If S = z ∈ C : z - i z + 2 i ∈ R , then

S is a straight line in the complex plane

S is a circle in the complex plane

S contains exactly two elements

MEDIUM

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Let $A, B$ be two sets of complex numbers as defined below

$A = \{z : I m (z) = |R e (z)|\}$

$B = \{z : a r g (\frac{z - 2}{z + 2}) = \frac{π}{2}\}$

The numbers of elements in set $A \cap B$ is

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Important Questions on Complex Numbers

HARD

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

z_{1}, z_{2}

and

z_{3}

are any three distinct complex numbers such that

|z_{1}| = 1, |z_{2}| = 2, |z_{3}| = 4, \arg z_{2} = \arg z_{1} - π

and

\arg z_{3} = \arg z_{1} + \frac{π}{2},

then

z_{2} z_{3}

is equal to

EASY

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

Amplitude of the complex number

i \sin (\frac{π}{19})

MEDIUM

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

z

and

ω

are two complex numbers such that

|z ω| = 1

and

\arg (z) - \arg (ω) = \frac{3 π}{2}

, then

\arg (\frac{1 - 2 \bar{z} ω}{1 + 3 \bar{z} ω})

is:
(Here

\arg (z)

denotes the principal argument of complex number

z

)

HARD

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

Let

z_{1}

and

z_{2}

be any two non-zero complex numbers such that

3 |z_{1}| = 4 |z_{2}| .

z = \frac{3 z_{1}}{2 z_{2}} + \frac{2 z_{2}}{3 z_{1}}

then maximum value of

|z|

EASY

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

Let

z

be a complex number such that the principal value of argument,

\arg z > 0 .

Then,

\arg z - \arg (- z)

HARD

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

What will be the argument of the complex number

z = \frac{i - 1}{e^{\frac{i π}{3}}} ?

MEDIUM

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

Let

z_{1}

and

z_{2}

be two complex numbers such that

{\bar{z}}_{1} + i z_{2} = 0

and

\arg (z_{1} {\bar{z}}_{2}) = π

, then

\arg ({\bar{z}}_{2})

is equal to

EASY

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

z

is a point on the argand plane such that

|z - 1| = 1,

then

\frac{z - 2}{z}

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

The amplitude of the complex number

1 + \sin α - i \cos α

EASY

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

z = \sqrt{20 i - 21} + \sqrt{20 i + 21}

then the principal value of

\arg (z)

can be

MEDIUM

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

What will be the argument of the complex number

z = 1 - \cos p + i \sin p

HARD

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

The principal argument of the complex number

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

EASY

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

z

is a complex number such that

\arg (z) < 0 .

What will be the value of

\arg (- z) - \arg (z) ?

EASY

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

The polar form of the complex number

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

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

S = \{z \in C : \frac{z - i}{z + 2 i} \in R\}

, then

HARD

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

x = 9^{1 / 3} 9^{1 / 9} 9^{1 / 27} ... \infty

$y = 4^{1 / 3} 4^{- 1 / 9} 4^{1 / 27} ... \infty$

$z = \sum_{r = 1}^{\infty} {(1 + i)}^{- r}$ ,

then $\arg (x + y z)$ will be

MEDIUM

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

The value of

{(1 + i)}^{8} + {(1 - i)}^{8}

MEDIUM

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

z_{1}, z_{2}

are two non-zero complex numbers such that

|z_{1} + z_{2}| = |z_{1}| + |z_{2}|

, then

\arg (z_{1}) - \arg (z_{2})

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

If the amplitude of

(z - 1 - 2 i)

\frac{π}{3}

, then the locus of

z

MEDIUM

Mathematics>Algebra>Complex Numbers>Properties of Conjugate, Modulus and Argument of Complex Numbers

\frac{3 + i s i n θ}{4 - i c o s θ}, θ \in [0, 2 π],

is a real number, then an argument of

s i n θ + i c o s θ

Let A, B be two sets of complex numbers as defined below A=z:Imz=Rez B=z:argz-2z+2=π2 The numbers of elements in set A∩B is

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Let $A, B$ be two sets of complex numbers as defined below

$A = \{z : I m (z) = |R e (z)|\}$

$B = \{z : a r g (\frac{z - 2}{z + 2}) = \frac{π}{2}\}$

The numbers of elements in set $A \cap B$ is