Express the following expression in the form $A + iB$: $(\cos 2\theta + i\sin 2\theta )^{ - 5} (\cos 3\theta - i\sin 3\theta )^6 (\sin \theta - i\cos \theta )^3$.

If $1, \omega, \omega^2$ are the cube roots of unity,$k$ is a positive integer and $(1-\omega+\omega^2)^{3k} + (1-\omega^2+\omega)^{3k} = (1-\omega+\omega^2)^{3k+1} + (1+\omega-\omega^2)^{3k+1}$,then $k=$

If the complex number $a$ is such that $|a|=1$ and $\arg (a)=\theta$,then the roots of the equation $\left(\frac{1+i z}{1-i z}\right)^4=a$ are $z=$

If $\omega (\neq 1)$ is a cube root of unity and $(1 + \omega)^7 = A + B\omega$,then $A$ and $B$ are equal to:

If $\alpha_1, \alpha_2, \alpha_3, \ldots, \alpha_n$ are real numbers,$\alpha_1 \neq 0$ and $z = \cos \theta + i \sin \theta$ is a root of the equation $\alpha_1 + \alpha_2 z + \alpha_3 z^2 + \ldots + \alpha_n z^{n-1} + z^n = 0$,then $\alpha_1 \cos n \theta + \alpha_2 \cos (n-1) \theta + \ldots + \alpha_n \cos \theta =$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 27 = 0$,find the quadratic equation whose roots are $\left( \frac{\gamma}{\alpha} \right)^2$ and $\left( \frac{\beta}{\alpha} \right)^2$.