Advertisements
Advertisements
प्रश्न
If , where α and β are the complex cube-roots of unity, show that xyz = a3 + b3.
Advertisements
उत्तर
x = a + b, y = αa + βb and z = aβ + bα
α and β are the complex cube roots of unity.
∴ α = `(-1 + isqrt3)/2` and β = `(-1 - isqrt3)/2`
∴ αβ = `((-1 + isqrt3)/2)((-1 - isqrt3)/2)`
= `((-1)^2 - (isqrt3)^2)/4`
= `(1-(-1)(3))/4` ...[∵ i2 = -1]
= `(1 + 3)/4`
= `4/4`
∴ αβ = 1
Also, α + β = `(-1 + isqrt3)/2 + (-1 - isqrt3)/2`
= `(-1 + isqrt3 -1 - isqrt3)/2`
= `-2/2`
α + β = −1
∴ xyz = (a + b)(αa + βb)(aβ + bα)
= (a + b)(αβa2 + α2ab + β2ab + αβb2)
= (a + b)[1.(a2) + (α2 +β2)ab + 1.(b2)]
= (a + b){a2 + [(α + β)2 − 2αβ]ab + b2}
= (a + b){a2 + [(−1)2 − 2(1)]ab + b2}
= (a + b)[a2 + (1 − 2)ab + b2]
= (a + b)(a2 − ab + b2)
= a3 + b3
APPEARS IN
संबंधित प्रश्न
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 (1 - ω - ω2)3 + (1 - ω + ω2)3
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: (a + b) + (aω + bω2) + (aω2 + bω) = 0.
Find the value of ω–105
If ω is a complex cube root of unity, show that (3 + 3ω + 5ω2)6 − (2 + 6ω + 2ω2)3 = 0
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) + (aω + bω2) + (aω2 + bω) = 0
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 α4 + β4 + α−1β−1 = 0
Find the equation in cartesian coordinates of the locus of z if `|("z" + 3"i")/("z" - 6"i")|` = 1
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 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 (1 + ω2)m = (1 + ω4)m and ω is an imaginary cube root of unity, then least positive integral value of m is ______.
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 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
If w is a complex cube root of unity, show that `((a + bω + cω^2))/(c + aω + bω^2) = ω^2`
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 + 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, 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
