Question

Define f(z)=z^2+bz−1=0 and g(z)=z^2+z+b=0. If there exists α satisfying f(α)=g(α)=0, which of the following cannot be a value of b?

(a) √3i

(b) -√3i

(c) 0

(d) √3i/2

I had been asked this question in quiz.

Asked question is from Functions of a Complex Variable topic in chapter Complex Function Theory of Engineering Mathematics

Answer 1

Right choice is (d) √3i/2

Explanation: α^2+bα−1=0 and α^2+α+b=0 ⇒ (b−1)α−1−b=0 ⇒ α=(b+1)/(b-1)

⇒ (b+1)^2/(b-1)^2+(b+1)/(b-1)+b=0 ⇒ b=√3i, -√3i, 0.