Complex Numbers
IThe complex plane is referred to a direct orthonormal system O; u, v . Consider the points A, B, C,
E, and F of respective affixes zA = 3 + i, zB = 2i, zC = 2 – 2i, zE = –1 + i, and
zF =
z − 2i
(z 
1 − 2 3 − i 3 . Let M be a point of affix z and M be a point of affix z such that z =
z
0).
1) Let z = x + iy and z = x+ iy.
a- Express x and y in terms of x and y.
b- Deduce the set (G) of points M when z is pure imaginary, then prove that E belongs
to (G).
2) Let (C) be the circle of center O and radius 1.
a- Give a geometric interpretation of | z |, then deduce the set (L) of points M when M
moves on the circle (C). Verify that A belongs to (L).
b- Give a geometric interpretation of arg(z), then deduce the set (L) of points M when
z is real.
zF − zB
3) Write
in exponential form and deduce the nature of triangle BCF.
zC − zB
4)
a- Write zC in exponential form.
b- For what values of the natural integer n is (zC)n real?
5) Let N be a point of affix Z and N be a point of affix Z such that NN = AC .
a- Express Z in terms of Z.
b- Calculate the affix of point D, so that the quadrilateral ABDC is a square.
(
)
IIIn the complex plane referred to a direct orthonormal system O; u, v , consider the points A, B, D,
z +1− i
M and M of respective affixes −3, −1 + i, −4 + 2i, z, and z such that z =
(z  −3).
z+3
1) Write z' in exponential form when z = −1 + 2i.
2)
a- Give a geometric interpretation of | z |, then deduce the set (L) of points M when |
z' | = 1.
b- Verify that I, the midpoint of [BD], belongs to (L).
3)
a- Prove that | z' – 1 |  | z + 3 | is a constant to be determined.
b- Let E be a point of affix 1 and (C) be the circle of center E and radius 5 . Prove
that, if M' moves on (C), then M moves on a circle whose center and radius are to
be determined.
4)
z − zA
a- Prove that D
is pure imaginary.
zB − zA
b- Deduce the nature of triangle ABD.
(
)
III-
 → →
In the complex plane referred to a direct orthonormal system  O; u , v  , consider the points M1,


M2, M3, and M4 of respective affixes:
3+i
z1 =
, z2 = z1 – i, z3 = z 1 − 1 , and z4 = eiθ.
4
1)
a- Determine the algebraic forms of z2 and z3.
b- Calculate the modulus and an argument of z1 and those of z2.
2)
a- Calculate | z3 – z2 | and | z3 – z1 |. Deduce that the triangle M1M2M3 is isosceles.
b- Verify that the triangle OM1M2 is right.
3)
a- Write z = z12z22 in algebraic and exponential form.
b- Find θ, so that zz4 is real.
4) Consider the point M of affix z (z  0 and z  1) and the point M of affix z = 1 + z2. Let A
z2
be the point of affix 1 and U be the complex number such that U =
.
z −1
 z − 1 
a- Define, geometrically, arg 
.
 z −1 
b- Deduce that U is real when the points A, M, and M are collinear. Construct M and
M when U = 2.
c- Find the set of points M so that AM2 = AM.
IV-
 → →
The plane is referred to a direct orthonormal system  O; u , v  . Let A be the point of affix −2i. For


every point M of affix z, we associate the point M of affix z where z = − 2z + 2i .
1) Consider the point B of affix b = 3 − 2i. Determine the algebraic forms of a and b, the
associated affixes of points A and B respectively.
2) Prove that, for every point M of affix z, | z + 2i | = 2 | z + 2i |. Interpret this result
geometrically.
3) For every point M distinct from A, we call  an argument of z + 2i.
a- Give a geometric interpretation of .
b- Show that (z + 2i)(z + 2i) is a negative or null real number.
c- Deduce, in terms of , an argument of z + 2i.
VIn the complex plane of a direct orthonormal system O; u, v , consider the point M of affix z.
2iz − i
For every complex number z  -1, we associate the complex number Z such that Z =
.
z +1
1) Let z = x + iy, where x and y are real numbers. Calculate Z , Re(Z), Im(Z), and | Z | in terms
of x and y.
2) Find the set of points M such that | Z | = 1.
3) Find the set of points M such that Z is pure imaginary.
VI → →
In the complex plane referred to a direct orthonormal system  O, u, v  , consider the points A and


B of affixes −1 and −i respectively. M being a point in the plane, of affix z (M is distinct from A),
− iz − 2
consider the point M´ of affix z´ such that: z´ =
.
z +1
(
)
−i

1) Write z´ in its algebraic form in the case where z = 2e 4 .
2) Let z = x + iy and z´ = x´ + iy´.
a- Express x´ and y´ in terms of x and y.
b- What is the set of points M when z' is a real number?
3)
a- Calculate z´ + i in terms of z. What can you say about | z´ + i |  | z + 1 |?
b- Suppose that M moves on a circle (C) of center A and radius 2. Show that M´ moves on
a circle (C´) whose center and radius are to determined.
VII- (4 points)
In the complex plane referred to a direct orthonormal system O; u, v , consider the point A of
affix 1, the point B of affix i, the circle (C) of center O and radius 1, and the straight-line (D) of
z−i
equation y = 1. M is a point of affix z (z  i) and M' is a point of affix z' such that z' =
.
z+i
(

