Question

If |z|=1 and arg $\left(z\right)=\frac{\pi }{4}$ then which of the following is /are true?

• A. Modulus of $(z^5-iz)$ is $\sqrt{2}$
• B. Modulus of $(z^4-iz)$ is 1
  
• C. Principal argument of $(z^5-iz)$ is $\frac{\pi }{4}$
  
• D. Principal argument is $(z^5-iz)$ is $\frac{\pi }{2}$

Answer 1

Answer: (A), (D)

The correct options are
A Modulus of $(z^5-iz)$ is $\sqrt{2}$
D Principal argument is $(z^5-iz)$ is $\frac{\pi }{2}$

Let $z=e^{i\frac{\pi }{4}}$

$z^5-iz=e^{i\frac{5\pi }{4}}-i.e^{i\frac{\pi }{4}}$

$=(-1-i)e^{i\frac{\pi }{4}}$

So $|z^5-iz|=\sqrt{2}$

arg $(z^5-iz)=\frac{-3\pi }{4}+\frac{\pi }{4}=-\frac{\pi }{2}$

$z^4-iz=e^{i\pi }-i.e^{i\frac{\pi }{4}}$
So $|z^5-iz|\ne \sqrt{2}$