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Ressource
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Out High
4
3
2
1
Grade 10
/ Algebraic calculation, first degree equation, problems
130 exercises (100% corrected)
a
Reminders
(3 exercices)
Development
(10 exercices)
Development: identification of terms
(3 exercices)
Factorisation : with common factor
(18 exercices)
Factoring and Product Equations
(1 exercice)
Product and 1st degree equation
(5 exercices)
Problems and Equations: Products
(11 exercices)
Functions and product equations
(8 exercices)
Factorization: recognizing common factors / multiples of each other
(22 exercices)
Factorization: recognizing common factors
(5 exercices)
Product equation: recognition of the common factor
(6 exercices)
Rational expression and equation
(7 exercices)
Algebraic expression: antecedents
(2 exercices)
Algebraic expression: antecedents and definition set
(1 exercice)
Algebraic expression: image and antecedents
(5 exercices)
Study of functions
(3 exercices)
Set of definitions
(4 exercices)
image PNG : /home/_math/_exercice/d4/4673/metapost/a-02.png
image PNG : /home/_math/_exercice/d4/4673/metapost/a-01.png
image PNG : /home/_math/_exercice/d4/4673/metapost/a-03.png
5(
x
2
x
3(
x
2
5
x
10
2
x
3
x
15
2
7
x
10
3
x
17
7
x
3
x
17
−
10
7
x
3
x
7
7
x
−
3
x
7
4
x
7
x
7
4
ChingQuizz
:
12
exercises
a
v
ailable
for
Quizz
assessmen
t
:
1.
R
eminders
E.4399
Sa
y
whether
the
follo
wing
equa-
tions
accept
for
solution
x
=
2
:
a
3
x
+
1
=
2
x
−
1
b
3(
x
+
1)
−
3(2
−
x
)
=
x
+
1
c
2
x
+
1
3
x
+
4
=
1
2
d
3
x
2
+
4
=
4
E.433
By
means
of
coun
ter-examples,
sho
w
that
the
follo
wing
equalities
are
false
:
a
3
x
+
1
=
4
x
b
(
x
+
1)
2
=
x
2
+
1
c
(
x
+
y
)
2
=
x
2
+
y
2
d
1
x
+
1
y
=
1
x
+
y
e
x
2
+
y
2
=
x
+
y
E.4401
The
diagram
b
elo
w
sho
ws
ho
w
to
solv
e
an
equation.
Complete
eac
h
of
the
lab
els
with
a
ˇ
mathematical
action
ı.
2.
Development
E.9384
Expand
the
expressions
b
elo
w
:
a
x
2
x
−
1
−
3
5
−
x
b
3
x
+
1
x
−
3
x
−
2
E.9383
Expand
the
follo
wing
expressions
:
a
3
x
−
5
−
2
x
1
−
2
x
b
3
x
+
2
−
4
2
−
2
x
E.9409
Dev
elop
and
reduce
the
follo
wing
pro
ducts
:
a
2
x
+
1
3
−
2
x
b
x
−
3
−
x
−
1
E.4382
Expand
and
giv
e
the
reduced
form
of
the
expressions
b
elo
w
:
a
(3
x
+
2)(5
−
2
x
)
b
(
x
−
1)(3
x
2
−
2)
E.9697
Expand
and
giv
e
the
reduced
form
of
the
expressions
b
elo
w
:
a
2(3
−
2
x
)
x
−
2(
x
−
2)
b
2
+
2(
x
−
5)
(
x
−
1)
c
(5
x
+
1)
2(
x
−
1)
−
5
x
E.9386
Expand
the
follo
wing
expressions
:
a
3
−
x
2
x
+1
+2
x
+2
b
x
−
1
2
x
−
1
−
3
3+2
x
E.4427
Expand
the
follo
wing
expressions
:
a
(5
x
+
1)(1
−
2
x
)
−
2(3
x
−
1)
b
(
x
+
2)(2
x
−
1)
−
(3
−
x
)(5
x
−
1)
c
(3
x
+
2)(5
x
+
1)
−
(5
x
−
1)
E.6558
Expand
and
reduce
the
follo
wing
expressions
:
a
−
5
−
2
x
+
x
+
3
2
x
+
1
b
x
1
+
x
−
x
+
2
3
−
x
E.9385
Expand
the
expressions
b
elo
w
:
a
x
2
x
+
3
1
−
x
b
x
x
+
1
4
−
2
x
E.9387
Expand
the
expression
:
x
+
1
2
x
−
1
3
−
x
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4399
sacados/4399
chapExoCorrec/433
sacados/433
chapExoCorrec/4401
sacados/4401
5(
x
2
x
3(
x
2
5
x
10
2
x
3
x
15
2
7
x
10
3
x
17
7
x
3
x
17
−
10
7
x
3
x
7
7
x
−
3
x
7
4
x
7
x
7
4
chapExoCorrec/9384
sacados/9384
chapExoCorrec/9383
sacados/9383
chapExoCorrec/9409
sacados/9409
chapExoCorrec/4382
sacados/4382
chapExoCorrec/9697
sacados/9697
chapExoCorrec/9386
sacados/9386
chapExoCorrec/4427
sacados/4427
chapExoCorrec/6558
sacados/6558
chapExoCorrec/9385
sacados/9385
chapExoCorrec/9387
sacados/9387
3.
Development
:
identific
ation
of
terms
E.8121
F
or
eac
h
of
the
questions
b
elo
w,
de-
termine
the
v
alues
of
the
reals
a
and
b
realizing
the
prop
osed
iden
tit
y
:
a
2
x
2
−
5
x
−
3
=
(2
x
+
1)(
a
×
x
+
b
)
b
−
2
x
2
+
5
x
−
3
=
(
x
−
1)(
a
×
x
+
b
)
c
−
x
2
−
3
x
+
4
=
(
−
x
+
a
)(
−
1
+
b
×
x
)
d
4
x
2
+
12
x
+
9
=
2
x
+
a
2
E.8155
1
Consider
the
algebraic
expression
−
4
x
2
+4
x
+3
.
Its
fac-
torized
form
is
one
of
the
four
expressions
b
elo
w.
Whic
h
is
it?
a
−
2
x
+
1
2
x
+
3
b
2
x
−
1
2
x
−
3
c
2
x
+
1
2
x
−
3
d
2
x
+
1
3
−
2
x
2
Determine
the
v
alues
of
the
n
um
b
ers
a
and
b
p
erforming
the
follo
wing
factorization
:
6
x
2
−
7
x
−
5
=
2
x
+
1
a
×
x
+
b
E.8160
1
Consider
the
algebraic
expression
−
4
x
2
−
4
x
+3
.
Its
fac-
torized
form
is
one
of
the
four
expressions
b
elo
w.
Whic
h
is
it?
a
−
2
x
+
1
2
x
+
3
b
2
x
−
1
2
x
−
3
c
2
x
+
1
2
x
−
3
d
2
x
+
1
3
−
2
x
2
Determine
the
v
alues
of
the
n
um
b
ers
a
and
b
p
erforming
the
follo
wing
factorization
:
8
x
2
−
2
x
−
3
=
2
x
+
1
a
×
x
+
b
4.
