If` 1, z_(1), z_(2), z_(3), …, z_(n-1)` are the nth roots of unity, then prove that `(1-z_(1)) (1-z_(2)) …(1-z_(n-1))=n`

If z_(1) and z_(2) are non -zero complex numbers, then show that (z_(1) z_(2))^(-1)=z_(1)^(-1) z_(2)^(-1)

If the complex numbers z _(1) , z _(2) and z _(3) denote the vertices of an isosceles triangle, right angled at z _(1), then (z _(1) - z _(2)) ^(2) + (z _(1) - z _(3)) ^(2) is equal to A)0 B) (z _(2) + z _(3)) ^(2) C)2 D)3

If arg (bar(z)_(1))= "arg" (z_(2)), (z ne 0) then

If z_(1), z_(2),……..,z_(n) are complex numbers such that |z_(1)| = |z_(2)| = …….. = |z_(n)| = 1 , then |z_(1) + z_(2) +……..+ z_(n)| is equal to a) |z_(1)z_(2)z_(3)…..z_(n)| b) |z_(1)|+|z_(2)|+…….+|z_(n)| c) |(1)/(z_(1)) + (1)/(z_(2)) + ……….+ (1)/(z_(n))| d)n

If |z_(1) + z_(2)|=|z_(1)-z_(2)| , then the difference in the amplitudes of z_(1) and z_(2) is

If z_(1) and z_(2) are two complex numbers satisfying the equation |(z_(1) + iz_(2))/(z_(1)-iz_(2))|=1 , then (z_(1))/(z_(2)) is

If z_(1) and z_(2) are complex numbers such that z_(1) ne z_(2) and |z_(1)|= |z_(2)| . If z_(1) has positive real part and z_(2) has negative imaginary part, then ((z_(1) +z_(2)))/((z_(1)-z_(2))) may be

If |z_(1)|= 2, |z_(2)|= 3 , then |z_(1) + z_(2) +5+12i| is less than or equal to