Advertisements
Advertisements
प्रश्न
If ω is a complex cube root of unity, show that `((a + bomega + comega^2))/("c" + aomega + bomega^2) = omega^2`.
Advertisements
उत्तर
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 + bomega + comega^2)/(c + aomega + bomega^2)`
= `(aomega^3 + bomega^4 + comega^2)/(c + aomega + bomega^2) ...[∵ omega^3 = 1, omega^4 = omega]`
= `(omega^2(c + aomega + bomega^2))/(c + aomega + bomega^2)`
= ω2
= R.H.S.
APPEARS IN
संबंधित प्रश्न
If ω is a complex cube root of unity, find the value of ω2 + ω3 + ω4.
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, 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) + (aω + bω2) + (aω2 + bω) = 0
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 ω2 + ω3 + ω4
If ω is a complex cube root of unity, find the value of (1 + ω2)3
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 ω(≠1) is a cube root of unity and (1 + ω)7 = A + Bω, then A and B are respectively the numbers ______.
Answer the following:
If ω is a complex cube root of unity, prove that (1 − ω + ω2)6 +(1 + ω − ω2)6 = 128
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, then prove the following.
(ω2 + ω − 1)3 = −8
