fonction
g(
x
M
,
y
M
,
z
M
,
x
N
,
y
N
,
z
N
,
x
P
,
y
P
,
z
P
)
d
←
x
N
−
x
M
e
←
y
N
−
y
M
f
←
z
N
−
z
M
g
←
x
p
−
x
M
h
←
y
p
−
y
M
i
←
z
P
−
z
M
k
←
d
×
g+e
×
h+f
×
i
...
Cop
y
and
complete
it
so
that
the
function
g
returns
the
v
alue
1
if
the
triangle
M
N
P
is
righ
t-angled
and
isosceles
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6265
sacados/6265
chapExoCorrec/6406
sacados/6406
A
B
C
D
E
F
G
H
A
B
C
D
O
S
at
M
and
the
v
alue
0
otherwise.
5
Consider
the
v
ector
−
→
n
5
;
−
8
;
4
normal
to
the
plane
(
M
N
P
)
.
a
Determine
a
standard
form
of
the
plane
(
M
N
P
)
.
b
Consider
the
line
(Δ)
passing
through
F
and
with
di-
rection
v
ector
−
→
n
.
Determine
a
parametric
represen
tation
of
the
line
(Δ)
.
6
Let
K
b
e
the
p
oin
t
of
in
tersection
of
the
plane
(
M
N
P
)
and
the
straigh
t
line
(Δ)
.
a
Demonstrate
that
the
co
ordinates
of
the
p
oin
t
K
are
4
7
;
24
35
;
23
35
.
b
W
e
giv
e
:
F
K
=
27
35
.
Calculate
the
v
olume
of
the
tetrahedron
M
N
P
F
.
E.6261
In
space,
consider
a
tetrahedron
AB
C
D
whose
faces
AB
C
,
AC
D
and
AB
D
are
righ
t-angled
and
isosceles
triangles
at
A
.
W
e
denote
b
y
E
,
F
and
G
the
resp
ectiv
e
middles
of
the
sides
[
AB
]
,
[
B
C
]
and
[
C
A
]
.
W
e
c
ho
ose
AB
as
the
unit
of
length
and
place
ourselv
es
in
the
orthonormal
reference
frame
A
;
−
−
→
AB
;
−
→
AC
;
−
−
→
AD
de
l’espace.
1
W
e
denote
b
y
P
the
plane
that
passes
through
A
and
is
orthogonal
to
the
line
(
D
F
)
.
Note
H
the
p
oin
t
of
in
tersection
of
the
plane
P
and
the
line
(
D
F
)
.
a
Giv
e
the
co
ordinates
of
p
oin
ts
D
and
F
.
b
Giv
e
a
parametric
represen
tation
of
the
line
(
D
F
)
.
c
Determine
a
standard
form
of
the
plane
P
.
d
Calculate
the
co
ordinates
of
the
p
oin
t
H
.
e
Demonstrate
that
the
angle
∠
E
H
G
is
a
righ
t
angle.
2
Denote
b
y
M
a
p
oin
t
on
the
line
(
D
F
)
and
b
y
t
the
real
suc
h
that
−
−
→
D
M
=
t
·
−
−
→
D
F
.
Note
¸
the
measure
in
radians
of
the
geometric
angle
∠
E
M
G
.
The
aim
of
this
question
is
to
determine
the
p
osition
of
p
oin
t
M
so
that
¸
is
maxim
um.
a
Demonstrate
that
:
M
E
2
=
3
2
·
t
2
−
5
2
·
t
+
5
4
b
Demonstrate
that
the
triangle
M
E
G
is
isosceles
at
M
.
En
déduire
que
:
M
E
·
sin
¸
2
=
1
2
·
√
2
.
c
Justify
that
¸
is
maximal
if,
and
only
if,
sin
¸
2
is
maximal.
Deduce
that
¸
is
maximal
if,
and
only
if,
M
E
2
is
min-
imal.
d
Conclude.
3.
Pr
oje
cte
d
ortho
gonal
E.6943
P
art
A
:
a
v
olume
calculation
without
a
b
enc
hmark
Consider
an
equilateral
p
yramid
S
AB
C
D
(squar
e-b
ase
d
pyr
a-
mid
whose
side
fac
es
ar
e
al
l
e
quilater
al
triangles)
sho
wn
op-
p
osite.
The
diagonals
of
the
square
AB
C
D
measure
24
cm
.
