Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

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

Complex Numbers and Quadratic Equations

Solution:

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 $| z |=1 z ^{2012}= z \overline{ z }=| z |^2=1 \Rightarrow 2012$ values of $z \Rightarrow $ Total $=2013$