If R e z - 1 2 z + i = 1 , where z = x + i y , then the point x , y lies on a

circle whose centre is at - 1 2 , - 3 2

straight line whose slope is - 2 3

straight line whose slope is 3 2

If A = z = x + i y / real part of z ¯ - 1 z - i = 2 then the locus of the point P x , y in the Cartesian plane is

a pair of lines passing through - 1 , 1

a circle of radius 1 2 and the centre - 1 2 , 3 2

a pair of lines passing through - 1 , - 2

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If complex number $z$ and its conjugate $\bar{z}$ satisfy $z \bar{z} + 2 i (z + \bar{z}) = 12 + 8 i$ then value of $z$ is

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Important Questions on Complex Numbers and Quadratic Equations

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

Simplified form of

\frac{10 i}{1 + 2 i}

is equal to

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

A value of

θ

for which

\frac{2 + 3 i \sin θ}{1 - 2 i \sin θ}

is purely imaginary, is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

R e (\frac{z - 1}{2 z + i}) = 1,

where

z = x + i y,

then the point

(x, y)

lies on a

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

Let

z = x + i y

be a non-zero complex number such that

z^{2} = i | z |^{2}

, where

i = \sqrt{- 1}

, then

z

lies on the

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

Let

w (I m w \neq0)

be a complex number. Then, the set of all complex numbers

z

satisfying the equation

w - \bar{w} z = k (1 - z)

, for some real number

k

, is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

{(\frac{1 + i}{1 - i})}^{\frac{m}{2}} = {(\frac{1 + i}{i - 1})}^{\frac{n}{3}} = 1, (m, n \in N)

then the greatest common divisor of the least values of

m

and

n

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

Let

z \in C

with

I m (z) = 10

and it satisfies

\frac{2 z - n}{2 z + n} = 2 i - 1

for some natural number

n .

Then

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

{(\frac{1 - i}{1 + i})}^{96} = a + i b

, then

(a, b)

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

\frac{3}{2 + \cos θ + i \sin θ} = a + i b

, then

[{(a - 2)}^{2} + b^{2}]

is equal to

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

Let

z_{1}

and

z_{2}

be the roots of the equation

z^{2} + p z + q = 0

where

p, q

are real. The points represented by

z_{1}, z_{2}

and the origin form an equilateral triangle, if

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

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

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

The area of the triangle in the complex plane formed by

z, i z

and

z + i z

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

Let

z \in C

and

i = \sqrt{- 1}

, if

a, b, c \in (0, 1)

be such that

a^{2} + b^{2} + c^{2} = 1

and

b + i c = (1 + a) z

, then

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

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

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 and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

A = \{z = x + i y / real part of \frac{\bar{z} - 1}{z - i} = 2\}

then the locus of the point

P (x, y)

in the Cartesian plane is

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

The conjugate of a complex number is

\frac{1}{1 - i}

. Then the complex number is

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

z = a + i b

, then

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

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

Let

z = 1 + a i

, be a complex number,

a > 0

, such that

z^{3}

is a real number. Then, the sum

1 + z + z^{2} + \dots . + z^{11}

is equal to :

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

For all complex numbers

z

of the form

1 + i α, α \in R,

z^{2} = x + i y,

then

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

The imaginary part of conjugate of

{(\frac{1 + i}{1 - i})}^{5}

If complex number z and its conjugate z¯ satisfy zz¯+2i z+z¯=12+8i then value of z is

Important Questions on Complex Numbers and Quadratic Equations

Learn and Prepare for any exam you want !

If complex number $z$ and its conjugate $\bar{z}$ satisfy $z \bar{z} + 2 i (z + \bar{z}) = 12 + 8 i$ then value of $z$ is