If $\omega$ is a complex root of the equation $z^3 = 1$,then $\omega + \omega^{\left( \frac{1}{2} + \frac{3}{8} + \frac{9}{32} + \frac{27}{128} + \dots \right)}$ is equal to
- A$-1$
- B$0$
- C$9$
- D$i$
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$\omega$ is a complex cube root of unity. Match the items of List-$I$ to the items of List-$II$.
List-$I$ (Expression) List-$II$ (Value) $A$. $\omega^{1010} + \omega^{2000}$ $I$. $0$ $B$. $(1 + \omega - \omega^2)(1 - \omega + \omega^2)$ $II$. $1$ $C$. $(2 + \omega^2 + \omega^4)^5$ $III$. $-1$ $D$. $(3 + 5\omega + 3\omega^2)^3$ $IV$. $4$ $V$. $8$
The correct match is:
| List-$I$ (Expression) | List-$II$ (Value) |
|---|---|
| $A$. $\omega^{1010} + \omega^{2000}$ | $I$. $0$ |
| $B$. $(1 + \omega - \omega^2)(1 - \omega + \omega^2)$ | $II$. $1$ |
| $C$. $(2 + \omega^2 + \omega^4)^5$ | $III$. $-1$ |
| $D$. $(3 + 5\omega + 3\omega^2)^3$ | $IV$. $4$ |
| $V$. $8$ |
The correct match is:
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