Advertisements
Advertisements
प्रश्न
If α and β are the complex cube root of unity, show that α4 + β4 + α−1β−1 = 0
Advertisements
उत्तर
α and β are the complex cube roots of unity
∴ α = `(-1 + "i"sqrt(3))/2` and β = `(-1 - "i"sqrt(3))/2`
∴ αβ = `((-1 + "i"sqrt(3))/2)((-1 - "i"sqrt(3))/2)`
= `((-1)^2 - ("i"sqrt(3))^2)/4`
= `(1 - (-1)(3))/4`
= `(1 + 3)/4`
∴ αβ = 1
Also, α + β = `(-1 + "i"sqrt(3))/2 + (-1 - "i"sqrt(3))/2`
= `(-1 + "i"sqrt(3) - 1 - "i"sqrt(3))/2`
= `(-2)/2`
∴ α + β = – 1
α4 + β4 + α−1β−1
= `α^4 + β^4 + 2α^2β^2 - 2α^2β^2 + 1/"αβ"` ...[Adding and subtracting 2α2β2]
= `(α^2 + β^2)^2 - 2α^2β^2 + 1/"αβ"`
= `[(α + β)^2 - 2αβ]^2 - 2(αβ)^2 + 1/"αβ"`
= `[(-1)^2 - 2(1)]^2 - 2(1)^2 + 1/1`
= (1 – 2)2 – 2 + 1
= (– 1)2 – 1
= 1 – 1
= 0
APPEARS IN
संबंधित प्रश्न
If ω is a complex cube root of unity, show that (2 + ω + ω2)3 - (1 - 3ω + ω2)3 = 65
If ω is a complex cube root of unity, find the value of `omega + 1/omega`
If ω is a complex cube root of unity, find the value of (1 - ω - ω2)3 + (1 - ω + ω2)3
If ω is a complex cube root of unity, then prove the following: (ω2 + ω - 1)3 = – 8
Find the value of ω18
Find the value of ω–105
If ω is a complex cube root of unity, show that (1 + ω − ω2)6 = 64
If ω is a complex cube root of unity, show that (2 + ω + ω2)3 − (1 − 3ω + ω2)3 = 65
If ω is a complex cube root of unity, show that (a − b) (a − bω) (a − bω2) = a3 − b3
If ω is a complex cube root of unity, find the value of `ω + 1/ω`
If ω is a complex cube root of unity, find the value of (1 − ω − ω2)3 + (1 − ω + ω2)3
If ω is a complex cube root of unity, find the value of (1 + ω)(1 + ω2)(1 + ω4)(1 + ω8)
If α and β are the complex cube root of unity, show that α2 + β2 + αβ = 0
Find the equation in cartesian coordinates of the locus of z if |z| = 10
Find the equation in cartesian coordinates of the locus of z if |z – 3| = 2
Find the equation in cartesian coordinates of the locus of z if |z − 5 + 6i| = 5
Find the equation in cartesian coordinates of the locus of z if |z + 8| = |z – 4|
Find the equation in cartesian coordinates of the locus of z if |z – 2 – 2i| = |z + 2 + 2i|
Select the correct answer from the given alternatives:
If ω is a complex cube root of unity, then the value of ω99+ ω100 + ω101 is :
If α and β are complex cube roots of unity, prove that (1 − α)(1 − β) (1 − α2)(1 − β2) = 9.
If α, β, γ are the cube roots of p (p < 0), then for any x, y and z, `(xalpha + "y"beta + "z"gamma)/(xbeta + "y"gamma + "z"alpha)` = ______.
If w is a complex cube root of unity, show that `((a+bw+cw^2))/(c+aw+bw^2) = w^2`
If w is a complex cube root of unity, show that
`((a + bw + cw^2)) /( c + aw + bw^2 )= w^2`
If w is a complex cube-root of unity, then prove the following:
(ω2 + ω − 1)3 = −8
If w is a complex cube root of unity, show that `((a + bw + cw^2))/(c+aw+bw^2) = w^2`
If w is a complex cube root of unity, show that `((a+bw+cw^2))/(c+aw+bw^2) =w^2`
If ω is a complex cube-root of unity, then prove the following:
(a + b) + (aω + bω2) + (aω2 + bω) = 0
If ω is a complex cube-root of unity, then prove the following:
(ω2 + ω − 1)3 = −8
Find the value of `sqrt(-3)xx sqrt (-6)`
If w is a complex cube root of unity, show that `((a + bw + cw^2))/(c + aw + bw^2) = w^2`
