$\frac{5+3\mathrm{i}}{\mathrm{i}(2\mathrm{i}+\mathrm{i}^2)+\mathrm{i}}$

Project ID: 
3000000002
SubArea: 
Complex numbers in algebraic and polar form
Question: 

Petr had to find the modulus of the complex number $$\frac{5+3\mathrm{i}}{\mathrm{i}(2\mathrm{i}+\mathrm{i}^2)+\mathrm{i}}.$$

In which step of his solution did Petr make a mistake?

(The step number is above the equality sign.)

$$ \left|\frac{5+3\mathrm{i}}{\mathrm{i}(2\mathrm{i}+\mathrm{i}^2)+\mathrm{i}}\right|\stackrel{(1)}=\left|\frac{5+3\mathrm{i}}{-2}\right|\stackrel{(2)}=\frac{|5+3\mathrm{i}|}{|-2|}\stackrel{(3)}=\frac{\sqrt{5^2+(3\mathrm{i})^2}}{2}\stackrel{(4)}=2$$

Answer 1: 

In step (1). The expression simplifies to $\left|\frac{5+3\mathrm{i}}{2}\right|$.

Answer 2: 

In step (2). It is not true that $\left|\frac{5+3\mathrm{i}}{-2}\right|=\frac{|5+3\mathrm{i}|}{|-2|}$.

Answer 3: 

In step (3). It holds that $|5+3\mathrm{i}|=\sqrt{5^2+3^2}$.

Answer 4: 

In step (4). The expression simplifies to $\frac{\sqrt{34}}{2}$.

Correct Answer: 
Answer 3