Let
O
b
e
the
cen
ter
of
the
square
AB
C
D
.
W
e’ll
assume
that
:
O
S
=
O
A
1
Without
using
a
reference
p
oin
t,
demonstrate
that
the
line
S
O
is
orthogonal
to
the
plane
AB
C
.
2
Deduce
the
v
olume,
in
cm
3
,
of
the
p
yramid
S
AB
C
D
.
P
art
B
:
in
a
landmark
On
considère
le
rep
ère
orthonormé
O
;
−
→
O
A
;
−
−
→
O
B
;
−
→
O
S
.
1
Note
P
and
Q
the
resp
ectiv
e
middles
of
segmen
ts
[
AS
]
and
[
B
S
]
.
a
Justify
that
−
→
n
1
;
1
;
−
3
is
a
normal
v
ector
to
the
plane
P
QC
.
b
Deduce
a
standard
form
of
the
plane
P
QC
.
2
Let
H
b
e
the
p
oin
t
on
the
plane
P
QC
suc
h
that
the
straigh
t
line
S
H
is
orthogonal
to
the
plane
P
QC
.
a
Giv
e
a
parametric
represen
tation
of
the
line
(
S
H
)
.
b
Calculate
the
co
ordinates
of
p
oin
t
H
.
c
Then
sho
w
that
the
length
S
H
,
in
units
of
length,
is
2
11
11
3
W
e’ll
admit
that
the
area
of
the
quadrilateral
P
QC
D
,
in
units
of
area,
is
equal
to
3
11
8
.
Calculate
the
v
olume
of
the
p
yramid
S
P
QC
D
,
in
units
of
v
olume.
P
art
C
:
equitable
sharing
F
or
the
birthda
y
of
her
t
win
daugh
ters
Anne
and
F
ann
y
.
Madame
No
v
a
has
made
a
prett
y
cak
e
in
the
shap
e
of
an
equi-
lateral
p
yramid
whose
base
square
diagonals
measure
24
cm
.
She
prepares
to
divide
it
in
t
w
o,
equally
,
b
y
placing
her
knife
on
the
ap
ex.
Then
Anne
stops
her
action
and
suggests
a
more
original
cut
:
ˇ
Place
the
blade
on
the
middle
of
an
edge,
parallel
to
one
side
of
the
base,
then
cut
a
w
a
y
to
w
ards
the
opp
osite
side
ı
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6261
sacados/6261
chapExoCorrec/6943
sacados/6943
A
B
C
D
O
S
H
G
F
E
D
C
B
A
F
ann
y
has
her
doubts,
the
shares
don’t
seem
fair
to
her.
Is
this
the
case?
Justify
y
our
answ
er.
E.3186
Consider
the
cub
e
AB
C
D
E
F
GH
sho
wn
on
the
attac
hed
sheet.
Throughout
the
exer-
cise,
space
is
referred
to
the
orthonormal
reference
frame
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
.
Note
I
the
p
oin
t
with
co
ordinates
1
3
;
1
;
1
.
1
Place
the
p
oin
t
I
on
the
figure.
2
The
plane
(
AC
I
)
in
tersects
the
line
(
E
H
)
at
J
.
Sho
w
that
the
straigh
t
lines
(
I
J
)
and
(
AC
)
are
parallel.
3
Note
R
the
orthogonal
pro
ject
of
I
on
to
the
line
(
AC
)
.
a
Justify
that
the
follo
wing
t
w
o
conditions
are
v
erified
:
i
There
exists
a
real
k
suc
h
that
:
−
→
AR
=
k
·
−
→
AC
ii
−
→
I
R
·
−
→
AC
=
0
b
Calculate
the
co
ordinates
of
p
oin
t
R
.
c
Deduce
that
the
distance
I
R
is
expressed
as
:
I
R
=
√
11
3
.
4
Demonstrate
that
the
v
ector
−
→
n
with
co
ordinates
3
;
−
3
;
2
is
normal
to
the
plane
(
AC
I
)
.
Deduce
a
Cartesian
equation
of
the
plane
(
AC
I
)
.
5
Demonstrate
that
the
distance
from
p
oin
t
F
to
plane
(
AC
I
)
is
5
√
22
.
E.3248
The
heigh
t
of
a
tetrahedron
is
an
y
straigh
t
line
con
taining
one
of
its
v
ertices
and
p
erp
endicular
to
the
plane
of
the
face
opp
osite
that
v
ertex.
