Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

HARD

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Let $z_{1}$ and $z_{2}$ be roots of the equation $z^{2} + 8 (i - 1) z + 63 - 16 i = 0$ where $i^{2} = - 1 .$ The area of triangle formed by $0, z_{1}$ and $z_{2}$ (where $0$ being origin) is equal to

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Important Questions on Complex Numbers

HARD

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

\cos (i \log (\frac{a - i b}{a + i b}))

is equal to

HARD

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

Let

n

denote the number of solutions of the equation

z^{2} + 3 \bar{z} = 0,

where

z

is a complex number. Then the value of

\sum_{k = 0}^{\infty} \frac{1}{n^{k}}

is equal to

HARD

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

If the roots of the equation

(z - 4)^{3} = 8 i

are

a - 2 i, b + i,

and

c + i,

then

\sqrt{a b c} =

MEDIUM

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

If

x + i y = \frac{1}{1 + \cos θ + i \sin θ}

, then show that

4 x^{2} - 1 = 0

.

HARD

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

If

z = a + i b

, then

i \log (\frac{\bar{z}}{z})

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

If $(x - i y)^{1 / 3} = a - i b$ , then show that:

$\frac{x}{a} - \frac{y}{b} = 4 (a^{2} - b^{2})$

HARD

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

If

S = \{z \in C : \bar{z} = i z^{2}\},

then the maximum value of

| z - \sqrt{3} - i |^{2}

on

S

is

HARD

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

Let

\bar{z}

denote the complex conjugate of a complex number

z

and let

i = \sqrt{- 1}

. In the set of complex numbers, the number of distinct roots of the equation

\bar{z} - z^{2} = i (\bar{z} + z^{2})

is _______.

HARD

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

If the least and the largest real values of

α,

for which the equation

z + α |z - 1| + 2 i = 0

(z \in C and i = \sqrt{- 1})

has a solution, are

p

and

q

respectively; then

4 (p^{2} + q^{2})

is equal to_______.

HARD

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

If

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

then

| z | =

HARD

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

The area of the polygon, whose vertices are the non-real roots of the equation

\bar{z} = i z^{2}

is

HARD

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

The imaginary part of

{(3 + 2 \sqrt{- 54})}^{\frac{1}{2}} - {(3 - 2 \sqrt{- 54})}^{\frac{1}{2}}

, can be

MEDIUM

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

Let

S = \{z \in ℂ : z^{2} + \bar{z} = 0\}

. Then

\sum_{z \in S} (Re (z) + Im (z))

is equal to _______.

HARD

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

The value of

|\sqrt{2 i} - \sqrt{- 2 i}|

is

MEDIUM

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

Let

z = a + i b, b \neq 0

be complex numbers satisfying

z^{2} = \bar{z} \cdot 2^{1 - |z|}

. Then the least value of

n \in N

, such that

z^{n} = {(z + 1)}^{n}

, is equal to _____ .

MEDIUM

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

\sqrt{(- 3 + 4 i) (8 + 6 i)} =

EASY

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

If

\frac{x - 1}{3 + i} + \frac{y - 1}{3 - i} = i

then the true statement among the following is

EASY

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

The solutions of the equation

z \bar{(z - 3 i)} = 2 (2 + 3 i)

are:

MEDIUM

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

If

z = x + i y, z^{\frac{1}{3}} = a - i b

, then

\frac{x}{a} - \frac{y}{b} = k (a^{2} - b^{2})

where

k

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Algebra of Complex Numbers

For all complex numbers

z

of the form

1 + i α, α \in R,

if

z^{2} = x + i y,

then