F
actorisation
:
with
c
ommon
factor
E.9666
F
actoring
expressions
:
a
(2
−
3
x
)(3
+
2
x
)
+
(3
x
+
2)(
−
6
x
−
9)
b
2
x
+
1
2
x
+
3
+
2
2
x
+
3
E.9389
F
actoring
expressions
:
a
3
x
+
2
2
−
2
x
+
3
x
+
2
x
+
4
b
x
−
1
2
x
−
2
+
2
x
−
2
5
−
2
x
E.8125
F
actoring
expressions
:
a
x
+
3
x
+
1
+
3
x
−
1
x
+
3
b
2
x
+
1
4
x
−
1
+
2
+
x
2
x
+
1
E.11606
F
actoring
expressions
:
a
(
x
+
1)(
x
+
2)
+
(
x
+
1)(
x
−
2)
b
(2
+
x
)(3
−
x
)
+
(5
−
2
x
)(3
−
x
)
E.4381
F
actor
the
follo
wing
expressions
:
a
(3
x
−
1)(2
x
+
1)
+
(5
−
x
)(2
x
+
1)
b
x
(2
−
x
)
+
(3
x
+
1)(2
−
x
)
E.9671
F
actoring
expressions
:
a
(5
x
+
1)(3
−
2
x
)
−
(5
x
+
1)(2
x
+
1)
b
(
x
+
1)(1
−
x
)
−
(
x
+
1)(2
x
+
1)
E.9608
F
actoring
expressions
:
a
4
−
3
x
x
+
5
−
4
−
3
x
x
+
2
b
2
x
+
5
x
+
2
−
2
x
+
5
3
x
−
2
E.9388
F
actoring
expressions
:
a
(5
x
+
2)(3
x
+
4)
+
(
x
−
2)(3
x
+
4)
b
(3
−
x
)(2
x
+
4)
−
(3
−
x
)(3
x
−
4)
E.9412
F
actoring
expressions
:
a
(
x
+
1)(
x
−
1)
−
(2
x
+
3)(
x
−
1)
b
(3
x
+
4)(2
x
−
1)
+
4(3
x
+
4)
E.11475
F
actoring
expressions
:
a
4
−
3
x
x
+
5
+
4
−
3
x
b
2
x
+
5
x
+
2
−
2
x
+
5
E.434
F
actoring
expressions
:
a
(
x
+
1)(2
x
−
1)
−
(2
x
−
1)
b
(
x
−
2)(
x
+
3)
−
(
x
−
2)
c
(
x
+
1)
×
x
+
2(
x
+
1)
E.9411
F
actoring
expressions
:
a
(2
x
+
4)(3
−
3
x
)
+
(2
x
+
4)
b
(5
x
+
1)(7
−
3
x
)
−
(5
x
+
1)
E.11609
F
actor
the
follo
wing
expressions
:
a
3(
x
+
1)(2
x
−
1)
+
(
x
+
1)(5
−
2
x
)
b
(
x
−
2)(
x
+
1)
−
2(
x
−
2)(2
x
+
3)
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8121
sacados/8121
chapExoCorrec/8155
sacados/8155
chapExoCorrec/8160
sacados/8160
chapExoCorrec/9666
sacados/9666
chapExoCorrec/9389
sacados/9389
chapExoCorrec/8125
sacados/8125
chapExoCorrec/11606
sacados/11606
chapExoCorrec/4381
sacados/4381
chapExoCorrec/9671
sacados/9671
chapExoCorrec/9608
sacados/9608
chapExoCorrec/9388
sacados/9388
chapExoCorrec/9412
sacados/9412
chapExoCorrec/11475
sacados/11475
chapExoCorrec/434
sacados/434
chapExoCorrec/9411
sacados/9411
chapExoCorrec/11609
sacados/11609
16
12
x
x
E.11610
F
actor
the
follo
wing
expressions
:
a
(2
x
−
1)
2
+
(2
x
−
1)(3
x
+
1)
b
(
x
+
3)
2
+
(
x
+
3)(2
x
−
4)
E.11612
F
actoring
expressions
:
a
(3
x
−
2)(
x
+
4)
+
(3
x
−
2)
2
b
(
x
+
2)
2
−
(
x
+
2)
E.9609
F
actoring
expressions
:
a
5
−
x
2
+
5
−
x
x
+
1
b
x
3
x
+
5)
+
3
x
+
5
2
E.9410
F
actoring
expressions
:
a
(
x
+
1)(3
−
2
x
)
+
(3
−
2
x
)
2
b
x
+
5
3
−
2
x
−
3
−
2
x
2
E.8159
F
actoring
expressions
:
a
(3
x
−
1)
2
+
(3
x
−
1)(5
x
+
4)
b
x
+
5
4
−
x
−
4
−
x
2
5.
F
actoring
and
Pr
o
duct
Equations
E.2110
Solv
e
b
y
the
metho
d
of
y
our
c
hoice
the
follo
wing
equations
:
a
(3
x
+
1)(2
−
3
x
)
−
(5
x
−
1)(3
x
+
1)
=
0
b
2(
x
+
2)(3
−
x
)
=
(
x
+
2)(5
x
−
7)
6.
Pr
o
duct
and
1st
de
gr
e
e
e
quation
E.2096
1
Expand
eac
h
of
the
follo
wing
expressions
:
a
x
(
x
−
3)
−
x
2
b
(6
x
+
1)
2
−
(12
x
+
2)(3
x
−
3)
2
Solv
e
the
follo
wing
equations
after
dev
elopmen
t
and
re-
duction
:
a
x
(
x
−
3)
−
x
2
=
0
b
(6
x
+
1)
2
=
(12
x
+
2)(3
x
−
3)
E.453
1
Using
men
tal
arithmetic,
whic
h
equations,
after
expan-
sion
and
reduction,
ha
v
e
no
terms
in
x
2
:
a
(
x
+
1)(
x
−
1)
−
(
x
+
1)(2
x
+
1)
=
0
b
x
2
−
8
=
(
x
+
3)(1
+
x
)
c
(3
x
−
2)
2
=
6
x
−
4
d
(2
x
+
1)(1
−
x
)
=
(3
x
−
3)(
x
+
2)
e
3
x
(4
x
−
1)
−
(2
x
−
5)(6
x
+
4)
=
0
2
Solv
e
the
equations
in
questions
b
and
e
.
E.9610
Solv
e
the
follo
wing
equations
using
the
metho
d
of
y
our
c
hoice
:
a
(2
x
+
3)(6
x
+
7)
+
(2
−
4
x
)(3
x
+
1)
=
3
x
−
7
b
2
x
+
1
x
−
2
+
3
x
−
5
2
x
+
1
=
0
E.8156
Solv
e
in
R
the
follo
wing
equations
:
a
2
x
+
1
4
−
x
+
4
x
−
1
2
x
+
1
=
0
b
−
(12
x
−
2)(2
−
3
x
)
=
36
x
2
−
12
x
+
1
E.443
Solv
e
the
follo
wing
equations
:
a
2
·
(6
x
+
4)(3
−
4
x
)
−
(8
x
−
6)
2
=
0
b
3
·
2
x
−
4
2
=
6
x
2
−
4
x
+
12
7.
Pr
oblems
and
Equations
:
Pr
o
ducts
E.4489
On
a
rectangular
former
w
asteland
of
length
16
m
and
12
m
,
the
m
unicipalit
y
wishes
to
build
a
kindergarten
with
a
path
w
a
y
running
around
the
pla
y
area
:
The
pla
yground
is
represen
ted
b
elo
w
b
y
the
hatc
hed
area
:
h
ttps:/
/c
hingmath.fr
chapExoCorrec/11610
sacados/11610
chapExoCorrec/11612
sacados/11612
chapExoCorrec/9609
sacados/9609
chapExoCorrec/9410
sacados/9410
chapExoCorrec/8159
sacados/8159
chapExoCorrec/2110
sacados/2110
chapExoCorrec/2096
sacados/2096
chapExoCorrec/453
sacados/453
chapExoCorrec/9610
sacados/9610
chapExoCorrec/8156
sacados/8156
chapExoCorrec/443
sacados/443
chapExoCorrec/4489
sacados/4489
16
12
x
x
9
m
5
m
x
A
B
C
D
4
·
x
−
1
x
A
B
C
D
H
S
x
x
−
1
6
x
2
3
E
F
G
I
J
K
L
A
B
C
D
E
F
G
x
4
m
1
Without
justification,
sp
ecify
the
p
ossible
v
alues
of
the
v
ariable
x
for
this
problem.
2
a
Justify
that
the
pla
ying
area
measures,
as
a
function
of
x
:
4
x
2
−
56
x
+
192
b
Justify
that
the
area
of
the
aisle
measures,
as
a
func-
tion
of
x
:
56
x
−
4
x
2
3
a
Establish
the
follo
wing
equalit
y
:
8
x
2
−
112
x
+
192
=
8(
x
−
12)(
x
−
2)
b
Determine
the
p
ossible
aisle
widths
so
that
the
pla
y
area
has
the
same
area
as
the
aisle.
E.8154
Bac
king
on
to
his
house,
Jean
has
a
rectangular-shap
ed
garden
with
dimensions
9
m
and
5
m
.
He
w
an
ts
to
build
a
driv
ew
a
y
on
three
of
the
sides
of
this
garden
with
the
same
width,
and
he
will
plan
t
la
wn
on
the
rest
of
the
garden.
He
prop
oses
the
diagram
b
elo
w
où
the
hatc
hed
area
is
the
la
wn
space
Ho
w
wide
m
ust
the
driv
ew
a
y
b
e
for
the
whole
la
wn
to
ha
v
e
a
surface
area
of
10
m
2
?
Hin
t
:
Use
one
of
the
factorized
forms
b
elo
w
:
2
x
2
−
18
x
+28
=
2
x
−
4
x
−
7
2
x
2
−
20
x
+32
=
2
x
−
4
x
−
8
2
x
2
−
19
x
+35
=
2
x
−
5
x
−
7
2
x
2
−
21
x
+40
=
2
x
−
5
x
−
8
E.9754
1
Establish
the
follo
wing
factorization
:
x
+
5
4
x
−
1
−
10
x
−
8
=
(
x
−
1)(4
x
+
13)
2
Consider
the
rectangle
AB
C
D
whose
dimensions
are
a
function
of
a
real
n
um
b
er
x
and
are
giv
en
in
cen
timeters
:
AB
=
4
x
−
1
;
AD
=
x
+
5
a
What
are
the
p
ossible
v
alues
of
parameter
x
.
b
Determine
the
v
alue(s)
of
x
so
that
the
p
erimeter,
ex-
pressed
in
cm
,
of
the
rectangle
AB
C
D
is
equal
to
the
area,
expressed
in
cm
2
,
of
the
rectangle
AB
C
D
.
E.6599
Consider
the
t
w
o
surfaces
AB
C
S
D
and
E
F
LK
J
I
sho
wn
b
elo
w
où
x
is
a
real
n
um
b
er.
AB
C
D
is
a
square
with
side
x
and
S
D
C
is
a
triangle
whose
heigh
t
[
S
H
]
has
measure
x
−
1
.
The
quadrilaterals
E
F
GI
and
GJ
K
L
are
t
w
o
rectangles.
