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

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Solution

The correct option is B $0$
We have $(k - w) (k - w)^{2} = k^{2} - k (w + w^{2}) + w^{3}$

$= k^{2} - k (- 1) + 1 = k^{2} + k + 1$

$\Rightarrow \sum_{k = 1} (k - w) (k - w)^{2} = \sum_{k = 1}^{10} (k^{2} + k + 1)$

$= \frac{10 \times 11 \times 21}{6} + \frac{10 \times 11}{2} + 10 = 385 + 55 + 10 = 450$

Thus, $c o s [{\sum_{k = 1}^{10} (k - w) (k - w)^{2}} \frac{π}{900}] = c o s (\frac{450}{900} π) = 0.$

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If ω is a complex cube root of unity, then value of expression cos[{(1−w)(1−w)2+....+(10−w)(10−w)2}π900]

The correct option is B 0We have(k−w)(k−w)2=k2−k(w+w2)+w3 =k2−k(−1)+1=k2+k+1 ⇒∑k=1(k−w)(k−w)2=∑10k=1(k2+k+1) =10×11×216+10×112+10=385+55+10=450 Thus, cos[{∑10k=1(k−w)(k−w)2}π900]=cos(450900π)=0.

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