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The number of solution(s) of the equation $z^{2} = 4 z + {|z|}^{2} + \frac{16}{{|z|}^{3}}$ is (where $z = x + i y, x, y \in R, i^{2} = - 1 a n d x \neq 2$ )

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Important Questions on Complex Numbers and Quadratic Equations

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If the equation

a z^{2} + 2 b z + a = 0

, where

a, b

are complex numbers, has both roots purely real, then

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

u + i v = \frac{3 i}{x + i y + 2},

then

y =

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If a complex number lies in the

{III}^{rd}

quadrant, then find the quadrant in which its conjugate would lie

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

z

is a complete number, then

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

represents

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

\frac{z - α}{z + α} (α \in R)

is a purely imaginary number and

|z| = 2

, then a value of

α

is :

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

Find the complex conjugate of

(2 + 5 i) (- 4 + 6 i)

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

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}}

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

z^{4} = 7 - 5 i

, then

I m ({(\bar{z})}^{4})

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

a > 0

and

z = \frac{{(1 + i)}^{2}}{a - i}

, has magnitude

\sqrt{\frac{2}{5}}

, then

\bar{z}

is equal to:

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If the equation

x^{2} + b x + 45 = 0, b \in R

has conjugate complex roots and they satisfy

|z + 1| = 2 \sqrt{10},

then

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

Find the complex conjugate of

(3 + 4 i) (2 - 3 i)

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If the conjugate of a complex number

z

\frac{1}{i - 1}

,

then

z

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

Find the real part of the complex number

{(1 - 2 i)}^{- 3}

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

z_{1} = 5 y^{2} - 3 - 8 i x, z_{2} = 4 y^{2} - 16 i

and

z_{1} = {\bar{z}}_{2}

then

z = x + i y

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

a

and

b

are unit vectors and

|a + b| = 1

, then

|a - b|

is equal to.

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

Let $α = 8 - 14 i, A = \{z \in ℂ : \frac{α z - \bar{α} \bar{z}}{z^{2} - (\bar{z})^{2} - 112 i} = 1\}$ and $B = \{z \in ℂ : |z + 3 i| = 4\}$

Then, $\sum_{z \in A \cap B} (Re z - I m z)$ is equal to ________

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If the point

(x, y)

satisfies the equation

\frac{x + i (x - 2)}{3 + i} - i = \frac{2 y + i (1 - 3 y)}{i - 3}

, then

x + y =

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If a complex number

z

is such that

(7 + i) (z + \bar{z}) - (4 + i) (z - \bar{z}) + 116 i = 0

, then

z \bar{z} =

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

a

and

b

are real numbers and

{(a + i b)}^{11} = 1 + 3 i

,

then

{(b + i a)}^{11}

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

Let

z = {(\frac{\sqrt{3}}{2} + \frac{i}{2})}^{5} + {(\frac{\sqrt{3}}{2} - \frac{i}{2})}^{5} .

R (z)

and

I (z)

respectively denote the real and imaginary parts of

z,

then

The number of solution(s) of the equation z2=4z+z2 +16z3 is (where z=x+iy, x,y∈R,i2=–1 and x≠2)

Important Questions on Complex Numbers and Quadratic Equations

Learn and Prepare for any exam you want !

The number of solution(s) of the equation $z^{2} = 4 z + {|z|}^{2} + \frac{16}{{|z|}^{3}}$ is (where $z = x + i y, x, y \in R, i^{2} = - 1 a n d x \neq 2$ )