1
What
are
the
p
ossible
v
alues
of
the
v
ariable
x
according
to
the
constrain
ts
of
the
figures?
2
Express
the
area
of
these
t
w
o
surfaces
as
a
function
of
x
.
3
a
Establish
factorization
:
3
2
·
x
2
−
9
2
·
x
−
6
=
3
2
·
x
−
6
x
+
1
b
Determine
the
p
ossible
v
alue(s)
of
the
v
ariable
x
to
obtain
equalit
y
of
areas
of
these
t
w
o
surfaces.
E.4422
A
field
is
made
up
of
t
w
o
squares
and
a
righ
t-angled
triangle.
This
field
is
sho
wn
in
the
figure
b
elo
w
:
1
Justify
the
follo
wing
factorization
:
x
2
+
2
x
−
168
=
(
x
+
14)(
x
−
12)
2
a
Justify
that
the
square
C
E
F
G
has
area
(
x
2
+
16)
m
2
.
b
Deduce
the
v
alue
of
the
length
x
so
that
the
total
area
of
the
field
is
200
m
2
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8154
sacados/8154
9
m
5
m
x
chapExoCorrec/9754
sacados/9754
A
B
C
D
4
·
x
−
1
x
chapExoCorrec/6599
sacados/6599
A
B
C
D
H
S
x
x
−
1
6
x
2
3
E
F
G
I
J
K
L
chapExoCorrec/4422
sacados/4422
A
B
C
D
E
F
G
x
4
m
20
m
15
m
x
x
A
B
C
D
M
N
P
Q
x
6
cm
A
B
C
D
E
F
16
cm
6
cm
x
x
E.1859
A
garden
is
rectangular,
with
a
length
of
20
m
and
a
width
of
15
m
.
T
w
o
paths,
eac
h
x
m
wide,
run
across
the
garden
;
the
rest
of
the
garden
will
b
e
plan
ted
with
grass.
A
fence
m
ust
b
e
installed
around
the
la
wn
:
it
is
sho
wn
as
a
dotted
line
in
the
diagram.
1
Indicate
the
p
ossible
v
alues
of
the
v
ariable
x
.
2
a
Determine
the
total
area
of
the
t
w
o
paths
in
terms
of
x
.
b
Determine,
in
terms
of
x
,
the
area
of
the
la
wn
in
this
garden.
3
a
Determine
the
v
alues
of
the
real
n
um
b
ers
a
and
b
that
satisfy
the
equation
:
2
·
x
2
−
70
·
x
+
300
=
(
x
−
30)(
a
·
x
+
b
)
b
The
arc
hitect
in
c
harge
of
designing
this
garden
de-
cides
to
c
ho
ose
the
width
of
the
path
so
that
the
areas
of
the
paths
and
the
la
wn
are
equal.
4
The
garden
o
wner
decides
to
in
v
est
5
600
euros
in
land-
scaping
the
garden.
m
2
of
la
wn
costs
7
e
;
m
2
of
the
w
o
o
d
used
for
the
w
alk-
w
a
y
costs
30
e
;
m
of
the
fence
costs
12
e
.
a
Establish
the
follo
wing
equation
:
23
x
2
−
757
x
+
2660
=
(
x
−
4)(23
x
−
665)
b
Use
this
to
determine
the
width
of
the
paths
that
form
the
o
wner’s
designs.
E.9700
Consider
the
square
AB
C
D
with
sides
6
cm
and
four
p
oin
ts
M
,
N
,
P
,
Q
suc
h
that
:
AM
=
B
N
=
C
P
=
D
Q
W
e
admit
that
M
N
P
Q
is
a
square
and
note
:
x
=
B
M
1
Justify
that
the
area
A
of
the
square
M
N
P
Q
has
the
v
alue
:
A
=
2
x
2
−
12
x
+
36
2
a
Andabir
factorization
:
2
x
2
−
12
x
+
27
2
=
1
2
2
x
−
9
2
x
−
3
b
Determine
the
v
alue(s)
of
x
so
that
the
area
of
the
square
M
N
P
Q
has
the
v
alue
5
8
of
that
of
the
square
AB
C
D
.
E.8158
Consider
the
figure
b
elo
w
consisting
of
a
rectangle
AB
C
D
of
dimension
16
cm
and
6
cm
and
the
t
w
o
p
oin
ts
E
and
F
b
elonging
resp
ectiv
ely
to
the
segmen
ts
[
B
C
]
and
[
C
D
]
suc
h
that
:
C
E
=
D
F
=
x
where
x
is
a
real
n
um
b
er.
Consider
the
hatc
hed
area
of
the
figure
defined
b
y
the
triangle
AE
F
.
1
Giv
e
the
set
of
p
ossible
v
alues
of
the
n
um
b
er
x
.
2
a
Justify
that
the
ˇ
blanc
he
ı
part
of
this
figure
has
area
A
whose
expression
as
a
function
of
x
is
:
A
=
−
1
2
x
2
+
3
x
+
48
b
Determine
the
area
A
of
the
part
ˇ
hac
h
urée
ı.
3
Determine
the
v
alue(s)
of
x
to
obtain
the
t
w
o
domains
ˇ
blancs
ı
and
ˇ
hac
h
urés
ı
of
the
same
area.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/1859
sacados/1859
20
m
15
m
x
x
chapExoCorrec/9700
sacados/9700
A
B
C
D
M
N
P
Q
x
6
cm
chapExoCorrec/8158
sacados/8158
A
B
C
D
E
F
16
cm
6
cm
x
x
A
B
C
D
E
F
18
cm
10
cm
x
x
32
cm
10
cm
6
cm
x
x
x
x
x
x
10
m
x
A
B
C
D
M
N
O
P
I
J
O
C
f
C
g
E.8157
Consider
the
figure
b
elo
w
consisting
of
a
rectangle
AB
C
D
of
dimension
18
cm
and
10
cm
and
the
t
w
o
p
oin
ts
E
and
F
b
elonging
resp
ectiv
ely
to
the
segmen
ts
[
B
C
]
and
[
C
D
]
suc
h
that
:
C
E
=
D
F
=
x
where
x
is
a
real
n
um
b
er.
Consider
the
hatc
hed
area
of
the
figure
defined
b
y
the
triangle
AE
F
.
1
Giv
e
the
set
of
p
ossible
v
alues
of
the
n
um
b
er
x
.
2
a
Justify
that
the
ˇ
blanc
he
ı
part
of
this
figure
has
area
A
whose
expression
as
a
function
of
x
is
:
A
=
−
1
2
x
2
+
5
x
+
90
b
Determine
the
area
A
of
the
ˇ
hac
h
urée
ı
part.
Hin
t
:
remem
b
er
to
factor
b
y
x
.
3
Determine
the
v
alue(s)
of
x
to
obtain
the
t
w
o
domains
ˇ
blancs
ı
and
ˇ
hac
h
urés
ı
of
the
same
area.
E.4527
A
w
orkshop
has
a
blo
c
k
of
marble
with
a
parallelepip
ed
shap
e
and
dimensions
32
cm
long,
10
cm
deep
and
6
cm
high.
W
e
wish
to
reco
v
er
the
ˇ
heart
ı
of
this
blo
c
k.
T
o
do
this,
w
e
plane
eac
h
side
of
this
righ
t
blo
c
k
b
y
a
length
of
x
cm
:
1
Giv
e
the
p
ossible
v
alues
tak
en
b
y
the
v
ariable
x
.
2
a
Determine
the
v
olume
of
the
ˇ
heart
ı
of
this
marble
blo
c
k.
b
Deduce
the
v
olume
of
the
planed
part.
3
a
Expand
:
−
16
·
(
x
−
1)(
x
−
8)(
x
−
15)
b
F
or
what
v
alue
of
x
,
the
v
olume
of
the
planed
part
is
equal
to
the
v
olume
of
the
ˇ
co
eur
ı
of
this
part.
E.8126
Consider
a
park
represen
ted
b
y
the
square
AB
C
D
opp
osite.
This
park
consists
of
a
w
o
o
ded
area
(in
gr
ey)
and
a
driv
ew
a
y
(in
white)
.
The
hexagon
AM
N
C
O
P
repre-
sen
ting
the
driv
ew
a
y
has
its
sides
[
O
P
]
and
[
M
N
]
parallel
to
the
di-
agonal
[
AC
]
of
the
square
AB
C
D
.
W
e
note
x
the
measure
of
segmen
t
[
C
N
]
and
w
e
ha
v
e
equalit
y
of
lengths
:
C
N
=
C
O
Determine
the
v
alue(s)
of
x
so
that
the
w
o
o
ded
area
measures
one-third
of
the
driv
ew
a
y
.
Hin
t
:
Y
ou
can
use
one
of
the
follo
wing
factorizations
:
x
2
−
20
x
+75
=
x
−
15
x
−
5
x
2
+20
x
+75
=
x
+5
x
+15
x
2
+25
x
+150
=
x
+10
x
+15
x
2
−
25
x
+150
=
x
−
10
x
−
15
8.
