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Question

If $$\omega$$ is an imaginary cube root of unity then find: $$(1-\omega+\omega^2)(1-\omega^2+\omega^4)(1-\omega^4+\omega^8)$$ ..... to 2n factors $$=$$


A
2n+1
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B
22n
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C
22n+1
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D
2n
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Solution

The correct option is B $$2^{2n}$$
$$(1-w+{ w }^{ 2 })(1-{ w }^{ 2 }+{ w }^{ 4 })(1-{ w }^{ 4 }+{ w }^{ 8 })...2n\quad terms\\ =(1-w+{ w }^{ 2 })(1-{ w }^{ 2 }+w)(1-w+{ w }^{ 2 })...\\ =(0-2w)(0-2{ w }^{ 2 })(0-2w)(0-2{ w }^{ 2 })...2n\quad terms\\ ={ (-2w) }^{ n }{ (-2{ w }^{ 2 }) }^{ n }={ (2w.2{ w }^{ 2 }) }^{ n }={ 4 }^{ n }={ 2 }^{ 2n }\\ $$
Hence, (b) is correct.

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