Advertisements
Advertisements
प्रश्न
If ω is a complex cube root of unity, show that (3 + 3ω + 5ω2)6 − (2 + 6ω + 2ω2)3 = 0
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.
(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
संबंधित प्रश्न
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"omega + "c"omega^2))/("c" + "a"omega + "b"omega^2) = omega^2`.
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 ω is a complex cube root of unity, find the value of (1 - ω - ω2)3 + (1 - ω + 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: (a + b) + (aω + bω2) + (aω2 + bω) = 0.
Find the value of ω18
Find the value of ω21
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, find the value of ω2 + ω3 + ω4
If ω is a complex cube root of unity, find the value of (1 + ω)(1 + ω2)(1 + ω4)(1 + ω8)
If , where α and β are the complex cube-roots of unity, show that xyz = a3 + b3.
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 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
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) ______.
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 w 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+bw+cw^2))/(c+aw+bw^2) =w^2`
If w is a complex cube root of unity, show that `((a + bω + cω^2))/(c + aω + bω^2) = ω^2`
Find the value of `sqrt(-3) xx sqrt(-6)`.
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 + bomega + comega^2))/(c + aomega + bomega^2)=omega^2`
