If ω is complex cube root of unity, then 1+5ω+9ω2ω2+5+9ω+2+3ω+5ω25+2ω+3ω2 is equal to

1+5ω+9ω2ω2+5+9ω=ωω2+5+9ωω2+5+9ω=ω2+3ω+5ω25+2ω+3ω2=ω22ω+3ω2+55+2ω+3ω2=ω2∴ ω+ω2=−1.

If ω is complex cube root of unity, then 1+5ω+9ω2ω2+5+9ω+2+3ω+5ω25+2ω+3ω2 is equal to

1+5ω+9ω2ω2+5+9ω=ωω2+5+9ωω2+5+9ω=ω2+3ω+5ω25+2ω+3ω2=ω22ω+3ω2+55+2ω+3ω2=ω2∴ ω+ω2=−1.

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If $ω$ is complex cube root of unity, then $\frac{1 + 5 ω + 9 ω^{2}}{ω^{2} + 5 + 9 ω} + \frac{2 + 3 ω + 5 ω^{2}}{5 + 2 ω + 3 ω^{2}}$ is equal to

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-1

$2 ω$

$- 2 ω$

answer is A.

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Detailed Solution

$\begin{array}{l} \frac{1 + 5 ω + 9 ω^{2}}{ω^{2} + 5 + 9 ω} = \frac{ω (ω^{2} + 5 + 9 ω)}{ω^{2} + 5 + 9 ω} = ω \\ \frac{2 + 3 ω + 5 ω^{2}}{5 + 2 ω + 3 ω^{2}} = \frac{ω^{2} (2 ω + 3 ω^{2} + 5)}{(5 + 2 ω + 3 ω^{2})} = ω^{2} \\ ∴ ω + ω^{2} = - 1 . \end{array}$