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Question

If arg(zωzω2)=0 then prove that Re(z)= 12 ( ω and ω2 are non-real cube roots of unity).

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Solution

Remember that if arg (number)=0
then number must be a real number
i.e Arg (zwzw2)=0zwzw2=m(R).....(1)
also w=1+3i2w2=13i2....(2)
from (1)
zw=mzw2m
z(1m)=ww2m
taking real parts
Re(z)(1m)=Re wRe w2m
from (2)
Re(z)(1m)=12(12)m
Re(z)(1m)=12(1m)
or, Re(z)=12

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