Question

Let `z_1, z_2,z_3` be complex numbers (not all real) such that `|z_1|=|z_2|=|z_3|=1 and 2(z_1+z_2+z_3)-3z_1 z_2 z_3` is real. Then, `Max (arg(z_1), arg(z_2), arg(z_3))` (Given that argument of `z_1, z_2, z_3` is possitive ) has minimum value as `(kpi)/6` where `(k+2)` is

Answer 1

Correct Answer - 3
Assume `alpha_(i) lt (pi)/6`
`2(sinalpha_(1)+sin alpha_(2)+sin alpha_(3))=3sin(alpha_(1)+alpha_(2)+alpha_(3))`
Also, `(sin alpha_(1)+sin alpha_(2)+sin alpha_(3))/3 le sin ((alpha_(1)+alpha_(2)+alpha_(3))/3)[alpha_(i) epsilon (0, (pi)/2)]`
`implies sin 3t le 2 sin t`
`implies 4 sin^(3)t- sin t ge0`
`impliessin^(2) t ge 1/4`, which contradicts assumption