F
unctions
and
pr
o
duct
e
quations
E.8162
Consider
the
t
w
o
second-degree
func-
tions
f
and
g
defined
on
R
b
y
the
algebraic
expressions
:
f
(
x
)
=
3
x
2
+
3
x
−
4
;
g
(
x
)
=
x
+
1
In
the
reference
frame
O
;
I
;
J
orthogonal
b
elo
w,
consider
the
represen
tativ
e
curv
es
C
f
and
C
g
giv
en
b
elo
w
:
1
Establish
the
factorization
:
f
(
x
)
−
g
(
x
)
=
x
−
1
3
x
+
5
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8157
sacados/8157
A
B
C
D
E
F
18
cm
10
cm
x
x
chapExoCorrec/4527
sacados/4527
32
cm
10
cm
6
cm
x
x
x
x
x
x
chapExoCorrec/8126
sacados/8126
10
m
x
A
B
C
D
M
N
O
P
chapExoCorrec/8162
sacados/8162
I
J
O
C
f
C
g
I
J
O
C
f
C
g
-4
-3
-2
-1
2
3
4
I
-2
-1
2
J
O
C
f
C
g
I
J
O
C
f
C
g
-3
-2
-1
2
3
4
5
6
7
I
-1
2
3
4
J
O
C
g
C
f
2
Deduce
the
co
ordinates
of
the
in
tersection
p
oin
ts
of
the
curv
es
C
f
and
C
g
.
E.8161
Consider
the
t
w
o
second-degree
func-
tions
f
and
g
defined
on
R
b
y
the
algebraic
expressions
:
f
(
x
)
=
3
x
2
+
2
x
−
1
;
g
(
x
)
=
x
+
1
In
the
reference
frame
O
;
I
;
J
orthogonal
b
elo
w,
consider
the
represen
tativ
e
curv
es
C
f
and
C
g
giv
en
b
elo
w
:
1
Establish
the
factorization
:
f
(
x
)
−
g
(
x
)
=
x
+
1
3
x
−
2
2
Deduce
the
co
ordinates
of
the
in
tersection
p
oin
ts
of
the
curv
es
C
f
and
C
g
.
E.8153
Consider
the
t
w
o
second-degree
func-
tions
f
and
g
defined
on
R
b
y
the
algebraic
expressions
:
f
(
x
)
=
3
x
2
+
2
;
g
(
x
)
=
x
2
+
x
+
5
Note
resp
ectiv
ely
C
f
and
C
the
represen
tativ
e
curv
es
of
the
functions
f
and
g
in
an
y
reference
frame.
1
Using
the
calculator,
dra
w
the
curv
es
C
f
and
C
g
and
conjecture
the
co
ordinates
of
their
p
oin
ts
of
in
tersection.
2
a
Establish
factorization
:
f
(
x
)
−
g
(
x
)
=
2
x
−
3
x
+
1
b
Determine
the
co
ordinates
of
the
iin
tersection
p
oin
ts
of
the
curv
es
C
f
and
C
g
.
E.9701
Consider
the
t
w
o
functions
f
and
g
defined
on
R
b
y
:
f
(
x
)
=
(
−
x
−
2)(
x
+
3)
;
g
(
x
)
=
3
x
2
+
3
x
−
3
In
the
plane
pro
vided
with
a
reference
frame
O
;
I
;
J
,
w
e
note
resp
ectiv
ely
C
f
and
C
g
the
represen
tativ
e
curv
es
of
the
functions
f
and
g
:
1
Establish
factorization
:
f
(
x
)
−
g
(
x
)
=
−
2
x
−
1
2
x
+
3
2
Deduce
the
abscissas
of
the
p
oin
ts
of
in
tersection
of
these
t
w
o
curv
es.
E.6600
Consider
the
t
w
o
functions
f
and
g
defined
on
R
b
y
the
relations
:
f
(
x
)
=
x
2
+
2
·
x
+
1
;
g
(
x
)
=
−
x
+
1
x
−
2
The
curv
es
C
f
and
C
g
represen
tativ
e
resp
ectiv
ely
of
the
func-
tions
f
and
g
are
giv
en
in
the
orthonormal
frame
O
;
I
;
J
.
Determine
the
co
ordinates
of
the
p
oin
ts
of
in
tersection
of
these
t
w
o
curv
es.
An
y
trace
of
researc
h,
ev
en
if
incomplete,
will
b
e
tak
en
in
to
accoun
t
in
the
assessmen
t.
E.4472
Consider
the
t
w
o
functions
f
and
g
whose
images
of
a
n
um
b
er
x
are
giv
en
b
y
the
relation
:
f
(
x
)
=
−
1
4
x
2
+
x
+
3
;
g
(
x
)
=
1
2
x
+
1
The
represen
tation
C
f
and
C
g
of
the
functions
f
and
g
are
giv
en
b
elo
w
in
the
frame
O
;
I
;
J
:
1
a
Determine
the
v
alues
of
the
reals
a
and
b
v
erifying
the
equalit
y
:
−
x
2
+
4
x
+
12
=
x
−
6
a
·
x
+
b
b
Deduce
the
solutions
of
the
equation
:
f
(
x
)
=
0
2
a
Establish
the
follo
wing
equalit
y
:
−
x
2
4
+
x
2
+
2
=
−
(
x
−
4)(
x
+
2)
4
b
Solv
e
the
equation
:
f
(
x
)
=
g
(
x
)
c
Deduce
the
co
ordinates
of
the
in
tersection
p
oin
ts
of
the
curv
es
C
f
and
C
g
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8161
sacados/8161
I
J
O
C
f
C
g
chapExoCorrec/8153
sacados/8153
chapExoCorrec/9701
sacados/9701
-4
-3
-2
-1
2
3
4
I
-2
-1
2
J
O
C
f
C
g
chapExoCorrec/6600
sacados/6600
I
J
O
C
f
C
g
chapExoCorrec/4472
sacados/4472
-3
-2
-1
2
3
4
5
6
7
I
-1
2
3
4
J
O
C
g
C
f
-6
-5
-4
-3
-2
-1
2
3
4
I
-2
-1
2
3
4
J
O
C
f
C
g
-4
-3
-2
-1
2
3
4
I
-8
-6
-4
-2
2
4
6
J
O
C
f
E.4488
Consider
the
t
w
o
functions
f
and
g
whose
image
of
a
n
um
b
er
x
is
giv
en
b
y
the
relation
:
f
(
x
)
=
1
4
x
2
+
1
4
x
−
3
2
;
g
(
x
)
=
−
1
4
x
+
9
4
W
e
giv
e
the
represen
tations
C
f
and
C
g
resp
ectiv
ely
of
f
and
g
in
the
frame
O
;
I
;
J
b
elo
w
:
1
Graphically
,
giv
e
the
an
teceden
ts
of
0
b
y
the
function
f
.
The
follo
wing
questions
m
ust
b
e
dealt
with
algebraically
:
2
Determine
the
an
teceden
ts
of
the
n
um
b
er
0
b
y
the
func-
tion
g
.
3
a
Determine
the
v
alue
of
the
n
um
b
ers
a
and
b
v
erify-
ing
the
follo
wing
factorization
:
x
2
+
2
x
−
15
=
(
x
−
3)(
a
·
x
+
b
)
b
Deduce
the
solutions
of
the
equation
:
f
(
x
)
=
g
(
x
)
c
Giv
e
the
co
ordinates
of
the
in
tersection
p
oin
ts
of
the
curv
es
C
f
and
C
g
.
E.4444
In
the
(
O
;
I
;
J
)
orthogonal
repre-
sen
ted
b
elo
w,
the
curv
e
C
f
is
the
graphical
represen
tation
of
a
function
f
defined
on
R
:
1
a
Determine
the
image
of
the
n
um
b
er
−
3
b
y
the
func-
tion
f
.
Justify
y
our
answ
er.
b
Solv
e,
graphically
,
the
equation
f
(
x
)
=
1
.
Justify
y
our
answ
er.
2
The
image
of
a
n
um
b
er
x
b
y
the
function
f
is
giv
en
b
y
the
relation
:
f
(
x
)
=
x
3
+
2
x
2
−
5
x
−
5
a
Justify
,
b
y
calculation,
the
v
alue
of
the
image
of
the
n
um
b
er
−
3
.
b
Establish
the
follo
wing
equalit
y
:
x
3
+
2
x
2
−
5
x
−
5
=
(
x
+
3)(
x
−
2)(
x
+
1)
+
1
c
Solv
e,
b
y
calculation,
the
equation
f
(
x
)
=
1
.
9.
