If - - are complex cube roots of unity- then the value of a - b - c - a - b - c can beA- -B- -C- -2D- 1-

Let α=ω β=ω^2 a+bα+cβaα+bβ+c=a+bω+cω^2aω+bω^2+c =1ω(aω+bω^2+caω+bω^2+c) =1ω=ω^2=β=α^2 Similarly if α=ω^2,β=ω a+bα+cβaα+bβ+c=a+bω^2+cωaω^2+bω+c =1ω^2(aω^2+bω+c ...

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Standard XII

Mathematics

Cube Root of a Complex Number

If α, β are c...

Question

If $α, β$ are complex cube roots of unity, then the value of $\frac{a + b α + c β}{a α + b β + c}$ can be

α

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β

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α^{2}

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\frac{1}{β}

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Solution

The correct options are
B $β$
C $α^{2}$
If $α, β$ are complex cube roots of unity, then
$α β = 1, α^{2} = β$ and $α^{3} = 1$

Now,
$\frac{a + b α + c β}{a α + b β + c} = \frac{α^{2} (a + b α + c β)}{α^{2} (a α + b β + c)}$
$= \frac{α^{2} (a + b α + c β)}{(a α^{3} + α (α β) b + c α^{2})} = \frac{α^{2} (a + b α + c β)}{(a + b α + c β)} = α^{2} = β$

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If α,β are complex cube roots of unity, then the value of a+bα+cβaα+bβ+c can be

The correct options are B β C α2If α,β are complex cube roots of unity, then αβ=1,α2=β and α3=1 Now, a+bα+cβaα+bβ+c=α2(a+bα+cβ)α2(aα+bβ+c) =α2(a+bα+cβ)(aα3+α(αβ)b+cα2)=α2(a+bα+cβ)(a+bα+cβ)=α2=β

If $α, β$ are complex cube roots of unity, then the value of $\frac{a + b α + c β}{a α + b β + c}$ can be