Question

Let f(z)=z−1/z+1

. The questions that follow will all revolve around this complex-valued function.

1.

 The value of 25f(2+4i) is equal to a+ib where a,b∈R, and i^2=−1. What is the value of 10a+b?

Answer 1

$$f(z)=z-\frac{1}{z}+1$$ $$f(2+4i)=2+4i-\frac{1}{2+4i}+1=$$

$$=2+4i-\frac{2-4i}{(2+4i)(2-4i)}+1=$$

$$=2+4i-\frac{2-4i} {4+16} +1=$$

$$=2+4i-\frac{2-4i}{20}+1=$$

$$=2+4i-0.1+0.2i+1=$$

$$=2.9+4.2i$$

$$25f(2+4i)=25(2.9+4.2i)=72.5+105i$$

$a=72.5, b=105$

$$10a+b=10\cdot{72.5}+105=725+105=830$$

Answer: 830