F
actorization
:
r
e
c
o
gnizing
c
ommon
factors
/
multiples
of
e
ach
other
E.2095
1
Find
an
algebraic
relationship
b
et
w
een
the
t
w
o
expres-
sions
:
3
x
−
2
;
6
x
−
4
2
Deduce
a
factorization
of
the
follo
wing
algebraic
expres-
sion
:
A
=
(
x
+
2)(3
x
−
2)
+
(5
x
−
2)(6
x
−
4)
E.11604
F
actoring
expressions
:
a
(2
x
+
1)(3
x
−
1)
−
(
x
+
3)(6
x
−
2)
b
(2
x
−
4)(3
x
+
1)
−
(6
x
+
2)(4
x
+
1)
E.450
F
actor
the
expressions
:
a
(7
x
−
1)(5
x
−
6)
−
(10
x
−
12)
b
(7
x
−
1)(9
x
−
3)
−
(3
x
−
1)
E.4446
F
actor
the
follo
wing
expressions
:
a
(3
x
+
2)(
x
+
4)
+
(6
x
+
4)(4
x
+
1)
b
(3
x
+
2)(2
x
−
1)
+
(4
x
−
2)(3
−
5
x
)
E.11599
F
actor
the
follo
wing
expressions
:
a
(
x
+
1)(3
x
+
2)
−
(2
x
−
2)(6
x
+
4)
b
(5
x
+
2)(6
x
+
3)
−
(3
−
2
x
)(2
x
+
1)
E.2109
F
actoring
expressions
:
a
(
x
−
1)(2
x
+
1)
−
(2
x
−
2)(5
−
2
x
)
b
(2
−
x
)(3
x
−
4)
+
2
−
3
2
x
(2
x
+
3)
E.11603
F
actoring
expressions
:
a
(6
x
−
4)(5
x
+
1)
−
(3
x
−
2)
2
b
6
x
−
3
2
x
+
1
−
2
2
x
−
1
2
E.9657
F
actor
the
follo
wing
expressions
:
a
(6
x
−
4)(5
x
+
1)
−
(3
x
−
2)
2
b
(
x
−
5)
2
+
(
x
+
3)(2
x
−
10)
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4488
sacados/4488
-6
-5
-4
-3
-2
-1
2
3
4
I
-2
-1
2
3
4
J
O
C
f
C
g
chapExoCorrec/4444
sacados/4444
-4
-3
-2
-1
2
3
4
I
-8
-6
-4
-2
2
4
6
J
O
C
f
chapExoCorrec/2095
sacados/2095
chapExoCorrec/11604
sacados/11604
chapExoCorrec/450
sacados/450
chapExoCorrec/4446
sacados/4446
chapExoCorrec/11599
sacados/11599
chapExoCorrec/2109
sacados/2109
chapExoCorrec/11603
sacados/11603
chapExoCorrec/9657
sacados/9657
E.11598
1
Find
an
algebraic
relationship
b
et
w
een
:
3
−
x
;
x
−
3
2
Deduce
a
factorization
of
the
follo
wing
algebraic
expres-
sion
:
B
=
(2
x
+
1)(3
−
x
)
−
(2
−
2
x
)(
x
−
3)
E.11597
1
Find
an
algebraic
relationship
b
et
w
een
:
2
x
−
1
;
2
−
4
x
?
2
Deduce
a
factorization
of
the
follo
wing
algebraic
expres-
sion
:
C
=
(5
−
2
x
)(2
x
−
1)
+
(2
−
4
x
)
E.4485
F
actor
the
follo
wing
expressions
:
a
(2
x
−
1)(5
−
3
x
)
+
(5
x
−
2)(3
x
−
5)
b
(3
x
−
1)(2
−
x
)
−
2(1
−
3
x
)(4
x
−
3)
E.11600
F
actor
the
follo
wing
expressions
:
a
(7
x
−
2)(5
−
x
)
+
(4
x
−
1)(
x
−
5)
b
(2
x
−
3)(4
−
7
x
)
−
(3
x
−
2)(7
x
−
4)
E.824
F
actoring
expressions
:
a
3(
x
−
2)
+
(
x
+
2)(2
−
x
)
b
(2
x
−
3)(5
x
+
4)
−
(2
x
−
3)(3
−
2
x
)
E.9667
F
actoring
expressions
:
a
3(2
x
−
2)
+
(
x
+
1)(1
−
x
)
b
(2
−
6
x
)
+
(
x
+
1)(3
x
−
1)
E.9665
F
actoring
expressions
:
a
(
x
−
1)(3
x
+
2)
+
(2
x
+
3)(1
−
x
)
b
(2
−
x
)(2
+
x
)
−
5(2
−
x
)
+
(
−
x
+
1)(
x
−
2)
E.6597
F
actor
the
follo
wing
expressions
:
a
(25
x
−
10)(
x
+
1)
+
(2
−
5
x
)(1
−
x
)
b
(
x
+
3)(2
x
−
1)
+
(2
−
4
x
)(
x
−
1)
E.9611
F
actor
the
follo
wing
expressions
:
a
x
(
x
−
5)
+
(10
−
2
x
)(3
x
−
1)
b
(12
x
−
3)(7
+
2
x
)
−
(5
−
2
x
)(1
−
4
x
)
E.9678
F
actoring
expressions
:
a
3
x
+
2
x
−
2
+
4
−
2
x
2
x
+
3
b
(3
−
x
)(7
x
+
1)
−
2(2
x
+
2)(3
x
−
9)
E.11601
F
actoring
expressions
:
a
(2
x
−
1)(3
x
+
2)
+
(2
x
+
3)(2
−
4
x
)
b
(3
x
+
1)(2
−
2
x
)
−
(5
−
4
x
)(
x
−
1)
E.5902
F
actor
the
follo
wing
expressions
:
a
(2
x
+
3)(1
−
x
)
+
(4
x
+
6)
2
b
(3
−
9
x
)
2
+
3(3
x
−
1)
E.11602
F
actoring
expressions
:
a
x
(
x
−
2)
+
(3
x
−
6)
2
b
(2
+
x
)(5
−
x
)
+
(2
x
+
4)
2
E.2850
F
actoring
expressions
:
a
x
+
1
5
−
2
x
3
x
−
4
+
3
2
x
−
5
6
x
−
8
10.
F
actorization
:
r
e
c
o
gnizing
c
ommon
factors
E.436
F
actorize
the
expressions,
if
p
ossible
:
a
3(4
+
2
x
)
−
(3
+
x
)(10
+
5
x
)
b
(6
x
−
9)(
x
+
1)
−
(4
x
−
6)
E.2099
F
actor
the
follo
wing
expressions
:
a
(3
x
−
3)(5
x
+
2)
−
(2
x
−
2)(3
x
−
1)
b
3(4
+
2
x
)
−
(3
+
x
)(10
+
5
x
)
E.10189
F
actor
the
follo
wing
expressions
:
a
(2
x
−
4)(
x
+
4)
+
(6
−
3
x
)(4
x
+
2)
b
2(4
x
+
6)(5
−
2
x
)
+
(
x
−
3)(6
x
+
9)
E.11611
F
actor
the
follo
wing
expressions
:
a
(6
x
+
2)(2
x
+
3)
+
(9
x
+
3)
2
b
(5
x
+
1)(2
x
−
4)
+
(3
x
−
6)
2
E.11605
F
actoring
expressions
:
a
(6
x
+
2)(2
x
+
3)
+
(9
x
+
3)
2
b
h
ttps:/
/c
hingmath.fr
chapExoCorrec/11598
sacados/11598
chapExoCorrec/11597
sacados/11597
chapExoCorrec/4485
sacados/4485
chapExoCorrec/11600
sacados/11600
chapExoCorrec/824
sacados/824
chapExoCorrec/9667
sacados/9667
chapExoCorrec/9665
sacados/9665
chapExoCorrec/6597
sacados/6597
chapExoCorrec/9611
sacados/9611
chapExoCorrec/9678
sacados/9678
chapExoCorrec/11601
sacados/11601
chapExoCorrec/5902
sacados/5902
chapExoCorrec/11602
sacados/11602
chapExoCorrec/2850
sacados/2850
chapExoCorrec/436
sacados/436
chapExoCorrec/2099
sacados/2099
chapExoCorrec/10189
sacados/10189
chapExoCorrec/11611
sacados/11611
chapExoCorrec/11605
sacados/11605
11.
Pr
o
duct
e
quation
:
r
e
c
o
gnition
of
the
c
ommon
factor
E.2823
1
a
F
actor
the
follo
wing
algebraic
expression
:
(3
x
+
2)(2
x
−
1)
+
(4
x
−
2)(3
−
5
x
)
b
Solv
e
the
follo
wing
equation
:
(3
x
+
2)(2
x
−
1)
+
(4
x
−
2)(3
−
5
x
)
=
0
2
a
F
actor
the
follo
wing
expression
:
(2
x
+
1)(3
−
2
x
)
−
(3
x
−
2)(2
x
−
3)
b
Solv
e
the
follo
wing
equation
:
(2
x
+
1)(3
−
2
x
)
=
(3
x
−
2)(2
x
−
3)
E.4388
1
a
Sho
w
that
the
follo
wing
t
w
o
equations
are
equiv
a-
len
t
:
x
2
=
x
;
x
(
x
−
1)
=
0
b
Deduce
the
solutions
of
the
equation
:
x
2
=
x
2
Solv
e
the
follo
wing
equations
:
a
(
x
−
2)(3
−
2
x
)
=
0
b
(5
x
−
1)(2
−
x
)
+
(2
x
−
4)(3
−
2
x
)
=
0
E.4560
Solv
e
the
follo
wing
equations
:
a
(
x
+
2)(3
−
x
)
+
2(
x
−
3)(2
x
−
5)
=
0
b
(6
−
2
x
)(3
x
+
2)
=
(3
x
−
9)(
x
+
2)
E.2851
Reducing
to
a
pro
duct
equation,
solv
e
the
follo
wing
equations
:
a
3
x
−
1
2
x
+
2
+
3
5
−
2
x
x
+
1
=
0
b
3
5
x
+
1
2
−
3
x
+
6
x
−
4
x
−
1
=
0
c
4
x
+
6
1
−
2
x
=
5
2
x
+
3
2
E.2875
1
F
actor
the
follo
wing
expressions
:
a
(5
x
+
1)(6
+
4
x
)
+
(3
x
+
9)(2
x
+
3)
b
(3
−
2
x
)(4
x
+
1)
+
4
x
2
−
12
x
+
9
2
Solv
e
the
follo
wing
equations
:
a
(4
−
2
x
)(3
x
+
2)
=
3(2
x
+
3)(
x
−
2)
b
(4
x
+
3)(2
−
3
x
)
+
(2
−
6
x
)(3
x
−
2)
=
0
E.4486
Solv
e
the
follo
wing
equations
:
a
(5
x
−
1)(3
x
−
2)
+
(6
x
−
4)(
x
−
1)
=
0
b
(3
−
2
x
)(2
x
+
4)
=
(2
x
−
3)(3
−
9
x
)
c
(6
x
−
3)(1
−
2
x
)
+
(3
−
4
x
)(2
−
3
x
)
=
0
12.
