Question

During the weekends, A delivers news paper in the complex plane. On Saturday, he being at z and delivers news paper to houses located at $z^3,z^5,z^7,\dots \dots ,z^{2017}$ in that order. On Sunday, he being at 1 and delivers news paper to houses located at $z^2,z^4,z^6,\dots \dots ..,z^{2016}$ in that order. Mr. A travels in a straight line between two houses. The distance he must travel from his starting point to the last house is $\sqrt{2016}$ on each of the two days.

• A. $|z|=1$
• B. $|z|=\frac{1}{2}$
• C. $\mathrm{Re}\left(z^2\right)=\frac{1008}{1009}$
• D. $\mathrm{Re}\left(z^2\right)=\frac{1007}{1008}$

Answer 1

Answer: (A), (D)

$\Theta \sum _{K=1}^{1008}\left|z^{2K+1}-z^{2K-1}\right|=\sum _{K=1}^{1008}\left|z^{2K}-z^{2K-2}\right|=\sqrt{2016}$
$\Rightarrow |z|=1\&\sum _{K=I}^{1008}\left|z^{2K}-z^{2K-2}\right|=\sqrt{2016}$
$\Rightarrow \left|z^2-1\right|\sum _{K=1}^{1008}|z{|}^{2K-2}=\sqrt{2016}$
$\Rightarrow 1008\left|z^2-1\right|=\sqrt{2016}$
$\Theta |z|=1$
$\therefore \text{ Let }z=\cos \theta +i\sin \theta$
$\therefore |\cos 2\theta +i\sin 2\theta -1|=\frac{\sqrt{2016}}{1008}$
$\Rightarrow \left|-2{\sin }^2\theta +i\cdot 2\sin \theta \cos \theta \right|=\frac{\sqrt{2}}{\sqrt{1008}}$
$\Rightarrow |\sin \theta |=\frac{1}{\sqrt{2016}}$
$\Theta \mathrm{Re}\left(z^2\right)=\cos 2\theta =1-2{\sin }^2\theta =1-\frac{1}{1008}=\frac{1007}{1008}$