Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

MEDIUM

Earn 100

If $z_{1} = - \sqrt{3} + i$ and $z_{2} = i \sqrt{3} - 1$ , show that $a r g (z_{1} z_{2}) \neq a r g (z_{1}) + a r g (z_{2})$

Important Questions on Complex Numbers

HARD

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

Let

A = \{θ \in (- \frac{π}{2}, π) : \frac{3 + 2 i s i n θ}{1 - 2 i s i n θ} is purely imaginary\} .

Then the sum of the elements in

A

is:

EASY

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

Amplitude of

\frac{1 + \sqrt{3} i}{\sqrt{3} + 1}

is

MEDIUM

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

The value of sum

\sum_{n = 1}^{13} (i^{n} + i^{n + 1}), where i = \sqrt{- 1},

equals

MEDIUM

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

If

z

and

ω

are two complex numbers such that

|z ω| = 1

and

a r g (z) - a r g (ω) = \frac{π}{2}

, then:

HARD

Mathematics>Algebra>Complex Numbers>Basics of 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>Basics of Complex Numbers

If

a > 0

and

z = \frac{{(1 + i)}^{2}}{a - i}

, has magnitude

\sqrt{\frac{2}{5}}

, then

\bar{z}

is equal to:

EASY

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

The value of

{(i^{18} + {(\frac{1}{i})}^{25})}^{3}

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Basics of 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>Basics of Complex Numbers

If

z_{1} = 2 - i

and

z_{2} = 1 + i

,

then

|\frac{z_{1} + z_{2} + 1}{z_{1} - z_{2} + i}|

is

MEDIUM

Mathematics>Algebra>Complex Numbers>Basics of 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>Basics of Complex Numbers

If the equation

x^{2} + b x + 45 = 0, b \in R

has conjugate complex roots and they satisfy

|z + 1| = 2 \sqrt{10},

then

EASY

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

If

\frac{z - α}{z + α} (α \in R)

is a purely imaginary number and

|z| = 2

, then a value of

α

is :

MEDIUM

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

If

z

is a complex number satisfying

|R e (z)| + |I m (z)| = 4,

then

|z|

cannot be

EASY

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

The set of all

α \in R

, for which

w = \frac{1 + (1 - 8 α) z}{1 - z}

is a purely imaginary number, for all

z \in C

satisfying

|z| = 1

and

R e (z) \neq 1

, is :

MEDIUM

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

Let

z

be a complex number such that

|\frac{z - i}{z + 2 i}| = 1

and

|z| = \frac{5}{2}

. Then, the value of

|z + 3 i|

is

HARD

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

If

z = x - i y

and

z^{\frac{1}{3}} = p + i q

, then

\frac{1}{p^{2} + q^{2}} (\frac{x}{p} + \frac{y}{q})

is equal to

HARD

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

The principal argument of the complex number

z = \frac{1 + \sin π - i \cos π}{1 + \sin π + i \cos π}

is

EASY

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

The modulus of

\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}

is

MEDIUM

Mathematics>Algebra>Complex Numbers>Basics of Complex Numbers

Let

z = {(\frac{\sqrt{3}}{2} + \frac{i}{2})}^{5} + {(\frac{\sqrt{3}}{2} - \frac{i}{2})}^{5} .

If

R (z)

and

I (z)

respectively denote the real and imaginary parts of

z,

then

HARD

Mathematics>Algebra>Complex Numbers>Basics of 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 = \frac{3 z_{1}}{2 z_{2}} + \frac{2 z_{2}}{3 z_{1}}

then maximum value of

|z|

is