`arg(z^n)=n.argz` where `n in ZZ`

Fill in the blanks. arg (z) + arg bar(z) " where " , (bar(z) ne 0) is …..

If arg(z-1)=arg(z+2i), then find (x-1):y, where z=x+iy

If arg(z-1)=arg(z+3i), then find (x-1):y, where z=x+iy

Prove that the general solution of cos theta=cos alpha is given by :theta=2n pi+-alpha where n in Z.

If z_(1),z_(2),z_(3),…,z_(n-1) are the roots of the equation z^(n-1)+z^(n-2)+z^(n-3)+…+z+1=0 , where n in N, n gt 2 and omega is the cube root of unity, then

If z_1,z_2,z_3,………..z_(n-1) are the roots of the equation 1+z+z^2+…….+z^(n-1)=0, where n epsilon N, ngt2 then (A) z_1,z_2, …z_(n-1) are terms of a G.P. (B) z_1,z_2,……,z_(n-1) are terms of an A.P. (C) |z_1|=|z_2|=|z_3|=.|z_(n-1)|!=1 (D) none of these

If the equation |z(z+1)^(8)=z^(8)|z+1| where z in C and z(z+1)!=0 has distinct roots z_(1),z_(2),z_(3),...,z_(n).( where n in N) then which of the following is/are true?

If f(z) = a_0z^n+a_1z^(n-1)+a_2z^(n-2)+...+a_n where z is a complex number and a_1 , a_2 , etc , are real , prove that barf(z)=f(barz) .

If z_(1),z_(2).........z_(n)=z, then arg z_(1)+arg z_(2)+......+argz_(n) and arg z differ by a

If arg(Z_(1)Z_(2))=arg Z_(1)+arg Z_(2)+n pi,n in Z and ifprinciple argument of Z in[-pi,pi) then Z_(1)=-2,Z_(2)=-3 then n=