Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 16

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The complex number $z_{1}, z_{2}$ satisfies the equation $z + 1 + 8 i = |z| (1 + i)$ , where $i = \sqrt{- 1}$ then the equation whose roots are $|z_{1}|$ and $|z_{2}|$ 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 17

The complex number

z

and

ω

are such that

|z| = |ω| = 1

. Then, the locus of

\frac{z + ω}{1 + z ω}

is

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 18

If

f (n, θ) = \sum_{r = 0}^{n} \frac{\cos (r θ)}{2^{r}}

, then the value of

\lim_{n \to \infty} f (n, \frac{π}{3})

is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 19

The number of common solutions of the equations

z^{n} = 1 + i

and

z^{m} = 2 - i, n, m \in N z \in C

is

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 20

Let

ω = \cos 3 ° + i \sin 3 °

then

\sum_{r = 1}^{10} Re (ω^{2 r - 1})

equals

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 21

If

1, α_{1}, α_{2}, α_{3}, \dots, α_{3 n}

are the roots of equation

z^{3 n + 1} = 1, (n \in N)

and

ω

is an imaginary cube root of unity. Then the value of

|(α_{1} - ω) (α_{2} - ω) (α_{3} - ω) \dots (α_{3 n} - ω)|

is

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

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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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 23

If

|z - 1| = 1

and

z \neq 0, 2

then the possible value of

\frac{z - 2}{z}

is

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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