Question

If $z_n=a_n+i{b}_n(n=1,2,3,\dots \dots ..,11)$ are the roots of the equation $z^{11}+2z^{10}+3z^9+4z^8+5z^7+6z^6+5z^5+4z^4+3z^3+2z^2+z=0$ then find the value of $\sum _{n=1}^{11}\left|a_n\right|$.

Answer 1

Answer: 6

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