If $i z^3+z^2-z+i=0$ (where $z$ is a complex number),then the value of $|z|$ is

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

Let the product of $\omega_1=(8+i) \sin \theta+(7+4 i) \cos \theta$ and $\omega_2=(1+8 i ) \sin \theta+(4+7 i ) \cos \theta$ be $\alpha+ i \beta$,where $i =\sqrt{-1}$. Let $p$ and $q$ be the maximum and the minimum values of $\alpha+\beta$ respectively. Then the value of $p+q$ is equal to

Let the set of all values of $k \in R$ such that the equation $z(\bar{z} + 2 + i) + k(2 + 3i) = 0, z \in C$, has at least one solution, be the interval $[\alpha, \beta]$. Then $9(\alpha + \beta)$ is equal to:

The $A + iB$ form of $\frac{(\cos x + i\sin x)(\cos y + i\sin y)}{(\cot u + i)(1 + i\tan v)}$ is

If ${e^{i\theta }} = \cos \theta + i\sin \theta $,then in $\Delta ABC$,the value of ${e^{iA}} \cdot {e^{iB}} \cdot {e^{iC}}$ is