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Mathematics
Cube roots of unity
If $\omega$ is an imaginary cube root of unity, then the value of $\sin \left[ {\left( {{\omega ^{10}} + {\omega ^{23}}} \right)\pi - \dfrac{\pi }{4}} \right]$ is
$\left( a \right)\dfrac{{ - \sqrt 3 }}{2}$
$\left( b \right)\dfrac{{ - 1}}{{\sqrt 2 }}$
$\left( c \right)\dfrac{1}{{\sqrt 2 }}$
$\left( d \right)\dfrac{{\sqrt 3 }}{2}$

Mathematics
Cube roots of unity
If $\omega$ is the imaginary cube root of the unity, then ${{(1+\omega -{{\omega }^{2}})}^{7}}$ equal
$A)128\omega$
$B)-128\omega$
$c)128{{\omega }^{2}}$
$D)-128{{\omega }^{2}}$
Mathematics
Cube roots of unity
If the cube roots of unity are $1,\omega ,{\omega ^2}$ , then the roots of the equation ${\left( {x - 1} \right)^3} + 8 = 0$ are
(a) $- 1,1 + 2\omega ,{\omega ^2}$
(b) $- 1,1 - 2\omega ,1 - 2{\omega ^2}$
(c) -1, -1, -1
(d) None of these
Mathematics
Cube roots of unity
If $\omega$ is a cube root of unity and $n$ is a positive integer satisfying $1 + {\omega ^n} + {\omega ^{2n}} = 0$ then, $n$ is of the type:
A.$3m$
B.$3m + 1$
C.$3m + 2$
D.None of these

Mathematics
Cube roots of unity
If omega ($\omega$) is an imaginary cube root of unity and $x = a + b$, $y = a\omega+ b{\omega ^2}$, $z = a{\omega ^2} + b\omega$, then ${x^2} + {y^2} + {z^2}$ is equal to
1). $6ab$
2). $3ab$
3). $6{a^2}{b^2}$
4). $3{a^2}{b^2}$

Mathematics
Cube roots of unity
If $\omega$ (not equal to $1$) is a cube root of unity and ${\left( {1 + {\omega ^2}} \right)^n} = {\left( {1 + {\omega ^4}} \right)^n}$, then the least positive value of $n$ is
1) $2$
2) $3$
3) $5$
4) $6$
Mathematics
Cube roots of unity
: Find the cube root of $1$ .
(A) $1$
(B) $2$
(C) ${\text{Does not exist}}$
(D) ${\text{None of these}}$
Mathematics
Cube roots of unity
If the a, b and c are cube root of unity,
$\left| \begin{matrix} {{e}^{a}} & {{e}^{2a}} & {{e}^{3a}} \\ {{e}^{b}} & {{e}^{2b}} & {{e}^{3b}} \\ {{e}^{c}} & {{e}^{2c}} & {{e}^{3c}} \\ \end{matrix} \right|-\left| \begin{matrix} {{e}^{a}} & {{e}^{2a}} & 1 \\ {{e}^{b}} & {{e}^{2b}} & 1 \\ {{e}^{c}} & {{e}^{2c}} & 1 \\ \end{matrix} \right|$
1. $0$
2. $e$
3. $e^2$
4. $e^3$
Mathematics
Cube roots of unity
The minimum value of $\left| a+b\omega +c{{\omega }^{2}} \right|$ where $a,b,c$ are all not equal integers and $\omega \left( \ne 1 \right)$ is a cube root of unity, is
1) $\sqrt{3}$
2) $\dfrac{1}{2}$
3) $1$
4) $0$
Mathematics
Cube roots of unity
If $w$ is the complex cube root of unity, then show that: ${(1 - w + {w^2})^5} + {(1 + w - {w^2})^5} = 32$.

Mathematics
Cube roots of unity
If w is a complex cube root of unity then show that $\left( {2 - w} \right)\left( {2 - {w^2}} \right)\left( {2 - {w^{10}}} \right)\left( {2 - {w^{11}}} \right) = 49$?

Mathematics
Cube roots of unity
If ${{x}^{2}}+x+1=0$, then the value of ${{\left( x+\dfrac{1}{x} \right)}^{2}}+{{\left( {{x}^{2}}+\dfrac{1}{{{x}^{2}}} \right)}^{2}}+.....+{{\left( {{x}^{27}}+\dfrac{1}{{{x}^{27}}} \right)}^{2}}$ is
A $27$
B $72$
C $45$
D $54$
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