Advertisements
Advertisements
प्रश्न
If ω is a complex cube root of unity, show that (1 + ω)3 − (1 + ω2)3 = 0
Advertisements
उत्तर
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.
(1 + ω)3 – (1 + ω2)3
= (– ω2)3 – (– ω)3
= – ω6 – (– ω3)
= – (ω3)2 + ω3
= – (1)2 + 1
= – 1 + 1
= 0
APPEARS IN
संबंधित प्रश्न
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 `((a + bomega + comega^2))/("c" + aomega + bomega^2) = omega^2`.
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, 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 (2 + ω + ω2)3 − (1 − 3ω + ω2)3 = 65
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 (1 − ω − ω2)3 + (1 − ω + ω2)3
If α and β are the complex cube root of unity, show that α2 + β2 + αβ = 0
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 + 8| = |z – 4|
If ω is the cube root of unity then find the value of `((-1 + "i"sqrt(3))/2)^18 + ((-1 - "i"sqrt(3))/2)^18`
Which of the following is the third root of `(1 + i)/sqrt2`?
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 ______.
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:
(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
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, 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
