If the cube roots of unity are $1, \omega, \omega^2$,then the roots of the equation $(x - 1)^3 + 8 = 0$ are
- A$ - 1, 1 + 2\omega, 1 + 2\omega^2$
- B$ - 1, 1 - 2\omega, 1 - 2\omega^2$
- C$ - 1, - 1, - 1$
- DNone of these
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If $\alpha, \beta$ are the roots of the equation $x+\frac{4}{x}=2 \sqrt{3}$,then $\frac{2}{\sqrt{3}}\left|\alpha^{2024}-\beta^{2024}\right|=$
If $x_n = \cos \frac{\pi}{2^n} + i \sin \frac{\pi}{2^n}$,then $\prod_{n=1}^{\infty} x_n$ is equal to
DifficultTS EAMCET 2004
View SolutionIf $\alpha, \beta$ are the roots of the equation $x^2-2x+4=0$,then $\alpha^n+\beta^n = \ldots \cos \left(\frac{n\pi}{3}\right)$ for any $n \in N$.
DifficultAP EAMCET 2020
View SolutionLet $z_k = \cos \left(\frac{2k\pi}{10}\right) + i \sin \left(\frac{2k\pi}{10}\right); k = 1, 2, \ldots, 9$.
List-$I$ List-$II$ $P.$ For each $z_k$ there exists a $z_j$ such that $z_k \cdot z_j = 1$ $1.$ True $Q.$ There exists a $k \in \{1, 2, \ldots, 9\}$ such that $z_1 \cdot z = z_k$ has no solution $z$ in the set of complex numbers. $2.$ False $R.$ $\frac{|1-z_1||1-z_2| \ldots |1-z_9|}{10}$ equals $3.$ $1$ $S.$ $1 - \sum_{k=1}^9 \cos \left(\frac{2k\pi}{10}\right)$ equals $4.$ $2$
Codes: $P \quad Q \quad R \quad S$
| List-$I$ | List-$II$ |
| $P.$ For each $z_k$ there exists a $z_j$ such that $z_k \cdot z_j = 1$ | $1.$ True |
| $Q.$ There exists a $k \in \{1, 2, \ldots, 9\}$ such that $z_1 \cdot z = z_k$ has no solution $z$ in the set of complex numbers. | $2.$ False |
| $R.$ $\frac{|1-z_1||1-z_2| \ldots |1-z_9|}{10}$ equals | $3.$ $1$ |
| $S.$ $1 - \sum_{k=1}^9 \cos \left(\frac{2k\pi}{10}\right)$ equals | $4.$ $2$ |
Codes: $P \quad Q \quad R \quad S$
If $\omega$ is a complex cube root of unity,then the value of $\omega^{99} + \omega^{100} + \omega^{101}$ is
Medium
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