Question

Number of ordered pairs(s) (a, b) of real numbers such that $(a+ib{)}^{2011}=a-ib$ holds good, is

• A. 2011
• B. 2012
• C. 2013
• D. 1

Answer 1

Answer: (C)

Let $z=a+ib\Rightarrow \overline{z}=a-ib$
hence we have $z^{2008}=\bar{z}\left[z^{2011}=\bar{z}\right]$
$\therefore |z{|}^{2011}=|\bar{z}|=|z||z|\left[|z{|}^{2010}-1\right]=0$
$|z|=0$ or $|z|=1;$ if $|z|=0\Rightarrow z=0\Rightarrow (0,0)$
if $|z|=1z^{2012}=z\overline{z}=|z{|}^2=1\Rightarrow 2012$ values of $z\Rightarrow$ Total $=2013$