Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 22

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If there exist exactly one complex number $z$ satisfying the conditions $|z - k i| \leq \sqrt{3}$ and arg $(z) \leq \frac{π}{6}$ (where $k \in R^{+}$ and $i = \sqrt{- 1}$ ), then the value of $k$ is

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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 24

The number of values of

z

satisfying the equations

1 + z + z^{2} + z^{3} + \dots + z^{17} = 0

and

1 + z + z^{2} + \dots + z^{11} = 0

simultaneously is

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 25

If

{[\frac{1 + \cos θ + i \sin θ}{\sin θ + i (1 + \cos θ)}]}^{4} = \cos n θ + i \sin n θ

then value of

n

is

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

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

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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 42

The algebraic sum of perpendicular distances from the points

1, α_{1}, α_{2}, α_{3}, α_{4}, α_{5}, α_{6}, α_{7}

, where

α_{i}

for

i = 1, 2, \dots, 7

, are complex

8^{th}

roots of unity, from the line

(3 + 2 i) z + (3 - 2 i) \bar{z} + 5 \sqrt{13} = 0

, is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 45

If the equation of all the circles which are orthogonal to

|z| = 1

and

|z - 1| = 4

is

|z + 7 - i k| = \sqrt{λ + k^{2}}, i = \sqrt{- 1}

and

k \in R

, then the value of

λ

must be

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 3

The complex numbers

z_{1}, z_{2}

and

z_{3}

satisfying

\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - i \sqrt{3}}{2}

are the vertices of a triangle which is

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

If

z^{2} + z |z| + {|z|}^{2} = 0

, then the locus of

z

is