Question

Consider a function f(z) of degree two, having real coefficients. If z1 and z2 satisfying f(z1)=f(z2)=0 are such that Re z1=Re z2=0 and if z3 satisfies f(f(z3))=0, then select the correct statement.

(a) Re z3=0

(b) Im z3=0

(c) Re z3×Imz3≠0

(d) Re z3=0 and Im z3=0

This question was posed to me in a national level competition.

This question is from Functions of a Complex Variable topic in section Complex Function Theory of Engineering Mathematics

Answer 1

Correct option is (c) Re z3×Imz3≠0

To elaborate: f(z)=az^2+b, with a, b of same sign ⇒ f(f(z))=a(az^2+b)^2+b

If z∈R or iz∈R ⇒ z^2∈R ⇒ f(z)∈R ⇒ f(f(z))≠0 ⇒ Hence real or purely imaginary number cannot satisfy f(f(z))=0.