For a non-zero complex number z , let arg ( z ) denote the principal argument with - π < a r g z ≤ π then, which of the following statement(s) is (are) FALSE?

For any three given distinct complex numbers z 1 , z 2 & z 3 , the locus of the point z satisfying the condition arg z - z 1 z 2 - z 3 z - z 3 z 2 - z 1 = π lies on a straight line

For any two non-zero complex numbers z 1 & z 2 , a r g z 1 z 2 - a r g z 1 + a r g z 2 is an integer multiple of 2 π

If z 1 = - 1 and z 2 = i , then find Arg z 1 z 2

Given, z 1 = - 1 and z 2 = i We know that, argument of purely real number with negative real part is π . Argument of purely imaginary part with positive imaginary part is π 2 . Now, Arg z 1 z 2 = Arg z 1 - Arg z 2 = π - π 2 = π 2 ∴ Arg z 1 z 2 = 90 °

EASY

Earn 100

Find the argument of a conjugate of a complex number $z = 3 + 2 i$ .

Important Questions on Complex Numbers

HARD

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

For a non-zero complex number

z

, let

\arg (z)

denote the principal argument with

- π < a r g (z) \leq π

then, which of the following statement(s) is (are) FALSE?

HARD

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

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>Amplitude of a Complex Number

The value of

amp (i ω) + amp (i ω^{2})

, where

i = \sqrt{- 1}

and

ω = \sqrt[3]{1} =

non-real is

MEDIUM

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

The amplitude of the complex number

1 + \sin α - i \cos α

HARD

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

z

is a complex number of unit modulus and argument

θ

, then arg

(\frac{1 + z}{1 + \bar{z}})

can be equal to

(given z \neq - 1)

EASY

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

z_{1} = - 1

and

z_{2} = i

, then find

Arg (\frac{z_{1}}{z_{2}})

MEDIUM

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

A point

z

moves in the complex plane such that

\arg (\frac{z - 2}{z + 2}) = \frac{π}{4},

then the minimum value of

| z - 9 \sqrt{2} - 2 i |^{2}

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

If the amplitude of

(z - 1 - 2 i)

\frac{π}{3}

, then the locus of

z

MEDIUM

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

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>Amplitude of a Complex Number

1 + i = (x + i y) (u + i v)

, then

\tan^{- 1} (\frac{y}{x}) + \cot^{- 1} (\frac{u}{v})

has the value

MEDIUM

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

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>Amplitude of a Complex Number

z = a + i b

satisfies

\arg (z - 1) = \arg (z + 3 i)

, then

(a - 1) : b =

EASY

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

Let

z

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

\arg z > 0 .

Then,

\arg z - \arg (- z)

MEDIUM

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

\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

HARD

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

Let

z

and

ω

be two complex numbers such that

|z| \leq 1, |ω| \leq 1

and

|z + i ω| = |z - i \bar{ω}| = 2

then

z

equals

MEDIUM

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

Represent the union of two sets by Venn diagram for each of the following.

$X = {x | x$ is a prime number between $80$ and $100}$

$Y = {y | y$ is an odd number between $90$ and $100}$

HARD

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

A man walks a distance of

3

units from the origin towards the North-East (

N

45 °

E

) direction. From there, he walks a distance of

4

units towards the North-West (

N

45 °

W

) direction to reach a point

P

. Then, the position of

P

in the argand plane is

HARD

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

A particle

P

starts from the point

z_{0} = 1 + 2 i

, where

i = \sqrt{- 1}

. It moves first horizontally away from origin by

5

units and then vertically away from origin by

3

units to reach a point

z_{1}

. From

z_{1}

the particle moves

\sqrt{2}

units in the direction of the vector

\hat{i} + \hat{j}

and then it moves through an angle

\frac{π}{2}

in anticlockwise direction on a circle with centre at origin, to reach a point

z_{2}

. The point is given by

EASY

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

\arg (z) < 0,

then

\arg (- z) - \arg (z) =

HARD

Mathematics>Algebra>Complex Numbers>Amplitude of a Complex Number

The shaded region, where

P = (- 1, 0), Q = (- 1 + \sqrt{2}, \sqrt{2}), R = (- 1 + \sqrt{2}, - \sqrt{2}), S = (1, 0)

is represented by

Question Image

Find the argument of a conjugate of a complex number z=3+2i.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Find the argument of a conjugate of a complex number $z = 3 + 2 i$ .