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 $z ^3, z ^5, z ^7, \ldots \ldots, z ^{2017}$ in that order. On Sunday, he being at 1 and delivers news paper to houses located at $z ^2, z ^4, z ^6, \ldots \ldots . ., z ^{2016}$ 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. $\operatorname{Re}\left(z^2\right)=\frac{1008}{1009}$
• D. $\operatorname{Re}\left(z^2\right)=\frac{1007}{1008}$

Answer 1

Answer: (A), (D)

$\Theta \displaystyle\sum_{ K =1}^{1008}\left| z ^{2 K +1}- z ^{2 K -1}\right|=\displaystyle\sum_{ K =1}^{1008}\left| z ^{2 K }- z ^{2 K -2}\right|=\sqrt{2016}$
$\Rightarrow | z |=1 \& \displaystyle\sum_{ K = I }^{1008}\left| z ^{2 K }- z ^{2 K -2}\right|=\sqrt{2016}$
$\Rightarrow \left|z^2-1\right| \displaystyle\sum_{K=1}^{1008}|z|^{2 K-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}} $
$\Theta \operatorname{Re}\left(z^2\right)=\cos 2 \theta=1-2 \sin ^2 \theta=1-\frac{1}{1008}=\frac{1007}{1008}$