Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 58

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Let $z_{1}, z_{2}, z_{3}$ be three distinct complex numbers satisfying, $|z_{1} - 1| = |z_{2} - 1| = |z_{3} - 1| = 1 .$ Let $A, B & C$ be the points representing vertices of equilateral triangle in the Argand plane corresponding to $z_{1}, z_{2}$ and $z_{3},$ respectively. Which of the following are true

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Important Questions on Complex Numbers

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 59

If

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

be the

n^{th}

roots of unity, then which of the following are true

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 60

Which of the following are true.

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 61

If

α, β, γ

are distinct roots of

x^{3} - 3 x^{2} + 3 x + 7 = 0

and

ω

is non-real cube root of unity, then the value of

\frac{α - 1}{β - 1} + \frac{β - 1}{γ - 1} + \frac{γ - 1}{α - 1}

can be equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 62

If

z

is a complex number then the equation

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

is satisfied by (

ω

and

ω^{2}

are imaginary cube roots of unity).

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 63

If

α

is imaginary

n^{th}, (n \geq 3)

root of unity. Which of the following is/are true.

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 64

Which of the following is true?

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-3>Q 1

Let

z = \cos θ + i \sin θ

. Then the value of

\sum_{m = 1}^{15} Im (z^{2 m - 1})

at

θ = 2^{°}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-3>Q 2

Let

z = x + i y

be a complex number where,

x

and

y

are integers. Then, the area ( in square units ) of the rectangle whose vertices are the root of the equation

z {\bar{z}}^{3} + \bar{z} z^{3} = 350,

is