A
tetrahedron
is
ortho
cen
tric
if
its
four
heigh
ts
are
concur-
ren
t.
P
art
A
Consider
a
tetrahedron
AB
C
D
and
note
H
the
orthogonal
pro
ject
of
the
p
oin
t
A
on
to
the
plane
(
B
C
D
)
.
Sho
w
that,
if
the
heigh
ts
of
the
tetrahedron
AB
C
D
arising
from
the
p
oin
ts
A
and
B
are
concurren
t,
then
the
line
(
B
H
)
is
a
heigh
t
of
the
triangle
B
C
D
.
P
art
B
In
space
pro
vided
with
an
orthonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
,
w
e
giv
e
the
p
oin
ts
:
A
3
;
2
;
−
1
;
B
−
6
;
1
;
1
;
C
4
;
−
3
;
3
;
D
−
1
;
−
5
;
−
1
1
a
V
erify
that
a
standard
form
of
the
(
B
C
D
)
plane
is
:
−
2
x
−
3
y
+
4
z
−
13
=
0
b
Determine
the
co
ordinates
of
the
p
oin
t
H
,
orthogonal
pro
jected
from
the
p
oin
t
A
on
the
plane
(
B
C
D
)
.
c
Calculate
the
scalar
pro
duct
:
−
−
→
B
H
·
−
−
→
C
D
d
Is
the
tetrahedron
AB
D
ortho
cen
tric?
2
W
e
define
the
p
oin
ts
I
1
;
0
;
0
,
J
0
;
1
;
0
,
K
0
;
0
;
1
.
Is
the
tetrahedron
O
I
J
K
ortho
cen
tric?
E.3139
Space
is
pro
vided
with
an
or-
thonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
Let
(
P
1
)
b
e
the
plane
with
Cartesian
equation
−
2
x
+
y
+
z
−
6
=
0
and
(
P
2
)
b
e
the
plane
with
Cartesian
equation
x
−
2
y
+4
z
−
9
=
0
.
1
Sho
w
that
(
P
1
)
and
(
P
2
)
are
p
erp
endicular.
Recall
that
t
w
o
planes
are
p
erp
endicular
if,
and
only
if,
a
non-zero
normal
v
ector
to
one
is
orthogonal
to
a
non-zero
normal
v
ector
to
the
other.
2
Let
(
D
)
b
e
the
line
of
in
tersection
of
(
P
1
)
and
(
P
2
)
.
Sho
w
that
a
parametric
represen
tation
of
(
D
)
is
:
x
=
−
7
+
2
t
y
=
−
8
+
3
t
z
=
t
where
t
∈
R
3
Let
M
b
e
an
y
p
oin
t
of
(
D
)
of
parameter
t
and
let
A
b
e
the
p
oin
t
of
co
ordinates
−
9
;
−
4
;
−
1
.
a
V
erify
that
A
b
elongs
neither
to
(
P
1
)
,
nor
to
(
P
2
)
.
b
Express
AM
2
in
terms
of
t
.
c
Let
f
b
e
the
function
defined
on
R
b
y
f
(
t
)
=
2
t
2
−
2
t
+3
.
Study
the
v
ariations
of
f
.
F
or
whic
h
p
oin
t
M
,
is
the
distance
AM
minimal?
In
the
follo
wing,
w
e’ll
refer
to
this
p
oin
t
as
I
.
Sp
ecify
the
co
ordinates
of
the
p
oin
t
I
.
4
Let
(
Q
)
b
e
the
plane
orthogonal
to
(
D
)
passing
through
A
.
a
Determine
an
equation
of
(
Q
)
.
b
Demonstrate
that
I
is
the
orthogonal
pro
jected
of
A
on
to
(
D
)
.
h
ttps:/
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hingmath.fr
chapExoCorrec/3186
sacados/3186
H
G
F
E
D
C
B
A
chapExoCorrec/3248
sacados/3248
chapExoCorrec/3139
sacados/3139
E.3172
Space
is
pro
vided
with
an
or-
thonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
P
art
A
(this
p
art
c
onstitutes
an
or
ganize
d
r
eturn
of
know
l-
e
dge)
Let
a
,
b
,
c
and
d
b
e
real
n
um
b
ers
suc
h
that
:
(
a
;
b
;
c
)
=
0
;
0
;
0
.
Let
P
b
e
the
plane
of
equation
:
a
·
x
+
b
·
y
+
c
·
z
+
d
=
0
.
