Question

If $z _{ n }= a _{ n }+ ib _{ n }( n =1,2,3, \ldots \ldots . ., 11)$ are the roots of the equation $z^{11}+2 z^{10}+3 z^9+4 z^8+5 z^7+6 z^6+5 z^5+4 z^4+3 z^3+2 z^2+z=0$ then find the value of $\displaystyle\sum_{n=1}^{11}\left|a_n\right|$.

Answer 1

$z^{11}+2 z^{10}+3 z^9+\ldots \ldots \ldots . .+2 z^2+z=0 $
$\Rightarrow z \left( z ^{10}+2 z ^9+3 z ^8+\ldots \ldots \ldots . .+2 z +1\right)=0 $
$\Rightarrow z \left(1+ z + z ^2+ z ^3+ z ^4+ z ^5\right)^2=0 $
$\Rightarrow z\left(\frac{1-z^6}{1-z}\right)^2=0 (z \neq 1) $
$\Rightarrow z\left(1-z^6\right)^2=0$
$\therefore z =0$ and every root of the equation $z ^6-1=0(\operatorname{except} z =1)$ two times repeated are the 11 roots of the given equation.
$z=(1)^{\frac{1}{6}} =\cos \frac{2 m \pi}{6}+i \sin \frac{2 m \pi}{6}, m=1,2,3,4,5 $
$\therefore \displaystyle\sum_{n=1}^{11}\left|a_n\right|=0+2\left(\displaystyle\sum_{m=1}^5\left|\cos \frac{m \pi}{3}\right|\right) $
$=2\left(\left|\cos \frac{\pi}{3}\right|+\left|\cos \frac{2 \pi}{3}\right|+|\cos \pi|+\left|\cos \frac{4 \pi}{3}\right|+\left|\cos \frac{5 \pi}{3}\right|\right)$
$=2\left(\frac{1}{2}+\frac{1}{2}+1+\frac{1}{2}+\frac{1}{2}\right)=6$