1.

If `|z_(1)| = sqrt(2), |z_(2)| = sqrt(3) and |z_(1) + z_(2)| = sqrt((5-2sqrt(3)))` then arg `((z_(1))/(z_(2)))` (not neccessarily principal)A. `(3pi)/(4)`B. `(2pi)/(3)`C. `(5pi)/(4)`D. `(5)/(2)`

Answer» Correct Answer - A::C
`|z_(1)+z_(2)|^(2) = |z_(1)|^(2) + |z_(2)|^(2)+ z_(1)barz_(2)+ barz_(1)z_(2)`
`rArr 5-2sqrt(3) = 2+ 3 + 2|z_(1)||z_(2)|cos(theta_(1)-theta_(2))`,
where `arg(z_(1))= theta_(1) and arg(z_(2))= theta_(2)`
`rArr -2sqrt(3) = 2 + 3+ 2|z_(1)||z_(2)|cos(theta_(1)-theta_(2))`
`rArr cos(theta_(1)-theta_(2))= -(1)/(sqrt(2))`
`rArr theta_(1) - theta_(2) = (3pi)/(4),(5pi)/(4)`
`rArr arg(z_(1))-arg(z_(2)) = arg((z_(1))/(z_(2))) = (3pi)/(4),(5pi)/(4)`


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