Consider
the
p
oin
t
I
of
co
ordinates
(
x
I
;
y
I
;
z
I
)
and
the
v
ector
−
→
n
de
co
ordonnées
(
a
;
b
;
c
)
.
The
aim
of
this
part
is
to
demonstrate
that
the
distance
from
I
to
the
plane
P
is
equal
to
:
⏐
⏐
a
·
x
I
+
b
·
y
I
+
c
·
z
I
+
d
⏐
⏐
a
2
+
b
2
+
c
2
1
Let
Δ
b
e
the
straigh
t
line
passing
through
I
and
orthog-
onal
to
the
plane
P
.
Determine,
as
a
function
of
a
,
b
,
c
,
x
I
,
y
I
and
z
I
,
a
system
of
parametric
equations
of
Δ
.
2
Note
H
the
p
oin
t
of
in
tersection
of
Δ
and
P
.
a
Justify
that
there
exists
a
real
k
suc
h
that
:
−
→
I
H
=
k
−
→
n
.
b
Determine
the
expression
of
k
as
a
function
of
a
,
b
,
c
,
d
,
x
I
,
y
I
and
z
I
.
c
Deduce
that
:
I
H
=
⏐
⏐
a
·
x
I
+
b
·
y
I
+
c
·
z
I
+
d
⏐
⏐
a
2
+
b
2
+
c
2
P
art
B
The
plane
Q
of
equation
x
−
y
+
z
−
11
=
0
is
tangen
t
to
a
sphere
S
of
cen
ter
the
p
oin
t
Ω
of
co
ordinates
1
;
−
1
;
3
.
1
Determine
the
radius
of
the
sphere
S
.
2
Determine
a
system
of
parametric
equations
of
the
line
Δ
passing
through
Ω
and
orthogonal
to
the
plane
Q
.
3
Deduce
the
co
ordinates
of
the
p
oin
t
of
in
tersection
of
the
sphere
S
and
the
plane
Q
.
4.
Qcm,
affirmations...
E.3195
Let
O
;
−
→
i
;
−
→
j
;
−
→
k
b
e
an
or-
thonormal
space
frame.
Consider
the
p
oin
ts
:
A
2
;
4
;
1
;
B
0
;
4
;
−
3
;
C
3
;
1
;
−
3
D
1
;
0
;
−
2
;
E
3
;
2
;
−
1
;
I
3
5
;
4
;
−
9
5
F
or
eac
h
of
the
follo
wing
fiv
e
statemen
ts,
sa
y
,
without
justi-
fication,
whether
it
is
true
or
false.
F
or
eac
h
question,
one
p
oin
t
is
coun
ted
if
the
answ
er
is
correct
and
zero
otherwise.
1
An
equation
of
the
plane
(
AB
C
)
is
:
2
x
+
2
y
−
z
−
11
=
0
2
The
p
oin
t
E
is
the
orthogonal
pro
ject
of
D
on
to
the
plane
(
AB
C
)
.
3
The
straigh
t
lines
(
AB
)
and
(
C
D
)
are
orthogonal.
4
The
straigh
t
line
(
C
D
)
is
giv
en
b
y
the
follo
wing
paramet-
ric
represen
tation
:
(
C
D
)
:
x
=
−
1
+
2
t
y
=
−
1
+
t
z
=
1
−
t
où
t
∈
R
5
The
p
oin
t
I
is
on
the
straigh
t
line
AB
.
E.5531
F
or
eac
h
question,
four
prop
osed
answ
ers
are
giv
en,
only
one
of
whic
h
is
correct.
F
or
eac
h
ques-
tion,
indicate,
without
justification,
the
correct
answ
er
on
the
cop
y
.
A
correct
answ
er
earns
1
p
oin
t.
A
wrong
answ
er
or
the
absence
of
an
answ
er
neither
earns
nor
deducts
an
y
p
oin
ts.
The
same
applies
if
sev
eral
answ
ers
are
giv
en
for
the
same
question.
Space
is
referred
to
an
orthonormal
reference
frame.
t
and
t
denote
real
parameters.
The
plane
(
P
)
has
equation
:
x
−
2
y
+3
z
+5
=
0
The
(
S
)
plane
has
parametric
represen
tation
:
x
=
−
2
+
t
+
2
t
y
=
−
t
−
2
t
z
=
−
1
−
t
+
3
t
où
t
∈
R
,
t
∈
R
The
straigh
t
line
(
d
)
has
the
follo
wing
parametric
represen
ta-
tion
:
x
=
−
2
+
t
y
=
−
t
z
=
−
1
−
t
où
t
∈
R
The
space
p
oin
ts
M
−
1
;
2
;
3
and
N
1
;
−
2
;
9
are
giv
en.
