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E M B I B E
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
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EASY
Earn 100
The value of
1
+
i
2
+
i
4
+
i
6
+
…
+
i
2
n
is
(a)
Positive
(b)
Negative
(c)
Zero
(d)
Cannot be determined
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answered this correctly
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Important Questions on Complex Numbers
MEDIUM
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
If
z
and
ω
are two complex numbers such that
z
ω
=
1
and
a
r
g
z
-
a
r
g
(
ω
)
=
π
2
, then:
EASY
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
The value of
i
18
+
1
i
25
3
is equal to
MEDIUM
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
If
a
>
0
and
z
=
(
1
+
i
)
2
a
-
i
, has magnitude
2
5
, then
z
-
is equal to:
MEDIUM
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
If
z
is a complex number satisfying
R
e
z
+
I
m
z
=
4
,
then
z
cannot be
MEDIUM
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
For all complex numbers
z
of the form
1
+
i
α
,
α
∈
R
,
if
z
2
=
x
+
i
y
,
then
HARD
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
If the equation
x
2
+
b
x
+
45
=
0
,
b
∈
R
has conjugate complex roots and they satisfy
z
+
1
=
2
10
,
then
MEDIUM
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
If
a
and
b
are real numbers and
a
+
i
b
11
=
1
+
3
i
,
then
b
+
i
a
11
is equal to
HARD
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
If
z
=
x
-
i
y
and
z
1
3
=
p
+
i
q
, then
1
p
2
+
q
2
x
p
+
y
q
is equal to
HARD
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
For any real number
r
,
let
A
r
=
e
i
π
r
n
:
n
is a natural number
}
be a set of complex numbers. Then
MEDIUM
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
If
z
1
=
2
-
i
and
z
2
=
1
+
i
,
then
z
1
+
z
2
+
1
z
1
-
z
2
+
i
is
MEDIUM
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
The value of sum
∑
n
=
1
13
i
n
+
i
n
+
1
,
where
i
=
-
1
,
equals
EASY
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
If
z
-
α
z
+
α
α
∈
R
is a purely imaginary number and
z
=
2
, then a value of
α
is :
MEDIUM
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
If
2
+
i
is a root of
x
2
-
4
x
+
c
=
0
,
where
c
is a real number, then the value of
c
is
MEDIUM
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
Let
z
be a complex number such that
z
-
i
z
+
2
i
=
1
and
z
=
5
2
. Then, the value of
z
+
3
i
is
HARD
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
The principal argument of the complex number
z
=
1
+
sin
π
-
i
cos
π
1
+
sin
π
+
i
cos
π
is
HARD
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
The real part of
e
(
3
+
4
i
)
x
is
MEDIUM
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
Let
z
=
x
+
i
y
be a complex number such that
|
z
+
i
|
=
2
. Then the locus of
z
is a circle whose centre and radius are
EASY
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
The modulus of
1
+
i
1
-
i
-
1
-
i
1
+
i
is
MEDIUM
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
Let
z
=
3
2
+
i
2
5
+
3
2
-
i
2
5
.
If
R
z
and
I
z
respectively denote the real and imaginary parts of
z
,
then
HARD
Mathematics
>
Algebra
>
Complex Numbers
>
Introduction to Complex Numbers
Let
z
1
and
z
2
be any two non-zero complex numbers such that
3
z
1
=
4
z
2
.
If
z
=
3
z
1
2
z
2
+
2
z
2
3
z
1
then maximum value of
z
is