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

EASY

Earn 100

Find the multiplicative inverse of $2 - 3 i$ .

Important Questions on Complex Numbers and Quadratic Equations

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

The value of

{[\frac{5 i}{(1 - i) (2 - i) (3 - i)}]}^{50}

is equal to

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Multiplicative inverse of

- 1 + i

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Multiplicative inverse of the complex number

(\sin θ, \cos θ)

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

The imaginary part of conjugate of

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

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

z_{1}

and

z_{z}

are any two complex numbers, then

Re (z_{1} z_{2})

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Let

A = \{\frac{1967 + 1686 i \sin θ}{7 - 3 i \cos θ} : θ \in R\}

. If

A

contains exactly one positive integer

n

, then the value of

n

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Let

z_{1} = a + i b

and

z_{2} = c + i d

be complex numbers such that

|z_{1}| = |z_{2}| = 1

and

Re (z_{1} \bar{z_{2}}) = 0

. Then the complex numbers

w_{1} = a + i c

and

w_{2} = b + i d

always satisfy

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

(a + i b)^{\frac{1}{4}} = 2 + 3 i,

then

3 b - 2 a =

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Find the multiplicative inverse of :

7 + 24 i

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

A, B,

and

C

are angles of a triangle such that

x = cis A, y = cis B

z = cis C,

then find the value of

x y z

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

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>Algebra of Complex Numbers

If $z_{1} = 1 + 2 i, z_{2} = 1 - 3 i$ and $z_{3} = 2 + 4 i$ then, the points on the Argand diagram representing $z_{1} z_{2} z_{3}, 2 z_{1} z_{2} z_{3}, - 7 z_{1} z_{2} z_{3}$ are:

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Z_{1} = a + i b

and

Z_{2} = c + i d

are two complex numbers where

a, b, c, d

are all real numbers, then

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Let $a \neq b$ be two non-zero real numbers. Then the number of elements in the set $X = \{z \in C : Re (a z^{2} + b z) = a and Re (b z^{2} + a z) = b\}$ is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

x + i y = \frac{1 + 7 i}{{(2 - i)}^{2}};

then

cosec (\tan^{- 1} \frac{y}{x} - \frac{π}{4}) =

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

x = \frac{4}{5} + \frac{3}{5} i, y = \frac{\sqrt{3}}{\sqrt{8}} - \frac{\sqrt{5}}{\sqrt{8}} i,

then

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

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>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} =

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Write the complex number

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

in the form

A + i B

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

x + i y = cis α \cdot cis β

, then find the value of

x^{2} + y^{2}

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

x = \frac{1 + \sqrt{3} i}{2}

then the value of the expression

y = x^{4} - x^{2} + 6 x - 4,

equals

Find the multiplicative inverse of 2-3i.

Important Questions on Complex Numbers and Quadratic Equations

Learn and Prepare for any exam you want !

Find the multiplicative inverse of $2 - 3 i$ .