The equation | z - i | = | z - 1 | , i = - 1 , represents:

the line through the origin with slope 1

the line through the origin with slope - 1

If z is a complex number such that z ≥ 2 , then the minimum value of z + 1 2 :

Is strictly greater than 3 2 but less than 5 2

HARD

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Let $z & z_{2}$ be the complex roots of the equation $3 z^{2} + 3 z + b = 0 .$ If the origin, together with the points represented by $z_{1} & z_{2}$ form an equilateral triangle, then the value of $b$

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Important Questions on Complex Numbers and Quadratic Equations

HARD

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

z_{1}, z_{2}

and

z_{3}

represent the vertices of an equilateral triangle such that

|z_{1}| = |z_{2}| = |z_{3}|

then

MEDIUM

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

The equation

| z - i | = | z - 1 |, i = \sqrt{- 1},

represents:

HARD

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

If the four complex numbers

z, \bar{z}, \bar{z} - 2 Re (\bar{z})

and

z - 2 Re (z)

represent the vertices of a square of side

4

units in the Argand plane, then

|z|

is equal to :

MEDIUM

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

Z

is a complex number such that

|Z| \leq 2

and

- \frac{π}{3} \leq amp Z \leq \frac{π}{3}

. The area of the region formed by locus of

Z

is (in sq. units)

HARD

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

The equation

I m (\frac{i z - 2}{z - i}) + 1 = 0, z \in C, z \neq i

represents a part of a circle having radius equal to :

HARD

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

Suppose

z \in C

has argument

θ

such that

0 < θ < \frac{π}{2}

and satisfy the equation

| z - 3 i | = 3

, then what is the value of

\cot θ - \frac{6}{z} ?

HARD

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

Let

S

be the set of all complex numbers

z

satisfying

|z - 2 + i| \geq \sqrt{5} .

If the complex number

z_{0}

is such that

\frac{1}{|z_{0} - 1|}

is the maximum of the set

\{\frac{1}{|z - 1|} : z \in S\},

then the principal argument of

\frac{4 - z_{0} - \bar{z_{0}}}{z_{0} - \bar{z_{0}} + 2 i}

EASY

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

z_{1}

and

z_{2}

be two non-zero complex numbers such that

\frac{z_{1}}{z_{2}} + \frac{z_{2}}{z_{1}} = 1,

then the origin and the points represented by

z_{1}

and

z_{2}

MEDIUM

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

z

is a complex number such that

|z| \geq 2,

then the minimum value of

|z + \frac{1}{2}|

HARD

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

The point represented by

2 + i

in the Argand plane moves

1

unit eastwards, then

2

units northwards and finally from there

2 \sqrt{2}

units in the south-west wards direction. Then its new position in the Argand plane is at the point represented by :

MEDIUM

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

The points in the Argand plane given by

Z_{1} = - 3 + 5 i, Z_{2} = - 1 + 6 i, Z_{3} = - 2 + 8 i, Z_{4} = - 4 + 7 i

form a

HARD

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

The largest value of

r

, for which the region represented by the set

\{ω \in C| |ω - 4 - i| \leq r\}

is contained in the region represented by the set

\{z \in C| |z - 1| \leq |z + i|\}

, is equal to :

HARD

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

Let the point

P

represent

z = x + i y, x, y \in R in

the Argand plane. Let the curves

C_{1}

and

C_{2}

be the loci of

P

satisfying the conditions

(i)

\frac{2 z + i}{z - 2}

is purely imaginary and

(ii)

Arg (\frac{z + i}{z + 1}) = \frac{π}{2}

, respectively. Then the point of intersection of the curves

C_{1}

and

C_{2},

other than the origin, is

HARD

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

The value of

λ

for which the loci

\arg z = \frac{π}{6}

and

| z - 2 \sqrt{3} i | = λ

on the argand plane touch each other is

MEDIUM

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

Let

A (3 - i), B (2 + i)

be two points in the Argand plane. If the point

P

represents the complex number

z = x + i y,

which satisfies

|z - 3 + i| = |z - 2 - i|,

then the locus of the point

P

HARD

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

Let

z = x + i y

and

w = u + i v

be complex numbers on the unit circle such that

z^{2} + w^{2} = 1

. Then the number of ordered pairs

(z, w)

EASY

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

The equation

z \bar{z} + (2 - 3 i) z + (2 + 3 i) \bar{z} + 4 = 0

represents a circle of radius

HARD

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

Let

z \in C

be such that

|z| < 1 .

ω = \frac{5 + 3 z}{5 (1 - z)},

then:

MEDIUM

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

Let

z \in C,

the set of complex numbers. Then the equation,

2 |z + 3 i| - |z - i| = 0

represents:

MEDIUM

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

Let

z_{1}

and

z_{2}

be two complex numbers satisfying

| z_{1} | = 9

and

| z_{2} - 3 - 4 i | = 4

. Then the minimum value of

|z_{1} - z_{2}|

is :

Let z & z2 be the complex roots of the equation 3z2+3z+b=0. If the origin, together with the points represented by z1 & z2 form an equilateral triangle, then the value of b

Important Questions on Complex Numbers and Quadratic Equations

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Let $z & z_{2}$ be the complex roots of the equation $3 z^{2} + 3 z + b = 0 .$ If the origin, together with the points represented by $z_{1} & z_{2}$ form an equilateral triangle, then the value of $b$