1
A
parametric
represen
tation
of
the
(
P
)
plane
is
:
a
x
=
t
y
=
1
−
2
t
z
=
−
1
+
3
t
b
x
=
t
+
2
t
y
=
1
−
t
+
t
z
=
−
1
−
t
c
x
=
t
+
t
y
=
1
−
t
−
2
t
z
=
1
−
t
−
3
t
d
x
=
1
+
2
t
+
t
y
=
1
−
2
t
+
2
t
z
=
−
1
−
t
2
a
The
straigh
t
line
(
d
)
and
the
plane
(
P
)
are
secan
t
at
the
p
oin
t
A
−
8
;
3
;
2
.
b
The
line
(
d
)
and
the
plane
(
P
)
are
p
erp
endicular.
c
The
straigh
t
line
(
d
)
is
a
straigh
t
line
of
the
plane
(
P
)
.
d
The
straigh
t
line
(
d
)
and
the
plane
(
P
)
are
strictly
parallel.
h
ttps:/
/c
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chapExoCorrec/3172
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chapExoCorrec/3195
sacados/3195
France Juin 2006 4 points
chapExoCorrec/5531
sacados/5531
3
a
The
straigh
t
line
(
M
N
)
and
the
straigh
t
line
(
d
)
are
orthogonal.
b
The
line
(
M
N
)
and
the
line
(
d
)
are
parallel.
c
The
line
(
M
N
)
and
the
line
(
d
)
are
secan
t.
d
The
straigh
t
line
(
M
N
)
and
the
straigh
t
line
(
d
)
are
coinciden
t.
4
a
The
(
P
)
and
(
S
)
planes
are
parallel.
b
The
straigh
t
line
(Δ)
of
parametric
represen
tation
:
x
=
t
y
=
−
2
−
t
z
=
−
3
−
t
t
∈
R
is
the
line
of
in
tersection
of
the
planes
(
P
)
and
(
S
)
.
c
The
p
oin
t
M
b
elongs
to
the
in
tersection
of
the
planes
(
P
)
and
(
S
)
.
d
(
P
)
and
(
S
)
planes
are
p
erp
endicular.
E.3140
In
space
referred
to
an
O
;
−
→
i
;
−
→
j
;
−
→
k
orthonormal
reference
frame,
consider
the
p
oin
ts
:
A
co
ordinates
3
;
1
;
−
5
;
B
co
ordinates
0
;
4
;
−
5
;
C
co
ordinates
−
1
;
2
;
−
5
;
D
co
ordinates
2
;
3
;
4
.
F
or
eac
h
of
the
six
statemen
ts
b
elo
w,
state
whether
it
is
true
or
false.
No
justification
is
required.
The
candidate
m
ust
indi-
cate
on
his
cop
y
the
n
um
b
er
of
the
question
and
the
men
tion
ˇVRAIı
or
ˇF
A
UXı.
0.5
p
oin
ts
are
a
w
arded
for
eac
h
correct
answ
er
and
0.25
p
oin
t
is
deducted
for
eac
h
incorrect
answ
er.
The
absence
of
an
answ
er
is
not
p
enalized.
An
y
negativ
e
total
is
reduced
to
0.
1
The
p
oin
ts
A
,
B
and
D
are
aligned.
2
The
line
(
AB
)
is
con
tained
in
the
plane
of
Cartesian
equa-
tion
:
x
+
y
=
4
.
3
A
Cartesian
equation
of
the
(
B
C
D
)
plane
is
:
18
x
−
9
y
−
5
z
+
11
=
0
4
The
p
oin
ts
A
,
B
,
C
and
D
are
coplanar.
5
A
parametric
represen
tation
of
the
line
(
B
D
)
is
:
x
=
1
−
2
k
y
=
7
2
+
k
z
=
−
1
2
−
9
k
where
k
∈
R
E.3149
P
art
One
Space
is
referred
to
an
orthonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
Consider
:
P
oin
ts
:
A
0
;
0
;
3
;
B
2
;
0
;
4
;
C
−
1
;
1
;
2
;
D
1
;
−
4
;
0
.
plans
:
(
P
1
)
:
7
x
+4
y
−
3
z
+9
=
0
;
(
P
2
)
:
x
−
2
y
=
0
.
