Let a , b ∈ R and a 2 + b 2 ≠ 0 . Suppose S = z ∈ C : z = 1 a + i b t , t ∈ R , t ≠ 0 , where i = - 1 . If z = x + i y and z ∈ S , then x , y lies on

The y - axis for a = 0 , b ≠ 0

The circle with radius - 1 2 a and centre - 1 2 a , 0 for a < 0 , b ≠ 0

Let s , t , r be non-zero complex numbers and L be the set of solutions z = x + i y x , y ∈ ℝ , i = - 1 of the equation s z + t z ¯ + r = 0 , where z ¯ = x - i y . Then, which of the following statement(s) is (are) TRUE?

If L has more than one element, then L has infinitely many elements

If s = t , then L has infinitely many elements

Show that the points in the Argand plane represented by the complex numbers - 2 + 7 i . - 3 2 + 1 2 i , 4 - 3 i , 7 2 1 + i are the vertices of a rhombus.

Given, - 2 + 7 i , - 3 2 + 1 2 i , 4 - 3 i , 7 2 1 + i Let the four points represented by the complex numbers be A - 2 , 7 , B - 3 2 , 1 2 , C 4 , - 3 , D 7 2 , 7 2 The distance between the given points are, A B = - 2 + 3 2 2 + 7 - 1 2 2 = 1 4 + 169 4 = 170 2 units B C = 4 + 3 2 2 + - 3 - 1 2 2 = 121 4 + 49 4 = 170 2 units C D = 4 - 7 2 2 + - 3 - 7 2 2 = 1 4 + 169 4 = 170 2 units A D = - 2 - 7 2 2 + 7 - 7 2 2 = 121 4 + 49 4 = 170 2 units So, all sides are equal, A B = B C = C D = A D . Length of diagonal A C = - 2 - 4 2 + 7 + 3 2 = 36 + 100 = 136 units Length of diagonal B D = - 3 2 - 7 2 2 + 1 2 - 7 2 2 = 100 4 + 36 4 = 136 4 units We know that the diagonals of a rhombus are not equal. ∴ A B C D is a rhombus.

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

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

Show that the points in the Argand diagram represented by the complex numbers 2 + 2 i , - 2 - 2 i , - 2 3 + 2 3 i are the vertices of an equilateral triangle.

Given, 2 + 2 i , - 2 - 2 i , - 2 3 + 2 3 i Let A ( 2 , 2 ) , B ( - 2 , - 2 ) , C ( - 2 3 , 2 3 ) be points represents given complex numbers in the Argand plane. A B = ( 2 + 2 ) 2 + ( 2 + 2 ) 2 ⇒ A B = 4 2 B C = ( - 2 + 2 3 ) 2 + ( - 2 - 2 3 ) 2 ⇒ B C = 4 + 12 - 8 3 + 4 + 12 + 8 3 ⇒ B C = 4 2 A C = ( 2 + 2 3 ) 2 + ( 2 - 2 3 ) 2 ⇒ A C = 4 + 12 + 8 3 + 4 + 12 - 8 3 ⇒ A C = 4 2 ∴ A B = A C = B C So, △ A B C is equilateral.

HARD

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Suppose z₁ + z₂ + z₃ + z₄ = 0 and |z₁| = |z₂| = |z₃| = |z₄| = 1. If z₁, z₂, z₃, z₄ are the vertices of a quadrilateral, then the quadrilateral must be a

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Important Questions on Complex Numbers

HARD

Mathematics>Algebra>Complex Numbers>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>Geometry of Complex Numbers

Let

a, b \in R

and

a^{2} + b^{2} \neq 0

. Suppose

S = \{z \in C : z = \frac{1}{a + i b t}, t \in R, t \neq 0\},

where

i = \sqrt{- 1} .

z = x + i y

and

z \in S,

then

(x, y)

lies on

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let

s, t, r

be non-zero complex numbers and

L

be the set of solutions

z = x + i y (x, y \in ℝ, i = \sqrt{- 1})

of the equation

s z + t \bar{z} + r = 0

, where

\bar{z} = x - i y .

Then, which of the following statement(s) is (are) TRUE?

HARD

Mathematics>Algebra>Complex Numbers>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 :

EASY

Mathematics>Algebra>Complex Numbers>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>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>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 :

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let

z \in C

be such that

|z| < 1 .

ω = \frac{5 + 3 z}{5 (1 - z)},

then:

EASY

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let

z_{1}, z_{2}

be the roots of the equation

z^{2} + a z + 12 = 0

and

z_{1}, z_{2}

form an equilateral triangle with origin. Then, the value of

|a|

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Show that the points in the Argand plane represented by the complex numbers $- 2 + 7 i . \frac{- 3}{2} + \frac{1}{2} i, 4 - 3 i, \frac{7}{2} (1 + i)$ are the vertices of a rhombus.

MEDIUM

Mathematics>Algebra>Complex Numbers>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>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

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

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

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let complex numbers

α

and

\frac{1}{\bar{α}}

lie on circles

{(x - x_{0})}^{2} + {(y - y_{0})}^{2} = r^{2}

, and

{(x - x_{0})}^{2} + {(y - y_{0})}^{2} = 4 r^{2}

, respectively. If

z_{0} = x_{0} + i y_{0}

satisfies the equation

2 {|z_{0}|}^{2} = r^{2} + 2

, then

|α| =

HARD

Mathematics>Algebra>Complex Numbers>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 :

HARD

Mathematics>Algebra>Complex Numbers>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}

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

The equation

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

represents:

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Show that the points in the Argand diagram represented by the complex numbers

2 + 2 i, - 2 - 2 i, - 2 \sqrt{3} + 2 \sqrt{3} i

are the vertices of an equilateral triangle.

MEDIUM

Mathematics>Algebra>Complex Numbers>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>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 :

Suppose z1 + z2 + z3 + z4 = 0 and |z1| = |z2| = |z3| = |z4| = 1. If z1, z2, z3, z4 are the vertices of a quadrilateral, then the quadrilateral must be a

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Suppose z₁ + z₂ + z₃ + z₄ = 0 and |z₁| = |z₂| = |z₃| = |z₄| = 1. If z₁, z₂, z₃, z₄ are the vertices of a quadrilateral, then the quadrilateral must be a