Advertisements
Advertisements
Question
If ω is a complex cube root of unity, show that (3 + 3ω + 5ω2)6 − (2 + 6ω + 2ω2)3 = 0
Solution
ω is a complex cube root of unity.
∴ ω3 = 1 and 1 + ω + ω2 = 0.
∴ ω + ω2 = – 1, 1 + ω = – ω2 and 1 + ω2 = – ω.
(3 + 3ω + 5ω2)6 − (2 + 6ω + 2ω2)3
= [3(1 + ω) + 5ω2]6 – [2(1 + ω2) + 6ω]3
= [3(– ω2) + 5ω2]6 - [2(– ω) + 6ω)3
= (2ω2)6 – (4ω)3
= 64ω12 – 64ω3
= 64(ω3)4 – 64ω3
= 64(1)4 – 64(1)
= 0
APPEARS IN
RELATED QUESTIONS
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 + ω2)3
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 x = a + b, y = αa + βb and z = aβ + bα, where α and β are the complex cube roots of unity, show that xyz = a3 + b3.
If ω is a complex cube root of unity, then prove the following: (ω2 + ω - 1)3 = – 8
Find the value of ω–105
If ω is a complex cube root of unity, show that (2 − ω)(2 − ω2) = 7
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 (1 + ω)3 − (1 + ω2)3 = 0
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 `ω + 1/ω`
If ω is a complex cube root of unity, find the value of ω2 + ω3 + ω4
If ω is a complex cube root of unity, find the value of (1 − ω − ω2)3 + (1 − ω + ω2)3
If , where α and β are the complex cube-roots of unity, show that xyz = a3 + b3.
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" + 3"i")/("z" - 6"i")|` = 1
Answer the following:
If ω is a complex cube root of unity, prove that (1 − ω + ω2)6 +(1 + ω − ω2)6 = 128
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 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) ______.
The value of the expression 1.(2 – ω) + (2 – ω2) + 2.(3 – ω)(3 – ω2) + ....... + (n – 1)(n – ω)(n – ω2), where ω is an imaginary cube root of unity is ______.
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
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 ω is a complex cube-root of unity, then prove the following:
(ω2 + ω −1)3 = −8
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+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, show that `((a + b\omega + c\omega^2))/(c + a\omega + b\omega^2) = \omega^2`
