Advertisements
Advertisements
Question
If α and β are the complex cube root of unity, show that α2 + β2 + αβ = 0
Advertisements
Solution
α and β are the complex cube roots of unity.
∴ α = `(-1 + "i"sqrt(3))/2` and β = `(-1 - "i"sqrt(3))/2`
∴ α + β = `(-1 + "i"sqrt(3))/2 + (-1 - "i" sqrt(3))/2`
= `(-1 + "i"sqrt(3) - 1 - "i"sqrt(3))/2`
= `(-2)/2`
= – 1
and αβ = `((-1 + "i"sqrt(3))/2)((-1 - "i"sqrt(3))/2)`
= `((-1)^2 - ("i"sqrt(3))^2)/4`
= `(1 - 3"i"^2)/4`
= `(1 + 3)/4`
= 1 ..........[∵ i2 = – 1]
α2 + β2 + aβ = (α + β)2 – αβ
= (– 1)2 – 1
= 1 – 1
= 0
APPEARS IN
RELATED QUESTIONS
If `omega` is a complex cube root of unity, find the value of `(1 + omega)(1 + omega^2)(1 + omega^4)(1 + omega^8)`
If α and β are the complex cube roots of unity, show that α2 + β2 + αβ = 0.
If ω is a complex cube root of unity, then prove the following: (ω2 + ω - 1)3 = – 8
Find the value of ω18
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 `("a" + "b"ω + "c"ω^2)/("c" + "a"ω + "b"ω^2)` = ω2
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, show that (a + b)2 + (aω + bω2)2 + (aω2 + bω)2 = 6ab
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)
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|
If α and β are complex cube roots of unity, prove that (1 − α)(1 − β) (1 − α2)(1 − β2) = 9.
Answer the following:
If ω is a complex cube root of unity, prove that (1 − ω + ω2)6 +(1 + ω − ω2)6 = 128
If (1 + ω2)m = (1 + ω4)m and ω is an imaginary cube root of unity, then least positive integral value of m is ______.
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)` = ______.
Let α be a root of the equation 1 + x2 + x4 = 0. Then the value of α1011 + α2022 – α3033 is equal to ______.
If 1, α1, α2, ...... αn–1 are the roots of unity, then (1 + α1)(1 + α2) ...... (1 + αn–1) is equal to (when n is even) ______.
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, 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.
(ω2 + ω −1)3 = −8
If w is a complex cube root of unity, show that `((a + bω + cω^2))/(c + aω + bω^2) = ω^2`
If ω 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, then prove the following.
(w2 + w - 1)3 = - 8
If w is a complex cube root of unity, show that `((a + bomega + comega^2))/(c + aomega + bomega^2) = w^2`
If ω is a complex cube root of unity, show that `((a + b\omega + c\omega^2))/(c + a\omega + b\omega^2) = \omega^2`
If ω is a complex cube-root of unity, then prove the following.
(ω2 + ω − 1)3 = −8
