A complex number z is said to be unimodular if z = 1 . Let z 1 and z 2 are complex numbers such that z 1 - 2 z 2 2 - z 1 z 2 is unimodular and z 2 is not unimodular, then the point z 1 lies on a

straight line parallel to x -axis

straight line parallel to y -axis

All the points in the set S = α + i α - i , α ∈ R , i = - 1 lie on a

straight line whose slope is - 1

Let a be a fixed non-zero complex number with | a | < 1 and w = z - a 1 - a - z , where z is a complex number. Then

| w | < 1 for all z such that | z | < 1

There exists a complex number z with | z | < 1 such that | w | > 1

| w | > 1 for all z such that | z | < 1

There exists z such with | z | < 1 and | w | = 1

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Consider a square $O A B C$ in argand plane, where $O$ is origin and $A$ be complex number $z_{0}$ . Then the equation of the circle that can be inscribed in this square is (Vertices of square are given in anticlockwise order and $i = \sqrt{- 1}$ )

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Important Questions on Complex Numbers

EASY

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

z = a + i b

satisfies

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

, then

(a - 1) : b =

HARD

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

A complex number

z

is said to be unimodular if

|z| = 1

. Let

z_{1}

and

z_{2}

are complex numbers such that

\frac{z_{1} - 2 z_{2}}{2 - z_{1} z_{2}}

is unimodular and

z_{2}

is not unimodular, then the point

z_{1}

lies on a

EASY

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

z_{1} = 2 + 3 i

and

z_{2} = 3 + 2 i

, then

|z_{1} + z_{2}|

is equal to

HARD

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

For any non-zero complex number

z,

the minimum value of

| z | + | z - 1 |

MEDIUM

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

z

is a complex number satisfying

|z^{3} + z^{- 3}| \leq 2,

then the maximum possible value of

|z + z^{- 1}|

is-

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)

MEDIUM

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

Let $z_{1}$ and $z_{2}$ be complex numbers satisfying $|z_{1}| = |z_{2}| = 2$ and $|z_{1} + z_{2}| = 3$ . Then $|\frac{1}{z_{1}} + \frac{1}{z_{2}}| =$

MEDIUM

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

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

, then

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

has the value

EASY

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

A real value of

x

will satisfy the equation

(\frac{3 - 4 i x}{3 + 4 i x}) = α - i β (α, β a r e r e a l)

, if

MEDIUM

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

All the points in the set

S = \{\frac{α + i}{α - i}, α \in R\}, i = \sqrt{- 1}

lie on a

EASY

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

Let

z_{1}

and

z_{2}

be complex numbers such that

z_{1} \neq z_{2}

and

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

Re (z_{1}) > 0

and

Im (z_{2}) < 0,

then

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

HARD

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

Let

a

be a fixed non-zero complex number with

| a | < 1

and

w = (\frac{z - a}{1 - \bar{a} z}),

where

z

is a complex number. Then

HARD

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

Suppose

z

is any root of

11 z^{8} + 21 i z^{7} + 10 i z - 22 = 0

where

i = \sqrt{- 1} .

Then,

S = | z |^{2} + | z | + 1

satisfies

MEDIUM

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

a, b

are the least and the greatest values respectively of

|z_{1} + z_{2}|,

where

z_{1} = 12 + 5 i

and

|z_{2}| = 9,

then

a^{2} + b^{2} =

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

Let a complex number be

w = 1 - \sqrt{3} i

. Let another complex number

z

be such that

| z w | = 1

and

\arg (z) - \arg (w) = \frac{π}{2}

. Then the area of the triangle (in sq. units) with vertices origin,

z

and

w

is equal to

MEDIUM

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

Let

z

and

w

be two complex numbers such that

w = z \bar{z} - 2 z + 2, |\frac{z + i}{z - 3 i}| = 1

and

Re (w)

has minimum value. Then, the minimum value of

n \in N

for which

w^{n}

is real, is equal to _______.

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

EASY

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

If the conjugate of a complex number

z

\frac{1}{i - 1}

,

then

z

HARD

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

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)

Consider a square OABC in argand plane, where O is origin and A be complex number z0. Then the equation of the circle that can be inscribed in this square is (Vertices of square are given in anticlockwise order and i=-1)

Important Questions on Complex Numbers

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Consider a square $O A B C$ in argand plane, where $O$ is origin and $A$ be complex number $z_{0}$ . Then the equation of the circle that can be inscribed in this square is (Vertices of square are given in anticlockwise order and $i = \sqrt{- 1}$ )