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

HARD

Earn 100

If $z$ is a complex number satisfying $| z^{2} - 1 | = | z | + 2$ , then

50% studentsanswered this correctly

Important Questions on Complex Numbers

HARD

Mathematics>Algebra>Complex Numbers>Triangle Inequality

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

HARD

Mathematics>Algebra>Complex Numbers>Triangle Inequality

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

HARD

Mathematics>Algebra>Complex Numbers>Triangle Inequality

For any non-zero complex number

z,

the minimum value of

| z | + | z - 1 |

EASY

Mathematics>Algebra>Complex Numbers>Triangle Inequality

Let a complex number

z, |z| \neq 1

, satisfy

\log_{\frac{1}{\sqrt{2}}} (\frac{|z| + 11}{{(|z| - 1)}^{2}}) \leq 2

. Then, the largest value of

|z|

is equal to _________.

MEDIUM

Mathematics>Algebra>Complex Numbers>Triangle Inequality

All the points in the set

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

lie on a

MEDIUM

Mathematics>Algebra>Complex Numbers>Triangle Inequality

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>Triangle Inequality

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} =

EASY

Mathematics>Algebra>Complex Numbers>Triangle Inequality

For the complex numbers

z_{1}, z_{2}

satisfying

|z_{1}| = 12

and

|z_{2} - 3 - 4 i| = 5

, the minimum value of

|z_{1} - z_{2}|

MEDIUM

Mathematics>Algebra>Complex Numbers>Triangle Inequality

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

EASY

Mathematics>Algebra>Complex Numbers>Triangle Inequality

z_{1} = 2 + 3 i

and

z_{2} = 3 + 2 i

, then

|z_{1} + z_{2}|

is equal to

HARD

Mathematics>Algebra>Complex Numbers>Triangle Inequality

i z^{3} + z^{2} - z + i = 0

(where

z

is a complex number), then the value of

|z|

MEDIUM

Mathematics>Algebra>Complex Numbers>Triangle Inequality

z

is a complex number satisfying

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

then the maximum possible value of

|z + z^{- 1}|

is-

MEDIUM

Mathematics>Algebra>Complex Numbers>Triangle Inequality

The maximum value of

|z|

when

z

satisfies the condition

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

is:

MEDIUM

Mathematics>Algebra>Complex Numbers>Triangle Inequality

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>Triangle Inequality

|z - i| \leq 2

and

z_{0} = 5 + 3 i

, then the maximum value of

|i z + z_{0}|

EASY

Mathematics>Algebra>Complex Numbers>Triangle Inequality

We are given that

p

and

q

are two complex numbers such that

| p + q | = | p | + | q | .

The value of

\arg (p) - \arg (q)

EASY

Mathematics>Algebra>Complex Numbers>Triangle Inequality

z = x + i y

and

ω = \frac{1 - i z}{z - i}

, then

|ω| = 1

implies that, in the complex plane

HARD

Mathematics>Algebra>Complex Numbers>Triangle Inequality

Let

S

be the set of all complex numbers

z

satisfying

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

. Then which of the following statements is/are TRUE?

EASY

Mathematics>Algebra>Complex Numbers>Triangle Inequality

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

HARD

Mathematics>Algebra>Complex Numbers>Triangle Inequality

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

If z is a complex number satisfying |z2−1|=|z|+2 , then

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

If $z$ is a complex number satisfying $| z^{2} - 1 | = | z | + 2$ , then