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If $I_{m} (\frac{z - 1}{2 z + 1}) = - 4,$ then locus of $z$ is-

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Important Questions on Complex Numbers

HARD

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

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

MEDIUM

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

z

is a complex number satisfying

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

then

|z|

cannot be

HARD

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

The principal argument of the complex number

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

HARD

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

If $z_{1}, z_{2}$ are complex numbers such that $Re (z_{1}) = |z_{1} - 1|$ and $Re (z_{2}) = |z_{2} - 1|$ and $\arg (z_{1} - z_{2}) = \frac{π}{6}$ , then $I m (z_{1} + z_{2})$ is equal to :

EASY

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

The modulus of

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

HARD

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

Let

u = \frac{2 z + i}{z - k i}, z = x + i y

and

k > 0

. If the curve represented by

Re (u) + Im (u) = 1

intersects the

y

-axis at points

P

and

Q

where

PQ = 5

, then the value of

k

MEDIUM

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

2 + i

is a root of

x^{2} - 4 x + c = 0,

where

c

is a real number, then the value of

c

MEDIUM

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

When

n = 8

{(\sqrt{3} + i)}^{n} + {(\sqrt{3} - i)}^{n} =

EASY

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

The value of

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

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

The value of sum

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

equals

HARD

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

a

and

b

are real numbers such that

{(2 + α)}^{4} = a + b α

, where

α = \frac{- 1 + i \sqrt{3}}{2}

, then

a + b

is equal to:

EASY

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

The value of

i^{2 n} + i^{2 n + 1} + i^{2 n + 2} + i^{2 n + 3},

where

i = \sqrt{- 1},

MEDIUM

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

m

be the least value of

| z - 3 + i 4 |^{2} + | z - 5 - i 2 |^{2}, z \in ℂ

attained at

z = z_{0},

then the ordered pair

(|z_{0}|, m)

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

Let

z = x + i y

be a complex number such that

| z + i | = 2

. Then the locus of

z

is a circle whose centre and radius are

HARD

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

If the point

(\frac{k - 1}{k}, \frac{k - 2}{k})

lies on the locus of

z

satisfying the inequality

|\frac{z + 3 i}{3 z + i}| < 1,

then the interval in which

k

lies is

EASY

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

z = x + i y

, then the equation

|z + 1| = |z - 1|

represents

MEDIUM

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

If the amplitude of

(z - 1 - 2 i)

\frac{π}{3}

, then the locus of

z

MEDIUM

Mathematics>Algebra>Complex Numbers>Introduction to 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|

MEDIUM

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

z_{1} = 2 - i

and

z_{2} = 1 + i

,

then

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

EASY

Mathematics>Algebra>Complex Numbers>Introduction to Complex Numbers

The value of

{(\frac{- 1 + i \sqrt{3}}{2})}^{n} + {(\frac{- 1 - i \sqrt{3}}{2})}^{n}

where

n

is not a multiple of

3

and

i = \sqrt{- 1},

If Imz-12z+1=-4, then locus of z is-

Important Questions on Complex Numbers

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If $I_{m} (\frac{z - 1}{2 z + 1}) = - 4,$ then locus of $z$ is-