R
ational
expr
ession
and
e
quation
E.7002
Establish
the
follo
wing
iden
tities
:
a
3
x
+
1
x
+
1
+
3
x
−
1
=
3
x
2
+
x
+
2
(
x
+
1)(
x
−
1)
b
2
−
x
3
x
+
1
+
x
+
1
2
=
3
x
2
+
2
x
+
5
2
3
x
+
1
E.5904
Solv
e
the
follo
wing
equations
:
a
2
x
+
1
−
3
2
x
−
1
=
0
b
2
x
−
1
4
x
+
1
−
3
x
6
x
−
1
=
0
E.9753
Solv
e
the
equation
:
4
x
2
x
+
1
−
2
x
−
3
x
+
1
=
0
E.9698
Solv
e
the
follo
wing
equations
:
a
2
x
4
x
+
1
=
x
+
1
2
x
−
1
b
1
−
x
2
−
x
=
x
+
3
x
−
1
E.9658
Solv
e
the
equation
:
3
5
x
+
3
−
2
−
5
x
x
+
2
=
0
E.2750
Solv
e
equation
:
1
2
x
2
+
5
x
+
3
−
1
2
x
2
+
3
x
+
1
=
0
E.2734
Solv
e
the
follo
wing
equation
:
(
E
)
:
1
3
x
2
−
8
x
+
4
+
2
3
x
2
+
10
x
−
8
=
0
13.
A
lgebr
aic
expr
ession
:
ante
c
e
dents
E.6498
1
Consider
the
follo
wing
three
functions
:
f
(
x
)
=
x
+1
1
−
x
2
;
g
(
x
)
=
1+
x
2
x
−
2
;
h
(
x
)
=
3
−
2
·
x
+1
Determine
the
image
of
the
n
um
b
er
1
b
y
eac
h
of
these
three
functions.
2
Consider
the
follo
wing
three
functions
:
j
(
x
)
=
1
1
−
x
;
k
(
x
)
=
x
2
−
x
+
1
x
+
1
;
‘
(
x
)
=
3
·
x
−
1
2
−
3
·
x
Determine
the
an
teceden
ts
of
the
n
um
b
er
−
1
b
y
eac
h
of
these
three
functions.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/2823
sacados/2823
chapExoCorrec/4388
sacados/4388
chapExoCorrec/4560
sacados/4560
chapExoCorrec/2851
sacados/2851
chapExoCorrec/2875
sacados/2875
chapExoCorrec/4486
sacados/4486
chapExoCorrec/7002
sacados/7002
chapExoCorrec/5904
sacados/5904
chapExoCorrec/9753
sacados/9753
chapExoCorrec/9698
sacados/9698
chapExoCorrec/9658
sacados/9658
chapExoCorrec/2750
sacados/2750
chapExoCorrec/2734
sacados/2734
chapExoCorrec/6498
sacados/6498
E.8026
Consider
the
three
functions
f
,
g
and
h
defining
the
image
of
the
n
um
b
er
x
as
follo
ws
:
f
(
x
)
=
3
x
−
2
;
g
(
x
)
=
x
2
;
h
(
x
)
=
2
3
x
−
1
1
Solv
e
the
follo
wing
three
equations
:
(
E
)
:
3
x
−
2
=
1
2
;
(
F
)
:
x
2
=
2
;
(
G
)
:
2
3
x
−
1
=
−
1
2
Using
the
previous
question,
determine
the
sets
b
elo
w
:
The
set
of
an
teceden
ts
of
1
2
b
y
f
;
The
set
of
an
teceden
ts
of
2
b
y
g
;
The
set
of
an
teceden
ts
of
−
1
b
y
h
.
14.
A
lgebr
aic
expr
ession
:
ante
c
e
dents
and
definition
set
E.8022
1
Consider
the
function
f
square
whose
algebraic
expres-
sion
is
:
f
(
x
)
=
x
2
Do
es
the
function
f
admit
one
or
more
an
teceden
ts
of
the
n
um
b
er
−
4
?
Justify
y
our
answ
er.
2
Consider
the
function
g
whose
expression
is
giv
en
b
y
the
relation
:
g
(
x
)
=
1
x
2
+1
What
can
b
e
said
ab
out
the
set
of
an
teceden
ts
of
the
n
um
b
er
2
b
y
the
function
g
?
3
Consider
the
function
h
defined
b
y
the
expression
:
h
(
x
)
=
3
·
x
+
1
x
Determine
the
set
of
an
teceden
ts
of
the
n
um
b
er
3
b
y
the
function
h
.
15.
A
lgebr
aic
expr
ession
:
image
and
ante
c
e
dents
E.6562
1
Let
f
b
e
a
function
realizing
the
relation
:
f
(2)
=
5
a
T
ranslate
this
relationship
in
to
a
sen
tence
using
the
w
ord
ˇ
image
ı.
b
T
ranslate
this
relationship
in
to
a
sen
tence
using
the
w
ord
ˇ
an
teceden
t
ı.
2
Let
g
b
e
a
function
suc
h
that
the
equation
g
(
x
)
=
1
ad-
mits
for
solution
the
n
um
b
ers
−
1
and
2
.
T
ranslate
this
prop
ert
y
in
to
a
sen
tence
using
the
w
ord
ˇ
an
teceden
t
ı.
E.362
Consider
the
follo
wing
t
w
o
functions
:
f
:
x
↦−
→
1
1
+
x
;
g
:
x
↦−
→
3
−
2
x
1
Determine
the
image
of
the
n
um
b
er
1
for
eac
h
of
these
functions.
2
Determine
the
set
of
an
teceden
ts
of
the
n
um
b
er
4
for
eac
h
of
these
t
w
o
functions.
E.367
W
e
define
six
functions
and,
for
eac
h
of
them,
t
w
o
n
umerical
v
alues
:
a
f
(
x
)
=
3
x
+
5
;
a
=
2
;
b
=
−
1
b
g
(
x
)
=
−
2
x
−
2
;
a
=
1
;
b
=
8
c
h
(
x
)
=
x
2
;
a
=
5
;
b
=
9
d
j
(
x
)
=
3
x
2
;
a
=
−
3
;
b
=
−
1
e
k
(
x
)
=
3
x
+
1
x
+
1
;
a
=
2
;
b
=
1
f
‘
(
x
)
=
2
x
−
2
x
+
ı
;
a
=
1
;
b
=
2
1
F
or
eac
h
question,
determine
the
image
of
the
n
um
b
er
a
b
y
the
asso
ciated
function.
2
F
or
eac
h
question,
determine
the
set
of
an
teceden
ts
of
the
n
um
b
er
b
b
y
the
asso
ciated
function.
E.4673
1
Belo
w
are
three
functions
that
ha
v
e
b
een
en
tered
on
a
calculator
:
a
b
c
Rewrite
these
three
functions
on
y
our
cop
y
using
the
usual
presen
tation
of
mathematical
expressions.
2
F
or
eac
h
of
the
functions
b
elo
w,
write
the
c
haracters
y
ou
w
ould
en
ter
in
a
calculator
to
insert
:
a
f
:
x
↦→
1
+
3
+
x
x
2
−
3
x
b
g
:
x
↦→
(1
−
2
x
)
×
3
x
−
1)
c
h
:
x
↦→
√
x
+
1
x
+
1
E.6497
1
Consider
the
follo
wing
three
functions
:
f
(
x
)
=
x
2
−
x
+
2
;
g
(
x
)
=
2
·
x
−
1
3
−
x
;
h
(
x
)
=
20
−
3
·
x
2
Determine
the
image
of
the
n
um
b
er
2
b
y
eac
h
of
these
three
functions.
