If $x = -2 + \sqrt{-3}$,then the value of $2x^4 + 5x^3 + 7x^2 - x + 38$ is equal to

Let $z \in \mathbb{C}$ and $i=\sqrt{-1}$. If $a, b, c \in (0,1)$ are such that $a^2+b^2+c^2=1$ and $b+ic=(1+a)z$,then $\frac{1+iz}{1-iz}=$

If $z_1, z_2, z_3, z_4$ are the roots of the equation $z^4 + z^3 + z^2 + z + 1 = 0$,then $\prod_{i=1}^{4} (z_i + 2)$ is equal to:

If $f(x)$ is a polynomial of degree $n$ with rational coefficients and $1+2i, 2-\sqrt{3}$ and $5$ are three roots of $f(x)=0$,then the least value of $n$ is

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 $e^{it} = \cos t + i \sin t$ and $e^{-it} = \cos t - i \sin t$, then $\cosh(x + iy) - \cosh(x - iy) =$