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Question:

For a non-zero complex number z, let arg(z) denote the principal argument with −π < arg(z) ≤ π. Then, which of the following statement(s) is (are) FALSE?

arg(−i) = π/4, where i = √−1

For any three given distinct complex numbers z1, z2 and z3, the locus of the point z satisfying the condition arg((z−z1)(z2−z3)(z−z3)(z2−z1)) = π, lies on a straight line

The function f:R→(−π,π], defined by f(t)=arg(1+it) for all t∈R, is continuous at all points of R, where i = √−1

For any two non-zero complex numbers z1 and z2, arg(z1z2)−arg(z1)+arg(z2) is an integer multiple of 2π

Solution:

(A) arg(−i) = −π/2, (B) f(t) = arg(1+it) = π + tan⁻¹(t), t≥0; −π + tan⁻¹(t), t<0. Discontinuous at t=0 (C) arg(z1z2)−arg(z1)+arg(z2) = arg(z1) + arg(z2) −arg(z1) + arg(z2) = 2nπ (D) arg((z−z1)(z2−z3)(z−z3)(z2−z1)) = π ⇒ (z−z1)(z2−z3)(z−z3)(z2−z1) is real. ⇒ z, z1, z2, z3 are concyclic.