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If $x$ is an imaginary number satisfying the equation $x^{6} = 2 x^{3} - 1$ , then the value of the $Σ_{r = 1}^{50} {(x^{r} + x^{2 r})}^{3}$ is equal to

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Important Questions on Complex Numbers

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

ω = \frac{- 1 + \sqrt{3} i}{2}

, then

{(3 + ω + 3 ω^{2})}^{4}

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

ω

is a cube root of unity, then

{(3 + 5 ω + 3 ω^{2})}^{2} + {(3 + 3 ω + 5 ω^{2})}^{2}

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Roots of Unity

Let

ω \neq 1

be a cube root of unity. Then the minimum of the set

{{|a + b ω + c ω^{2}|}^{2} : a, b, c

are distinct non-zero integers

}

equals ________

MEDIUM

Mathematics>Algebra>Complex Numbers>Roots of Unity

Imaginary part of

(\sqrt{3} - i)^{2016} + (- \sqrt{3} - i)^{2019}

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

The imaginary part of

{(i - \sqrt{3})}^{13}

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

ω

is a complex cube root of unity, then

{(1 - ω + ω^{2})}^{6} + {(1 - ω^{2} + ω)}^{6} =

MEDIUM

Mathematics>Algebra>Complex Numbers>Roots of Unity

z = \frac{\sqrt{3}}{2} + \frac{i}{2} (i = \sqrt{- 1}),

then

{(1 + i z + z^{5} + i z^{8})}^{9}

is equal to:

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

α

and

β

be the roots of the equation

x^{2} + x + 1 = 0

. Then, the equation whose roots are

α^{19}

and

β^{7}

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

z = e^{\frac{i 4 π}{3}}

,

then

{(z^{192} + z^{194})}^{3}

is equal to

HARD

Mathematics>Algebra>Complex Numbers>Roots of Unity

Let

ω

be a cube root of unity not equal to

1

. Then the maximum possible value of

|a + b w + c w^{2}|

where

a, b, c \in \{1, - 1\}

MEDIUM

Mathematics>Algebra>Complex Numbers>Roots of Unity

On any given arc of positive length on the unit circle

| z | = 1

in the complex plane.

MEDIUM

Mathematics>Algebra>Complex Numbers>Roots of Unity

Let

ω

be a complex number such that

2 ω + 1 = z

where

z = \sqrt{- 3}

. If

|\begin{matrix} 1 & 1 & 1 \\ 1 & - ω^{2} - 1 & ω^{2} \\ 1 & ω^{2} & ω^{7} \end{matrix}| = 3 k,

Then

k

can be equal to:

MEDIUM

Mathematics>Algebra>Complex Numbers>Roots of Unity

ω

is a complex cube root of unity, then the value of

\frac{p + q ω + r ω^{2}}{r + p ω + q ω^{2}} + \frac{p + q ω + r ω^{2}}{q + r ω + p ω^{2}}, (p, q, r \in R)

is equal to

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

The

n^{th}

roots of unity are in

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

ω

is an imaginary cube root of unity, then

{(1 + ω - ω^{2})}^{7}

equals

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

ω

is an imaginary cube root of unit, then

{(1 + ω - ω^{2})}^{7}

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Roots of Unity

The sum of

162^{t h}

power of the roots of the equation

x^{3} - 2 x^{2} + 2 x - 1 = 0

is ______.

MEDIUM

Mathematics>Algebra>Complex Numbers>Roots of Unity

α, β \in C

are the distinct roots of the equation

x^{2} - x + 1 = 0,

then

α^{101} + β^{107}

is equal to

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

The least positive integer

n

for which

{(\frac{1 + i \sqrt{3}}{1 - i \sqrt{3}})}^{n} = 1

MEDIUM

Mathematics>Algebra>Complex Numbers>Roots of Unity

Let

z_{0}

be a root of quadratic equation,

x^{2} + x + 1 = 0 .

z = 3 + 6 i z_{0}^{81} - 3 i z_{0}^{93}

, then

a r g

(z)

is equal to:

If x is an imaginary number satisfying the equation x6=2x3-1, then the value of the Σr=150xr+x2r3 is equal to

Important Questions on Complex Numbers

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If $x$ is an imaginary number satisfying the equation $x^{6} = 2 x^{3} - 1$ , then the value of the $Σ_{r = 1}^{50} {(x^{r} + x^{2 r})}^{3}$ is equal to