HARD

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Let $w \neq 1$ and $w^{13} = 1,$ then find the Quadratic equation whose roots are $w + w^{3} + w^{4} + w^{- 4} + w^{- 3} + w^{- 1}$ and $w^{2} + w^{5} + w^{6} + w^{- 6} + w^{- 5} + w^{- 2}$ .

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Important Questions on Complex Numbers

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

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

1, ω, ω^{2}

are the cube roots of unity, then find the value of

{(1 - ω + ω^{2})}^{3}

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

Let

z_{1}, z_{2}, \dots, z_{7}

be the vertices of a regular heptagon that is inscribed in the unit circle with centre at the origin in the complex plane. Let

w = \sum_{1 \leq i < j \leq 7} z_{i} z_{j}

, then

|w|

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Roots of Unity

x = a + b + c, y = a α + b β + c, z = a β + b α + c

where

α, β

are complex cube roots of unity and

a, b, c

are real, then

x y z

is equal to

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

z = e^{\frac{i 4 π}{3}}

,

then

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

is equal to

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:

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

If the fourth roots of unity are

z_{1}, z_{2}, z_{3}, z_{4}

then

z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2}

is equal to

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

If the cube roots of unity are

1, ω, ω^{2}

, then the roots of the equation

{(x + 1)}^{3} + 8 = 0

are

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

The value of

{[\frac{- 1 + i \sqrt{3}}{2}]}^{- 100} + {[\frac{- 1 - i \sqrt{3}}{2}]}^{100}

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 imaginary part of

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

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

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

ω = \frac{- 1 + \sqrt{3} i}{2}

, then

{(3 + ω + 3 ω^{2})}^{4}

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 ________

HARD

Mathematics>Algebra>Complex Numbers>Roots of Unity

Let

ω

be the complex number

\cos \frac{2 π}{3} + i \sin \frac{2 π}{3}

. Then the number of distinct complex numbers

z

satisfying

|\begin{matrix} z + 1 & ω & ω^{2} \\ ω & z + ω^{2} & 1 \\ ω^{2} & 1 & z + ω \end{matrix}| = 0

is equal to

EASY

Mathematics>Algebra>Complex Numbers>Roots of Unity

ω

is the cube root of unity, then the value of

(1 - ω) (1 - ω^{2}) (1 - ω^{4}) (1 - ω^{8})

HARD

Mathematics>Algebra>Complex Numbers>Roots of Unity

Solve:

x^{11} - x^{6} + x^{5} - 1 = 0

Let w≠1 and w13=1, then find the Quadratic equation whose roots are w+w3+w4+w-4+w-3+w-1 and w2+w5+w6+w-6+w-5+w-2.

Important Questions on Complex Numbers

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Let $w \neq 1$ and $w^{13} = 1,$ then find the Quadratic equation whose roots are $w + w^{3} + w^{4} + w^{- 4} + w^{- 3} + w^{- 1}$ and $w^{2} + w^{5} + w^{6} + w^{- 6} + w^{- 5} + w^{- 2}$ .