The
straigh
t
lines
(Δ
1
)
and
(Δ
2
)
defined
b
y
their
resp
ec-
tiv
e
parametric
equation
systems
:
x
=
−
1
+
t
y
=
−
8
+
2
t
z
=
−
10
+
5
t
where
t
∈
R
x
=
7
+
2
t
y
=
8
+
4
t
z
=
8
−
t
where
t
∈
R
F
or
eac
h
question,
only
one
of
the
four
prop
ositions
is
correct.
The
candidate
will
indicate
on
the
cop
y
the
n
um
b
er
of
the
question
and
the
letter
corresp
onding
to
the
c
hosen
answ
er.
No
justification
is
required.
A
correct
answ
er
earns
0.5
p
oin
t
;
an
incorrect
answ
er
deducts
0.25
p
oin
t
;
the
absence
of
an
answ
er
is
coun
ted
0
p
oin
t.
If
the
total
is
negativ
e,
the
score
is
reduced
to
0.
1
The
(
P
1
)
plane
is
:
a
le
plane
(
AB
C
)
b
le
plan
(
B
C
D
)
c
le
plan
(
AC
D
)
d
le
plan
(
AB
D
)
2
The
straigh
t
line
(Δ
1
)
con
tains
:
a
le
p
oin
t
A
b
le
p
oin
t
B
c
le
p
oin
t
C
d
le
p
oin
t
D
3
Relativ
e
p
osition
of
(
P
1
)
and
(Δ
2
)
:
a
(Δ
2
)
is
strictly
parallel
to
(
P
1
)
b
(Δ
2
)
is
included
in
(
P
1
)
c
(Δ
2
)
cut
(
P
1
)
d
(Δ
2
)
is
orthogonal
to
(
P
1
)
4
Relativ
e
p
osition
of
(Δ
1
)
and
(Δ
2
)
:
a
(Δ
1
)
is
strictly
parallel
to
(Δ
2
)
b
(Δ
1
)
and
(Δ
2
)
are
confondues
c
(Δ
1
)
and
(Δ
2
)
are
sécan
tes
d
(Δ
1
)
and
(Δ
2
)
are
non-coplanar.
5
The
in
tersection
of
(
P
1
)
and
(
P
2
)
is
a
line
whose
para-
metric
represen
tation
is
:
a
x
=
t
y
=
−
2
+
1
2
·
t
z
=
3
·
t
b
x
=
2
·
t
y
=
t
z
=
3
+
6
·
t
c
x
=
5
·
t
y
=
1
−
2
·
t
z
=
t
d
x
=
−
1
+
t
y
=
2
+
t
z
=
−
3
·
t
P
art
T
w
o
Space
is
referred
to
an
orthonormal
frame
of
reference
O
;
−
→
i
;
−
→
j
;
−
→
k
.
Consider
the
line
(
D
)
passing
through
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3140
sacados/3140
chapExoCorrec/3149
sacados/3149
H
G
F
E
D
C
B
A
I
J
A
0
;
0
;
3
and
whose
directing
v
ector
is
−
→
u
1
;
0
;
−
1
and
the
line
(
D
)
passing
through
B
2
;
0
;
4
and
one
of
whose
directing
v
ectors
is
−
→
v
0
;
1
;
1
.
The
aim
is
to
sho
w
that
there
is
a
single
line
p
erp
endicular
to
b
oth
(
D
)
and
(
D
)
,
to
determine
it
and
to
iden
tify
a
prop
ert
y
of
this
line.
1
Consider
a
p
oin
t
M
b
elonging
to
(
D
)
and
a
p
oin
t
M
b
elonging
to
(
D
)
defined
b
y
−
−
→
AM
=
a
·
−
→
u
et
−
−
−
→
B
M
=
b
·
−
→
v
,
where
a
and
b
are
real
n
um
b
ers.
Express
the
co
ordinates
of
M
,
M
and
then
the
v
ector
−
−
−
→
M
M
in
terms
of
a
and
b
.
2
Sho
w
that
the
line
(
M
M
)
is
p
erp
endicular
to
(
D
)
and
to
(
D
)
if,
and
only
if,
the
couple
(
a
;
b
)
is
solution
of
the
system
:
2
a
+
b
=
1
a
+
2
b
=
−
1
3
Solv
e
this
system.
