Advertisements
Advertisements
Question
If ω(≠1) is a cube root of unity and (1 + ω)7 = A + Bω, then A and B are respectively the numbers ______.
Options
0, 1
1, 1
1, 0
−1, 1
Advertisements
Solution
If ω(≠1) is a cube root of unity and (1 + ω)7 = A + Bω, then A and B are respectively the numbers 1, 1.
Explanation:
ω is a cube root of unity
⇒ 1 + ω + ω2 = 0
⇒ 1 + ω = – ω2
⇒ (1 + ω)7 = (– ω2)7
= – ω14
= – ω12 × ω2
= – ω2
= 1 + ω
= A + ω.B
A = 1, B = 1
APPEARS IN
RELATED QUESTIONS
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, 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.
Find the value of ω18
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 (2 + ω + ω2)3 − (1 − 3ω + ω2)3 = 65
If ω is a complex cube root of unity, show that `("a" + "b"ω + "c"ω^2)/("c" + "a"ω + "b"ω^2)` = ω2
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 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" + 3"i")/("z" - 6"i")|` = 1
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)` = ______.
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 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) ______.
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 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.
(w2 + w - 1)3 = - 8
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, then prove the following.
(ω2 + ω − 1)3 = −8
