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The real part of each root of the equation ${(z + 1)}^{6} + z^{6} = 0$ is: (where $z$ is a complex number)

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Important Questions on Complex Numbers

MEDIUM

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

Let

a = \cos 1 °

and

b = \sin 1 °

. We say that a real number is algebraic if it is a root of a polynomial with integer coefficients. Then,

HARD

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

z = \frac{\sqrt{3} + i}{2}

then

{(z^{101} + i^{103})}^{105} =

EASY

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

The number of complex numbers

p

such the

| p | = 1

and imaginary part of

p^{4}

0,

MEDIUM

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

Consider a regular

10 -

gon with its vertices on the unit circle. With one vertex fixed, draw straight lines to the other

9

vertices. Call them

L_{1}, L_{2}, \dots, L_{9}

and denote their lengths by

l_{1}, l_{2}, \dots, l_{9}

respectively. Then the product

l_{1} l_{2} \dots l_{9}

EASY

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

The value of

{(\frac{1 + \sin \frac{2 π}{9} + i \cos \frac{2 π}{9}}{1 + \sin \frac{2 π}{9} - i \cos \frac{2 π}{9}})}^{3}

MEDIUM

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

n

is a positive integer, then

{(1 + i)}^{n} + (1 - i)^{n}

is equal to

HARD

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

a = \cos (\frac{8 π}{11}) + i \sin (\frac{8 π}{11}),

then

Re (a + a^{2} + a^{3} + a^{4} + a^{5}) =

MEDIUM

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

(\sqrt{3} + i)^{100} = 2^{99} (p + i q),

then

p

and

q

are roots of the equation :

MEDIUM

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

The value of

{(\frac{- 1 + i \sqrt{3}}{1 - i})}^{30}

is :

MEDIUM

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

If $e^{i x}$ is a solution of the equation $z^{n} + p_{1} z^{n - 1} + p_{2} z^{n - 2} + \dots \dots + p_{n} = 0$ , where $p_{i}$ are real $(i = 1, 2, 3, \dots \dots \dots n)$ ,then

$p_{n} \sin n x + p_{n - 1} \sin (n - 1) x + \dots + p_{1} \sin x + 1 =$

MEDIUM

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

The product of all values of

(\cos α + i \sin α)^{3 / 5}

EASY

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

The value of

\frac{2 (\cos 75^{o} + i s i n 75^{o})}{0.2 (\cos 30^{o} + i s i n 30^{o})}

MEDIUM

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

The value of

(1 + i)^{8} + (1 - i)^{8} =

HARD

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

2 \cos \frac{7 π}{5}

is one of the values of

z^{\frac{1}{5}}

, then

z =

EASY

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

Which of the following can be the equal expression for

\cos (3 T) + isin (3 T)

using the De Moivre's theorem and binomial theorem concepts?

EASY

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

(\sin θ - i \cos θ)^{3}

is equal to

EASY

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

Real part of

(\cos 4 + i \sin 4 + 1)^{2020}

is_______

HARD

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

z = \cos 6 ° + i \sin 6 °,

then

\sum_{n = 1}^{20} Im (z^{2 n - 1}) =

HARD

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

1 + x^{2} = \sqrt{3} x

, then

\sum_{n = 1}^{24} {(x^{n} - \frac{1}{x^{n}})}^{2}

is equal to

HARD

Mathematics>Algebra>Complex Numbers>De Moivre's Theorem

z

is a non-real complex number, then the minimum value of

\frac{I m z^{5}}{{(I m z)}^{5}}

is (Where

I m z =

Imaginary part of

z

)

The real part of each root of the equation z+16+z6=0 is: (where z is a complex number)

Important Questions on Complex Numbers

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The real part of each root of the equation ${(z + 1)}^{6} + z^{6} = 0$ is: (where $z$ is a complex number)