2
Consider
the
follo
wing
three
functions
:
j
(
x
)
=
4
−
2
·
x
;
k
(
x
)
=
3
·
x
2
;
‘
(
x
)
=
2
−
x
2
·
x
+
1
Determine
the
an
teceden
ts
of
the
n
um
b
er
3
b
y
eac
h
of
these
three
functions.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8026
sacados/8026
chapExoCorrec/8022
sacados/8022
chapExoCorrec/6562
sacados/6562
chapExoCorrec/362
sacados/362
chapExoCorrec/367
sacados/367
chapExoCorrec/4673
sacados/4673
chapExoCorrec/6497
sacados/6497
-5
-4
-3
-2
-1
0
1
2
3
4
5
-3
-2
-1
1
2
3
i
j
C
f
C
f
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
y
-2
-1
1
2
3
4
C
g
16.
Study
of
functions
E.4451
1
In
the
reference
frame
b
elo
w,
is
represen
ted
the
represen-
tativ
e
curv
e
of
the
function
f
:
a
Giv
e
the
definition
set
of
the
function
f
.
b
Giv
e,
if
p
ossible,
the
images
of
the
follo
wing
n
um
b
ers
b
y
the
function
f
:
1
;
0
;
−
4.5
c
Giv
e
the
set
of
an
teceden
ts
for
eac
h
of
the
follo
wing
n
um
b
ers
:
2
;
−
1
;
0.75
2
Consider
the
function
g
whose
image
of
a
n
um
b
er
x
is
defined
b
y
:
g
(
x
)
=
2
x
2
+
3
x
−
2
x
2
+
1
a
Justify
that
the
function
g
is
defined
on
R
.
b
Determine
the
image
of
the
n
um
b
er
−
2
b
y
the
function
g
.
c
Determine
the
an
teceden
t
of
the
n
um
b
er
2
b
y
the
func-
tion
g
.
E.373
Consider
the
t
w
o
functions
f
and
g
:
the
function
f
defined
b
y
:
f
:
x
↦−
→
x
2
−
6
x
+2
.
The
function
g
is
defined
b
y
the
graph
sho
wn
b
elo
w
:
F
or
eac
h
of
the
follo
wing
questions,
only
one
of
the
four
p
os-
sible
answ
ers
is
correct
;
select
the
correct
answ
er.
1
The
image
of
1
under
the
function
f
is
:
a
1
b
0
c
−
1
d
−
3
2
The
domain
of
−
7
under
f
is
:
a
3
b
2
c
−
2
;
3
d
1
;
2
3
The
domain
of
the
function
g
is
:
a
−
1;
−
3
b
−
1;3
c
−
4;7
d
−
4;7
4
The
image
of
0
under
the
function
g
is
:
a
1
b
−
1
c
7
d
0
5
UWhic
h
p
oin
ts
do
not
b
elong
to
C
g
?
a
(
−
3;
−
1)
b
(
−
4;
1)
c
(6;
2)
d
(
−
2;
−
0
;
5)
E.2731
1
Consider
a
function
f
.
W
e
denote
(
C
)
the
represen
tativ
e
curv
e
of
the
function
f
.
Consider
the
follo
wing
prop
erties
of
the
curv
e
(
C
)
:
a
The
co
ordinate
p
oin
t
(
0
;
3
)
b
elongs
to
(
C
)
.
b
The
only
p
oin
t
of
(
C
)
with
ordinate
5
has
abscissa
−
1
.
c
No
p
oin
t
of
(
C
)
has
abscissa
−
2
.
d
There
is
no
(
C
)
p
oin
t
of
ordinate
6
.
T
ranslate
eac
h
of
these
sen
tences
in
to
a
sen
tence
describ-
ing
a
prop
ert
y
of
the
function
f
using,
eac
h
time,
at
least
one
of
the
follo
wing
w
ords
image
,
an
teceden
t
,
definitey
.
2
Let
g
b
e
the
defined
function
whose
image
of
a
n
um
b
er
x
is
defined
b
y
:
g
(
x
)
=
2
x
2
−
3
Let
(
C
g
)
b
e
the
represen
tativ
e
curv
e
of
the
function
g
.
a
A
is
a
p
oin
t
of
abscissa
2
of
(
C
g
)
.
What
is
the
ordinate
of
p
oin
t
A
?
b
B
is
a
p
oin
t
of
(
C
g
)
with
ordinate
−
3
.
Giv
e
the
ab-
scissa
of
the
p
oin
t
B
.
c
Ho
w
man
y
p
oin
ts
on
the
curv
e
(
C
g
)
ha
v
e
ordinates
−
1
?
Sp
ecify
,
if
they
exist,
the
co
ordinates
of
these
p
oin
ts.
d
Ho
w
man
y
p
oin
ts
on
the
curv
e
(
C
g
)
ha
v
e
ordinates
−
4
?
Sp
ecify
,
if
they
exist,
the
co
ordinates
of
these
p
oin
ts.
3
Consider
the
function
h
defined
b
y
the
relation
:
h
(
x
)
=
2
x
2
+
3
Let
(
C
h
)
b
e
the
represen
tativ
e
curv
e
of
the
function
h
.
a
Giv
e
the
ordinate
of
the
p
oin
t
of
(
C
h
)
abscissa
0
.
b
Ho
w
man
y
p
oin
ts
(
C
h
)
ha
v
e
ordinate
1
6
?
Giv
e,
if
they
exist,
the
co
ordinates
of
these
p
oin
ts.
17.
Set
of
definitions
E.363
Determine
the
definition
set
of
the
follo
wing
functions
:
1
f
:
x
↦−
→
2
x
+
5
2
g
:
x
↦−
→
1
x
3
h
:
x
↦−
→
1
2
x
+
5
4
j
:
x
↦−
→
x
+
1
2
x
+
5
5
k
:
x
↦−
→
x
6
‘
:
x
↦−
→
x
2
7
m
:
x
↦−
→
2
x
+
5
8
n
:
x
↦−
→
−
x
+
2
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4451
sacados/4451
-5
-4
-3
-2
-1
0
1
2
3
4
5
-3
-2
-1
1
2
3
i
j
C
f
C
f
chapExoCorrec/373
sacados/373
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
y
-2
-1
1
2
3
4
C
g
chapExoCorrec/2731
sacados/2731
chapExoCorrec/363
sacados/363
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
y
-3
-2
-1
1
2
3
C
f
C
f
h
B
b
A
B
C
D
F
G
H
E.377
Determine
the
definition
set
for
eac
h
of
the
functions
b
elo
w
:
1
f
:
x
↦−
→
1
3
x
+
7
2
g
:
x
↦−
→
1
−
6
x
E.2755
Consider
the
function
f
whose
image
of
a
n
um
b
er
x
is
defined
b
y
the
relation
:
f
(
x
)
=
2
x
+
1
x
−
4
1
Determine
the
definition
set
of
the
function
f
.
2
Determine
the
image
of
the
n
um
b
er
3
b
y
the
function
f
.
3
Determine
the
set
of
an
teceden
ts
of
0
b
y
the
function
f
.
E.2710
In
the
reference
frame
b
elo
w,
is
rep-
resen
ted
the
curv
e
of
the
function
f
1
Determine
the
images
of
the
follo
wing
n
um
b
ers
b
y
the
function
f
:
a
1
b
0
c
−
2
2
Determine
the
set
of
an
teceden
ts
for
eac
h
of
the
follo
wing
n
um
b
ers
:
a
2
b
−
2
3
a
Giv
e
t
w
o
n
um
b
ers
that
do
not
admit
images
b
y
the
function
f
.
b
Giv
e
a
n
um
b
er
that
admits
no
an
teceden
ts
b
y
the
func-
tion
f
.
18.
Unclassifie
d
exer
cises
E.1782
1
Establish
for
an
y
non-zero
natural
n
um
b
er
p
the
follo
w-
ing
equalit
y
:
1
p
−
1
p
+
1
=
1
p
(
p
+
1)
2
Deduce
the
v
alue
of
the
follo
wing
sum
:
S
=
1
1
×
2
+
1
2
×
3
+
1
3
×
4
+
·
·
·
+
1
2003
×
2004
+
1
2004
×
2005
E.455
The
follo
wing
equation
is
to
b
e
solv
ed
:
3
x
−
1
x
+
1
=
6
x
−
3
3
x
+
3
1
Giv
e
the
solution
set
for
the
ab
o
v
e
equation.
2
Solv
e
this
equation.
E.6598
Solv
e
the
follo
wing
equations
:
a
(3
x
+1)(5
x
−
2)
=
(6
x
+2)(1
−
x
)
b
1
x
+
1
+
2
x
−
1
=
0
E.6970
Solv
e
the
equations
:
a
x
3
x
+
2
=
5
x
2
x
+
1
b
4
x
−
2
2
x
−
3
−
6
x
−
2
3
x
+
1
=
0
E.6695
Reminder
:
Area
of
a
trap
e-
zoid
A
=
(
B
+
b
)
×
h
2
A
rectangular
pizza
AB
C
D
has
crust
on
t
w
o
consecutiv
e
sides,
[
D
A
]
and
[
AB
]
.
W
e
w
an
t
to
divide
the
pizza
in
to
three
equal
pieces
:
eac
h
slice
m
ust
ha
v
e
the
same
length
of
crust
and
the
same
area.
W
e
set
the
length
of
the
short
side
AD
=
1
.
In
the
sp
ecific
case
sho
wn
here,
w
e
assume
that
the
division
is
equal.
Determine
the
lengths
:
D
F
,
F
H
,
and
H
C
.
