Let $p(x) = a_0 + a_1 x + \ldots + a_n x^n$ be a non-zero polynomial with integer coefficients. If $p(\sqrt{2} + \sqrt{3} + \sqrt{6}) = 0$,then the smallest possible value of $n$ is

$\frac{{{{( - 1 + i\sqrt 3 )}^{15}}}}{{{{(1 - i)}^{20}}}} + \frac{{{{( - 1 - i\sqrt 3 )}^{15}}}}{{{{(1 + i)}^{20}}}} = \dots$

The least value of $|z|$ where $z$ is a complex number satisfying the inequality $\exp \left(\frac{(|z|+3)(|z|-1)}{|z|+1} \log _{ e } 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{7}+9 i |$,where $i=\sqrt{-1}$,is equal to:

If $(\cos \theta + i\sin \theta )(\cos 2\theta + i\sin 2\theta ) \dots (\cos n\theta + i\sin n\theta ) = 1$,then the value of $\theta$ is

If $z(1 + a) = b + ic$ and $a^2 + b^2 + c^2 = 1$,then $\frac{1 + iz}{1 - iz} = $

If $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely real,then $\cos ^3 \theta+\sin ^2 \theta+\cos \theta+1=$