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Question

The value of (a+bω+cω2)3+(a+bω2+cω)3 is/are
[Here ω is a complex cube root of unity]

A
(2abc)(2bac)(2cab)
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B
9abc ,if a+b+c=0
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C
27abc ,if a+b+c=0
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D
(2a+b+c)(2bac)(2cab)
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Solution

The correct options are
A (2abc)(2bac)(2cab)
C 27abc ,if a+b+c=0
Let a+bω+cω2=x and a+bω2+cω=y

(a+bω+cω2)3+(a+bω2+cω)3=x3+y3=(x+y)(x+ωy)(x+ω2y)

Now,
x+y=(a+bω+cω2)+(a+bω2+cω)
=2a+b(ω+ω2)+c(ω+ω2)
=(2abc)(1)

x+ωy=(a+bω+cω2)+ω(a+bω2+cω)
=(1+ω)a+(1+ω)b+2ω2c
=ω2(2cab)(2)

x+ω2y=(a+bω+cω2)+ω2(a+bω2+cω)
=ω(2bac)(3)

Hence from (1),(2) and (3)
x3+y3=(2abc)(2cab)(2bac)(4)

If
a+b+c=0b+c=a,c+a=b,a+b=c
Putting these values in the equation (4), we get
x3+y3=(3a)(3b)(3c)=27 abc

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