Deduce
the
co
ordinates
of
the
t
w
o
unique
p
oin
ts
M
and
M
,
whic
h
w
e
will
note
here
H
and
H
,
suc
h
that
the
straigh
t
line
(
H
H
)
is
indeed
common
p
erp
endicular
to
(
D
)
and
to
(
D
)
.
Sho
w
that
H
H
=
3
units
of
length.
4
Consider
an
y
p
oin
t
M
on
the
line
(
D
)
and
an
y
p
oin
t
M
on
the
line
(
D
)
.
a
Using
the
co
ordinates
obtained
in
question
1
,
demon-
strate
that
:
M
M
2
=
(
a
+
b
)
2
+
(
a
−
1)
2
+
(
b
+
1)
2
+
3
b
Deduce
that
the
distance
M
M
minim
um
when
M
is
in
H
and
M
is
in
H
.
E.3167
In
this
exercise,
an
answ
er
b
y
ˇ
VRAIı
or
ˇ
F
A
UXı,
without
justification,
is
requested
from
the
candidate
against
a
list
of
statemen
ts.
An
y
mathemat-
ically
correct
answ
er
giv
es
0.4
p
oin
t.
An
y
incorrect
answ
er
deducts
0.1
p
oin
t.
No
answ
er
is
not
coun
ted.
The
total
can-
not
b
e
negativ
e.
W
e
giv
e
the
cub
e
AB
C
D
E
F
GH
,
of
edge
length
1
,
and
the
middles
I
and
J
of
the
edges
[
AB
]
and
[
C
G
]
.
The
useful
elemen
ts
of
the
figure
are
giv
en
opp
osite.
The
candidate
is
ask
ed
to
judge
eac
h
of
the
follo
wing
10
statemen
ts.
Affirmation
V
rai
F
aux
1
−
→
AC
·
−
→
AI
=
1
2
2
−
→
AC
·
−
→
AI
=
−
→
AI
·
−
−
→
AB
3
−
−
→
AB
·
−
→
I
J
=
−
−
→
AB
·
−
→
I
C
4
−
−
→
AB
·
−
→
I
J
=
AB
×
I
C
×
cos
ı
3
W
e
no
w
use
the
orthonormal
reference
frame
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
5
A
parametric
represen
tation
of
the
line
(
I
J
)
is
:
x
=
t
+
1
y
=
2
t
z
=
t
where
t
∈
R
6
A
parametric
represen
tation
of
the
line
(
I
J
)
is
:
x
=
1
2
t
+
1
y
=
t
+
1
z
=
1
2
t
+
1
2
where
t
∈
R
.
7
6
x
−
7
y
+8
z
−
3=0
is
a
Cartesian
equation
of
the
line
(
I
J
)
.
8
The
in
tersection
of
the
planes
(
F
I
J
)
and
(
AB
C
)
is
the
straigh
t
line
passing
through
I
and
through
the
middle
of
the
edge
[
D
C
]
9
The
co
ordinate
v
ector
-4
1
2
is
a
v
ector
normal
to
the
plane
(
F
I
J
)
10
The
v
olume
of
the
tetrahedron
E
F
I
J
is
equal
to
1
6
5.
Unclassifie
d
financial
ye
ars
E.8137
W
e
place
ourselv
es
in
space
pro-
vided
with
an
orthonormal
reference
frame
whose
origin
is
the
p
oin
t
A
.
Consider
the
p
oin
ts
B
10
;
−
8
;
2
,
C
−
1
;
−
8
;
5
and
D
14
;
4
;
8
.
1
a
Determine
a
system
of
parametric
equations
of
eac
h
of
the
lines
(
AB
)
and
(
C
D
)
.
b
Chec
k
that
the
straigh
t
lines
(
AB
)
and
(
C
D
)
are
not
coplanar.
2
Consider
the
p
oin
t
I
on
the
line
(
AB
)
with
abscissa
5
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3167
sacados/3167
H
G
F
E
D
C
B
A
I
J
sacados/8137
(
C
D
I
J
M
P
(
AB
M
A
B
C
D
E
F
G
H
L
S
M
I
J
K
and
the
p
oin
t
J
on
the
line
(
C
D
)
with
abscissa
4
.
a
Determine
the
co
ordinates
of
p
oin
ts
I
and
J
and
de-
duce
the
distance
I
J
.
b
Demonstrate
that
the
line
(
I
J
)
is
p
erp
endicular
to
the
lines
(
AB
)
and
(
C
D
)
.
