lf w is a complex cube root of unity, then value of expression $\cos [\{(1 - w) (1 - w^{2}) + . . . . . . . + (10 - w) (10 - w^{2})\} \frac{π}{900}]$

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$\frac{\sqrt{3}}{2}$

answer is B.

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Detailed Solution

$\begin{array}{rcl} \cos ((1 - w) (1 - w^{2}) + . . . . + (10 - w) (10 - w^{2}) \frac{π}{900}) \end{array}$

$\begin{array}{rcl} \cos \sum_{r = 1}^{10} ((r - w) (r - w^{2})) \frac{π}{900} \end{array}$

$\begin{array}{rcl} \cos \sum_{r = 1}^{10} (r^{2} - r (w + w^{2}) + w^{3}) \frac{π}{900} \end{array}$

$\begin{array}{rcl} = & \cos \sum_{r = 1}^{10} (r^{2} + r + 1) \frac{π}{900} \end{array}$

$\begin{array}{rcl} \cos (\frac{10 \times 11 \times 21}{6} + \frac{10 (11)}{2} + 10) \frac{π}{900} \end{array}$

$\begin{array}{rcl} \cos \frac{450 π}{900} & = & \cos \frac{π}{2} = 0 \end{array}$

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