If - are complex cube roots of unity- then the value of a-b-c-a-b-c can be

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 ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XIII

Mathematics

Cube Root of a Complex Number

If α,β are co...

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

β

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

α^{2}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\frac{1}{β}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

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

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

Suggest Corrections

Similar questions

Q. 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

Q. 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

Q. If

α

and

β

are the complex cube roots of unity, then find

α^{2} + β^{2} + α β = 0

Q. If

α, β

are the complex cube roots of unity, then

α^{4} + β^{4} + α^{- 1} β^{- 1} =

Q. if

α a n d β

are complex cube root of unity then find the value of

α^{2} + β^{2} + α β

Join BYJU'S Learning Program

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

The correct option is 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