Question

Number of values of z satisfying both the equations$1+z+z^2+....+z^{17}=0and1+z+z^2+....+z^{13}=0$,
where $z$ is a complex number, is

• A. $0$
• B. $1$
• C. $2$
• D. $13$

Answer 1

The correct option is B $1$

$1+z+z^2+....+z^{17}=0$ and $1+z+z^2+....+z^{13}=0$
Simultaneously solving the equations , we get
$z^{14}+z^{15}+z^{16}+z^{17}=0$
$z^{14}(1+z+z^2.+z^3)=0$
$z^{14}=0$ or $(1+z+z^2+z^3)=0$
But $z=0$, does not satisfy any of the given equations.
Now taking,
$(1+z+z^2.+z^3)=0$
$(1+z)(z^2+1)=0$
$(z^2=-1)$ is not a solution to any equation.
Therefore, the only solution of the given equations is $z=-1$
Hence, the correct option is 'B'.