Advertisements
Advertisements
Question
If ω is a complex cube root of unity, show that `("a" + "b"ω + "c"ω^2)/("c" + "a"ω + "b"ω^2)` = ω2
Advertisements
Solution
ω is a complex cube root of unity
∴ ω3 = 1 and 1 + ω + ω2 = 0
Also, 1 + ω2 = − ω, 1 + ω = − ω2 and ω + ω2 = − 1
L.H.S. = `("a" + "b"ω + "c"ω^2)/("c" + "a"ω + "b"ω^2)`
= `("a"ω^3 + "b"ω^4 + "c"ω^2)/("c" + "a"ω + "b"ω^2)` ...[∵ ω3 = 1, ∴ ω4 = ω]
= `(ω^2("a"ω + "b"ω^2 + "c"))/("c" + "a"ω + "b"ω^2)`
= `(ω^2("c" + "a"ω + "b"ω^2))/("c" + "a"ω + "b"ω^2)`
= ω2
= R.H.S.
APPEARS IN
RELATED QUESTIONS
If `omega` is a complex cube root of unity, show that `(2 - omega)(2 - omega^2)` = 7
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 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: (a + b) + (aω + bω2) + (aω2 + bω) = 0.
Find the value of ω18
Find the value of ω21
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 (a + b) + (aω + bω2) + (aω2 + bω) = 0
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 `ω + 1/ω`
If ω is a complex cube root of unity, find the value of ω2 + ω3 + ω4
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 – 2 – 2i| = |z + 2 + 2i|
If ω(≠1) is a cube root of unity and (1 + ω)7 = A + Bω, then A and B are respectively the numbers ______.
If (1 + ω2)m = (1 + ω4)m and ω is an imaginary cube root of unity, then least positive integral value of m is ______.
Let z = `(1 - isqrt(3))/2`, i = `sqrt(-1)`. Then the value of `21 + (z + 1/z)^3 + (z^2 + 1/z^2) + (z^3 + 1/z^3)^3 + ...... + (z^21 + 1/z^21)^3` is ______.
If the cube roots of the unity are 1, ω and ω2, then the roots of the equation (x – 1)3 + 8 = 0, are ______.
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
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
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`
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 + bomega + comega^2))/(c + aomega + bomega^2) = w^2`
If ω is a complex cube root of unity, show that `((a + bomega + comega^2))/(c + aomega + bomega^2)=omega^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`