An
y
evidence
of
researc
h
and
initiativ
e,
ev
en
if
incom-
plete,
will
b
e
tak
en
in
to
accoun
t
in
the
ev
aluation.
E.4447
Expand
the
follo
wing
expressions
:
a
(2
x
+
1)(3
−
x
)
b
(5
−
2
x
)(3
−
x
)
−
3(3
−
2
x
)
c
4(
x
+
4)(5
−
2
x
)
d
(
x
−
2)(2
x
−
1)(5
−
x
)
h
ttps:/
/c
hingmath.fr
chapExoCorrec/377
sacados/377
chapExoCorrec/2755
sacados/2755
chapExoCorrec/2710
sacados/2710
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
y
-3
-2
-1
1
2
3
C
f
C
f
chapExoCorrec/1782
sacados/1782
chapExoCorrec/455
sacados/455
chapExoCorrec/6598
sacados/6598
chapExoCorrec/6970
sacados/6970
chapExoCorrec/6695
sacados/6695
h
B
b
A
B
C
D
F
G
H
chapExoCorrec/4447
sacados/4447
-3
-2
-1
2
3
4
I
-2
-1
2
3
4
J
O
C
f
C
g
E.6596
Expand
and
reduce
the
follo
wing
expressions
:
a
(2
x
+
1)(3
−
x
)
−
2(3
x
+
2)
b
(2
x
+
1)
2
c
(2
x
+
1)(1
−
x
)(
x
+
2)
E.4484
Definition
:
a
random
exp
erimen
t
is
said
to
b
e
equiprobable
if
eac
h
of
its
outcomes
has
the
same
probabilit
y
of
o
ccur-
ring.
The
probabilit
y
of
an
ev
en
t
is
equal
to
the
sum
of
the
probabilities
of
the
outcomes
that
define
it.
Prop
osition
:
F
or
an
equiprobable
random
exp
erimen
t
with
N
outcomes
and
for
an
ev
en
t
A
defined
b
y
n
outcomes,
the
probabilit
y
of
A
has
the
v
alue
:
n
N
Example
:
W
e
thro
w
a
balanced
die
where
the
six
faces
are
n
um
b
ered
from
1
to
6
.
Consider
the
ev
en
t
A
:
ˇ
the
face
obtained
is
strictly
greater
than
4
ı.
This
ev
en
t
is
comp
osed
of
2
out-
comes.
The
probabilit
y
of
the
ev
en
t
A
is
:
2
6
=
1
×
2
\
3
×
2
\
Jean
o
wns
365
comic
b
o
ok
albums.
In
order
to
sort
the
al-
bums
in
his
collection,
he
is
arranged
b
y
series
and
classifies
the
series
in
to
three
categories
:
franco-b
elges,
comics
and
manga
as
b
elo
w
:
Séries
franco-b
elges
Séries
de
comics
23
albums
ˇAstérixı
22
albums
ˇTin
tinı
45
albums
ˇLuc
ky-Luk
eı
35
albums
ˇBatmanı
90
albums
ˇSpider-Manı
Manga
series
85
albums
ˇOne-pièceı
65
albums
ˇNarutoı
He
c
ho
oses
an
album
at
random
from
all
those
in
his
collec-
tion.
1
What
is
the
probabilit
y
that
the
c
hosen
album
is
a
ˇ
Lucky-Luke
ı
album?
2
What
is
the
probabilit
y
that
the
c
hosen
album
is
a
comic
b
o
ok?
E.4448
Solv
e
the
follo
wing
equations
using
the
metho
d
of
y
our
c
hoice
:
a
(3
x
−
2)(10
x
+
4)
=
(3
−
2
x
)(5
x
+
2)
b
(5
−
2
x
)(3
x
+
4)
−
(5
−
2
x
)
=
0
E.4462
Solv
e
the
follo
wing
equations
using
the
metho
d
of
y
our
c
hoice
:
a
(
x
−
2)(3
x
+
1)
=
2(
x
−
2)(
x
−
5)
b
(5
−
2
x
)(3
x
+
1)
+
(4
x
+
10)(2
x
−
5)
=
0
c
(2
x
+
3)(8
x
−
3)
+
(3
−
4
x
)(4
x
+
1)
=
0
E.4443
Solv
e
the
equations
b
elo
w
:
a
10
x
2
−
15
x
=
(2
x
−
3)(3
x
+
1)
b
(3
−
x
)(4
x
+
2)
−
(6
x
+
3)(5
−
2
x
)
=
0
c
(3
−
2
x
)(
x
+
1)
+
(6
x
−
9)(3
−
4
x
)
=
0
E.9685
Solv
e
the
follo
wing
equations
:
a
x
+
1
x
−
1
=
3
x
x
+
1
b
x
2
x
−
1
−
2
x
−
1
3
−
x
=
0
E.4815
Solv
e
the
follo
wing
equations
:
a
(
x
+
1)
2
=
4
b
(
x
−
2)
2
+
4
=
7
c
(
x
+
2)
2
+
5
=
2
d
3
·
x
2
−
6
=
1
E.4461
F
actor
the
follo
wing
expressions
:
a
(5
x
−
1)(3
x
+
1)
+
(5
x
−
1)
2
b
(3
x
+
1)(2
−
3
x
)
+
(2
−
3
x
)
c
(
x
−
3)(7
−
x
)
+
(
x
−
3)(2
x
+
1)
d
(3
x
−
1)(
x
−
2)
−
(
x
−
2)(1
−
5
x
)
E.4449
Consider
the
t
w
o
functions
f
and
g
defined
b
y
the
relations
:
f
(
x
)
=
(2
x
+
1)(2
−
x
)
;
g
(
x
)
=
1
8
(2
x
+
1)
2
Their
graphical
represen
tations
are
giv
en
b
elo
w
in
the
refer-
ence
frame
O
;
I
;
J
orthonormal
:
1
Determine,
graphically
,
the
co
ordinates
of
the
in
tersec-
tion
p
oin
ts
of
the
curv
es
C
f
and
C
g
.
2
a
Solv
e
the
follo
wing
equation
:
(2
x
+
1)
2
=
8(2
x
+
1)(2
−
x
)
b
Determine,
b
y
calculation,
the
images
of
the
n
um
b
ers
−
1
2
and
3
2
b
y
the
functions
f
and
g
.
3
What
do
the
solution
n
um
b
ers
f
(
x
)
=
g
(
x
)
represen
t
for
the
t
w
o
curv
es
C
f
and
C
g
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6596
sacados/6596
chapExoCorrec/4484
sacados/4484
chapExoCorrec/4448
sacados/4448
chapExoCorrec/4462
sacados/4462
chapExoCorrec/4443
sacados/4443
chapExoCorrec/9685
sacados/9685
chapExoCorrec/4815
sacados/4815
chapExoCorrec/4461
sacados/4461
chapExoCorrec/4449
sacados/4449
-3
-2
-1
2
3
4
I
-2
-1
2
3
4
J
O
C
f
C
g
O
A
B
C
I
x
y
M
E.2867
w
est
Indies
ffl
2004
ffl
7
p
oin
ts
ffl
Com-
pulsory
The
figure
b
elo
w
is
a
diagram
of
a
car
jac
k.
This
consists
of
a
deformable
rhom
bus
O
AB
C
,
the
p
oin
t
O
b
eing
the
p
oin
t
of
supp
ort
on
the
ground
and
the
p
oin
t
B
b
eing
the
p
oin
t
through
whic
h
the
car
is
lifted.
With
eac
h
turn
of
the
crank
M
,
the
n
uts
A
and
C
mo
v
e
to-
w
ards
(or
away)
from
2
cm
,
whic
h
mo
v
es
(or
down)
the
B
supp
ort
up,
along
the
(
O
y
)
axis.
W
e
giv
e
:
O
A
=
O
C
=
AB
=
B
C
=
25
cm
In
the
orthonormal
frame
of
reference
(
O
;
x
;
y
)
of
unit
one
cen
timetre,
x
A
designates
the
abscissa
of
the
p
oin
t
A
and
v
aries
from
0
to
25.
The
ordinate
of
p
oin
t
B
is
denoted
y
B
:
F
or
x
A
=0
,
w
e
ha
v
e
:
y
B
=
50
;
F
or
x
A
=25
,
w
e
ha
v
e
:
y
B
=
0
.
1
Demonstrate
that
the
v
alues
x
A
and
y
B
v
erify
the
rela-
tionship
:
y
B
=
2
625
−
x
A
2
2
a
Determine
the
v
alue
of
y
B
when
x
A
is
equal
to
7
.
b
Determine
the
v
alue
of
x
A
when
y
B
equals
40
.
3
Assume
the
jac
k
is
closed
;
the
heigh
t
of
p
oin
t
B
is
then
0
cm
:
a
When
the
jac
k
is
fully
closed,
ho
w
man
y
crank
turns
are
required
to
reac
h
a
heigh
t
of
24
cm
for
p
oin
t
B
?
b
Ho
w
man
y
more
turns
do
es
it
tak
e
to
double
the
heigh
t
of
p
oin
t
B
?
h
ttps:/
/c
hingmath.fr
chapExoCorrec/2867
sacados/2867
O
A
B
C
I
x
y
M
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