The
line
(
I
J
)
is
called
the
common
p
erp
endicular
to
the
lines
(
AB
)
and
(
C
D
)
.
3
The
purp
ose
of
this
question
is
to
c
hec
k
that
the
dis-
tance
I
J
is
the
minim
um
distance
b
et
w
een
the
straigh
t
lines
(
AB
)
and
(
C
D
)
.
The
diagram
b
elo
w
sho
ws
the
straigh
t
lines
(
AB
)
and
(
C
D
)
,
the
p
oin
ts
I
and
J
,
and
the
line
Δ
parallel
to
the
line
(
C
D
)
passing
through
I
.
Consider
a
p
oin
t
M
on
the
line
(
AB
)
distinct
from
the
p
oin
t
I
.
Consider
a
p
oin
t
M
on
the
line
(
C
D
)
distinct
from
the
p
oin
t
J
.
a
Justify
that
the
parallel
to
the
line
(
I
J
)
passing
through
the
p
oin
t
M
in
tersects
the
line
Δ
at
a
p
oin
t
that
w
e’ll
denote
P
.
b
Sho
w
that
the
triangle
M
P
M
is
rectangular
at
P
.
c
Justify
that
M
M
>
I
J
and
conclude.
E.8141
An
artist
wishes
to
create
a
sculpture
comp
osed
of
a
tetrahedron
placed
on
a
cub
e
of
6
meters
edge.
These
t
w
o
solids
are
represen
ted
b
y
the
cub
e
AB
C
D
D
E
F
GH
and
the
tetrahe-
dron
S
E
LM
b
elo
w.
The
space
is
giv
en
the
orthonormal
reference
frame
A
;
−
→
AI
;
−
→
AJ
;
−
−
→
AK
suc
h
that
:
I
∈
[
AB
]
,
J
∈
[
AD
]
,
K
∈
[
AE
]
and
AI
=
AJ
=
AK
=
1
,
the
graphic
unit
represen
ting
1
meter.
The
p
oin
ts
L
,
M
and
S
are
defined
as
follo
ws
:
L
is
the
p
oin
t
suc
h
that
:
−
→
F
L
=
2
3
·
−
−
→
F
E
M
is
the
p
oin
t
of
in
tersection
of
the
plane
(
B
D
L
)
and
the
line
(
E
H
)
;
S
is
the
p
oin
t
of
in
tersection
of
the
straigh
t
lines
(
B
L
)
and
(
AK
)
.
1
Demonstrate,
without
calculating
co
ordinates,
that
the
straigh
t
lines
(
LM
)
and
(
B
D
)
are
parallel.
2
Sho
w
that
the
co
ordinates
of
the
p
oin
t
L
are
2
;
0
;
6
3
a
Giv
e
a
parametric
represen
tation
of
the
line
(
B
L
)
.
b
Chec
k
that
the
co
ordinates
of
p
oin
t
S
are
(
0
;
0
.
4
Let
−
→
n
b
e
the
co
ordinate
v
ector
3
;
3
;
2
.
a
Chec
k
that
−
→
n
is
normal
to
the
plane
(
B
D
L
)
.
b
Sho
w
that
a
standard
form
of
the
plane
(
B
D
L
)
is
:
3
x
+
3
y
+
2
z
−
18
=
0
c
W
e
admit
that
the
straigh
t
line
(
E
H
)
has
as
paramet-
ric
represen
tation
:
x
=
0
y
=
s
(
s
∈
R
)
z
=
6
Calculate
the
co
ordinates
of
the
p
oin
t
M
.
5
Calculate
the
v
olume
of
the
tetrahedron
S
E
LM
.
Recall
that
the
v
olume
V
of
a
tetrahedron
is
giv
en
b
y
the
fol-
lo
wing
form
ula
:
V
=
1
3
×
Aire
de
la
base
×
Hauteur
6
The
artist
w
an
ts
the
measure
of
the
angle
∠
S
LE
to
lie
b
et
w
een
55
o
and
60
o
.
Is
this
angle
constrain
t
resp
ected?
h
ttps:/
/c
hingmath.fr
(
C
D
I
J
M
P
(
AB
M
sacados/8141
Antilles-Guyane Juin 2018 5 points
A
B
C
D
E
F
G
H
L
S
M
I
J
K
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