− i
2e 6 .
)
1) Write z' in its algebraic form when z =
2) Determine the set of points M when z' = 1.
3) Prove that z   z  = 1 .Give a geometric interpretation of this result.
4)
z '−1
a- Prove that
is pure imaginary. Deduce that the two straight-lines (AM') and
z−i
(BM) are perpendicular.
b- M is a point that does not belong to (D). Construct, geometrically, the point M'.
c- Specify the position of point M' when point M belongs to the straight-line (D)
deprived of point B.
VIIIPart A
(
)
In the complex plane of a direct orthonormal system O; u, v , consider the point M of affix z.
2iz − i
For every complex number z  -1, we associate the complex number Z such that Z =
. Let
z +1
z = x + iy, where x and y are real numbers.
1) Calculate Z , Re(Z), Im(Z), and | Z | in terms of x and y.
2) Find the set of points M such that | Z | = 1.
3) Find the set of points M such that Z is pure imaginary.
Part B
1) Place, in the complex plane of a direct orthonormal system O; u, v , the points A, B, C,
(
)
and D of respective affixes z A = i 3 , z B = −i 3 , z C = 3 + 2i 3 , and z D = z C .
2)
a- Prove that A, B, C, and D belong to the same circle.

b- Let E be the symmetric of D with respect to O. Prove that
−i
zC − zB
=e 3.
zE − zB
c- Determine the nature of triangle of BEC.
IXIn the complex plane referred to a direct orthonormal system O; u, v , consider the points M of
− 1 + 2i
affix z (z  2) and M' of affix z' such that z  =
.
z−2
1+ i
1) Find the exponential form of z' when z =
.
2
2) Prove that: If M' moves on the circle of center O and radius 1, then M moves on a circle
whose radius and center are to be determined.
3) Let z = x + iy. Find the set of the points M when z' is a real number.
(
)
X-
z1 3 − z 2 = −2
1) Let z1 and z2 be two complex numbers such that 
. Find z1 and z2.
z1 − z 2 3 = −2i
(
)
2) In the complex plane of a direct orthonormal system O; u, v (unit: 2 cm), consider the
points A and B of respective affixes: zA = − 3 + i and zB = − 1+ i 3 . Write zA and zB in
the exponential form, then place the points A and B.
3)
a- Calculate the modulus and the argument of
zA
.
zB
(
)
b- Deduce the nature of triangle ABO and give a measure of angle OA; OB .
4) Determine the affix of point C such that the quadrilateral ACBO is a rhombus. Place C, then
calculate the area of triangle ABC in cm2.
z1
in the trigonometric and exponential forms.
z2
z
2) Determine the algebraic form of 1 .
z2
1) Write z1, z2, and
3) Deduce that: cos

=
12
6+ 2

=
and sin
4
12
6− 2
.
4
XIIn the table below only one, among the proposed answers to each question, is correct. Write down
the number of each question and give, with justification, the corresponding answer.
A
B
C
The exponential form of z
1)
2e( − )i
2e−i
2e( + )i
= −2ei is
If the exponential form of z
is
11
i
2e 12
and its algebraic
2)
−1− 3
3 −1
+
i,
form is
2
2
then
z = (− 1 + i )(2 sin  + 2i cos ) .
3 −i
3)
|z|=
z = (− 1 + i )(2 sin  + 2i cos ) .
3 −i
4)
An argument of z is


3 +1
cos =
12
2 2


sin  = 3 − 1
 12
2 2
 11
=
cos
12


sin 11 =
 12
2
2
−
3 +1
2 2
3 −1
2 2
 11 − 1 − 3
=
cos

12
2

sin 11 = 3 − 1

12
2
None
2
7
+
12
11
+
12
7
−
12
XII-
(
)
In the complex plane referred to a direct orthonormal system O; u, v , consider the points
A, I, and B of respective affixes 1, 2, and 3.
M being a point in the plane of affix z (M is distinct from I), consider the point M of affix z
1
+2.
such that z =
z−2
1) Determine the points M so that M and M are confounded.
2)
→
→
a- Calculate, in terms of z, the affixes of the vectors IM and IM  .
b- Deduce a relation between IM and IM, then a relation between the angles
→ → 
→ → 
 u , IM  and  u , IM   .








i

3
c- Place the point M0 of affix z0 = 2 + 2 e in the plane, then the point M0 using
the preceding part.
3) Suppose that M is a point in the plane different from A and B.
a- Calculate z − 1 and z − 3 in terms of z.
b- Verify that
1 − z
1− z
.
=−
3 − z
3−z
M A
MA
and
, then a relation between the angles
M B
MB
 → → 
 → → 
 M B, M A  and  MB, MA  .








d- Prove that, if M belongs to the perpendicular bisector of [AB], then M belongs
to the perpendicular bisector of [AB].
c- Deduce a relation between
XIII
(
)
1) In the complex plane referred to a direct orthonormal system O; u, v , consider the
1+ i
complex number z =
.
1− i 3
a- Determine the modulus and an argument of z.
b- For what values of the non-zero natural number n is zn real? Calculate the
smallest value of n.
2) Linearize cos3x.

3 1 
3) Write z =  −
+ i 
2
2 

2000
in its trigonometric form.