[" Therefore,"p/q" is a real number."],[" Example "8quad " If "|z^(3)+(1)/(z^(3))|<=2," then "|z+(1)/(z)|" cannot exceed "]

Determine the largest real number k such that |z_(1)z_(2)+Z_(2)z_(3)+z_(3)z_(1)|>=k|z_(1)+z_(2)+z_(3)| for complex numbers z_(1),z_(2),z_(3) with unit absolute value.From the given condition,

For a complex number z, the equation z^(2)+(p+iq)z+ r+" is "=0 has a real root (where p, q, r, s are non - zero real numbers and i^(2)-1 ), then

For a complex number z, the equation z^(2)+(p+iq)z r+" is "=0 has a real root (where p, q, r, s are non - zero real numbers and i^(2) = -1 ), then

If a point p dentes a complex number z=x+iy in the argand plane and if (z+1)/(z+i) is a purely real number , then the locus of p is

z_1,z_2,z_3 are complex number and p,q,r are real numbers such that: p/(|z_2-z_3|)= q/(|z_3-z_1|)= r/(|z_1-z_2|) . Prove that p^2/(z_2-z_3)= q^2/(z_3-z_1)= r^2/(z_1-z_2)=0

a z4 1 -250 i 6 The mumber of solutions of the equation z2 z 0 where z is a complex number, is (a 1 a 4 z In the quadratic equatian x +(p iq)x +3i 0, p ad ade real. If the sum of the saames of the roots is 8 then 3, q a p 3, q --1 t3, q

Let a(a != 0) is a fixed real number and (a-x)/(p x) = (a-y)/(qy) = (a-z)/(r z) . If p, q, r are in A.P., show that (1)/(x), (1)/(y), (1)/(z) are in A.P.

Let z=1+ ai be a complex number,a>0, such that z^(3) is a real number.Then the sum 1+z+z^(2)+...+z^(11) is equal to: