Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 32

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Let $z_{1} = 5 + i$ and $z_{2} = 2 - 4 i$ (where $i = \sqrt{- 1}$ ). Let $z$ be a complex number such that $\arg (\frac{z - z_{1}}{z_{2} - z}) = \frac{π}{2}$ , then $z$ satisfies

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Important Questions on Complex Numbers

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 33

If

z

and

\bar{z}

represent adjacent vertices of a regular polygon of

λ

sides whose centre is at origin and if

\frac{l m (z)}{Re (z)} = \sqrt{2} - 1

, then

λ

equals

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 34

If

1, ω, ω^{2}, . . ., ω^{n - 1}

are the

n, n^{th}

roots of unity, the value of

(7 - ω) (7 - ω^{2}) . . . (7 - ω^{n - 1})

equals

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 35

Let

u

and

v

are two complex such that

|u| \neq |v|

and

|u| = 1

and

z = \frac{a u + b}{u - v}

, where

z = x + i y

and

a, b

are fixed complex numbers then the locus of

z

is

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 36

If '

α

' denotes

2020^{th}

root of unity then

1 + 2 α + 3 α^{2} + . . .

upto

2020

terms is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 37

A complex number '

z

' is rotated in anticlockwise direction by an angle '

α

' we get

z_{1}

and if same complex number

z

is rotated by an angle

α

in clockwise direction we get

z_{2}

then

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 38

If

1, z_{1}, z_{2}, . . ., z_{n - 1}

be the

n^{th}

roots of unity and '

ω

' be a non-real complex cube root of unity, then sum of all possible values of

\prod_{r = 1}^{n - 1} (ω - z_{r})

will be equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 39

If

z_{1}, z_{2}, z_{3}

are vertices of an equilateral triangle with orthocentre

z_{0}

then

z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = k z_{0}^{2}

, where

k \in N

, then

k

is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 40

If a complex number

z

lies on a circle of radius

3

and centre at

z_{0} = 0

then the complex number

(- 3 + 9 z)

lies on a circle of radius