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Question

If ω is a cube root of unity, and the value of 
(1ω)(1ω2)(1ω4)(1ω8)=α, and
a+bω+cω2b+cω+aω2+a+bω+cω2c+aω+bω2=β then 
  1. α+β=8
  2.  αβ=9
  3.  αβ=9
  4.  α+β=9


Solution

The correct options are
A α+β=8
C  αβ=9
(1ω)(1ω2)(1ω4)(1ω8)
=(1ω)(1ω2)(1ω)(1ω2)  [ω3=1]
=(1ω2)2(1ω)2
=(1+ω42w2)(1+ω22ω)  [ω3=1]
=(1+ω2ω2)(ω2ω)   [1+ω+ω2=1]
=(3ω2)(3ω)
=9=α(1)

Now,
a+bω+cω2b+cω+aω2+a+bω+cω2c+aω+bω2
=aω3+bω+cω2b+cω+aω2+a+bω+cω2cω3+aω+bω2  [ω3=1]=ω+ω2   [1+ω+ω2=1]=1=β(2)

From (1) and (2)
α+β=8
and, α×β=9

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