Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 29

HARD

JEE Main/Advance

IMPORTANT

Earn 100

If $z_{1}, z_{2}, \dots, z_{100}$ are root of equation $1 + z + z^{2} + \dots + z^{100} = 0$ , then the value of $\sum_{r = 1}^{100} \frac{1}{z_{r} - 1}$ is equal to

50% studentsanswered this correctly

Important Questions on Complex Numbers

EASY

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 31

Value of

{(\frac{3}{2} + \frac{i \sqrt{3}}{2})}^{30}

equals

MEDIUM

JEE Main/Advance

IMPORTANT

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

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

HARD

JEE Main/Advance

IMPORTANT

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

EASY

JEE Main/Advance

IMPORTANT

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

MEDIUM

JEE Main/Advance

IMPORTANT

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

MEDIUM

JEE Main/Advance

IMPORTANT

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

EASY

JEE Main/Advance

IMPORTANT

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

HARD

JEE Main/Advance

IMPORTANT

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