Question

Let z is a complex number satisfying the equation $Z^6+Z^3+1=0$. If this equation has a root $r{e}^{i\theta }$ with ${90}^o<\theta <{180}^o$ then the value of ${}^{\prime }thet{a}^{\prime }$ is

• A. ${100}^o$
• B. ${110}^o$
• C. ${160}^o$
• D. ${170}^o$

Answer 1

The correct option is C ${160}^o$

Let $y=z^3$

Then Get $y^2+y+1=0$

it is cube Root of unity

Represented as :

$w=e\frac{2ni}{3}$ & $w^2=e\frac{4ni}{3},$

$y=e\frac{2kni}{3},k\ne 0,$

Take $z=e\frac{2k\pi i}{9}$

$z=e\frac{2\pi i}{9},e\frac{4\pi }{9},e\frac{8\pi }{9},e\frac{10\pi }{9}e\frac{14\pi }{9},e\frac{16\pi }{9}$

for $90<\theta <180$

$\theta =\frac{8\times 180}{9}=160$