If $1, \omega, \omega^2$ are the cube roots of unity and $(x+y)(x \omega+y \omega^2)(x \omega^2+y \omega)=f(x, y)$,then $f(2, 3)=$

If $\omega$ is a complex cube root of unity,then for a positive integral value of $n$,the product $\omega \cdot \omega^2 \cdot \omega^3 \cdots \omega^n$ will be:

If $2 \cos \frac{7 \pi}{5}$ is one of the values of $z^{\frac{1}{5}}$,then $z=$

Let $\alpha$ be a root of the equation $1+x^{2}+x^{4}=0$. Then the value of $\alpha^{1011}+\alpha^{2022}-\alpha^{3033}$ is equal to

Let $z_k = \cos \left(\frac{2k\pi}{10}\right) + i \sin \left(\frac{2k\pi}{10}\right); k = 1, 2, \ldots, 9$.

List-$I$	List-$II$
$P.$ For each $z_k$ there exists a $z_j$ such that $z_k \cdot z_j = 1$	$1.$ True
$Q.$ There exists a $k \in \{1, 2, \ldots, 9\}$ such that $z_1 \cdot z = z_k$ has no solution $z$ in the set of complex numbers.	$2.$ False
$R.$ $\frac{|1-z_1||1-z_2| \ldots |1-z_9|}{10}$ equals	$3.$ $1$
$S.$ $1 - \sum_{k=1}^9 \cos \left(\frac{2k\pi}{10}\right)$ equals	$4.$ $2$

Codes: $P \quad Q \quad R \quad S$

If $\alpha$ is a non-real root of $x^6=1$,then $\frac{\alpha^5+\alpha^3+\alpha+1}{\alpha^2+1}$ is equal to