Roots of unity

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Let $\mathbb R$ denote the set of all real numbers and let $i=\sqrt{-1}$. Consider the matrices
$$S=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\quad\text{and}\quad T=\begin{bmatrix}1&1\\0&1\end{bmatrix}.$$
Let $a,b,c,d$ be real numbers such that
$$ST=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$$
Let
$$H=\{x+iy:\ x,y\in\mathbb R\ \text{and}\ y>0\}.$$
Then which of the following statements is (are) TRUE ?

For a non-zero complex number $z$, let $\arg(z)$ denote the principal argument of $z$, with $-\pi < \arg(z) \le \pi$. Let $\omega$ be the cube root of unity for which $0 < \arg(\omega) < \pi$. Let

$$\alpha = \arg\left(\sum_{n=1}^{2025}(-\omega)^n\right).$$

Then the value of $\dfrac{3\alpha}{\pi}$ is _____.

Let $A_1, A_2, A_3, \ldots, A_8$ be the vertices of a regular octagon that lie on a circle of radius 2. Let $P$ be a point on the circle and let $PA_i$ denote the distance between the points $P$ and $A_i$ for $i = 1, 2, \ldots, 8$. If $P$ varies over the circle, then the maximum value of the product $PA_1\cdot PA_2\cdots PA_8$, is _____.

Let $\bar z$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt{-1}$. In the set of complex numbers, the number of distinct roots of the equation $$\bar z - z^2 = i(\bar z + z^2)$$ is _____.

If $z^2+z+1=0$, $z\in\mathbb C$, then $\left|\displaystyle\sum_{n=1}^{15}\left(z^n+(-1)^n\frac1{z^n}\right)^2\right|$ is equal to ______.

The value of $\cos\left(\dfrac{2\pi}7\right)+\cos\left(\dfrac{4\pi}7\right)+\cos\left(\dfrac{6\pi}7\right)$ is equal to :

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 :

If $\alpha,\beta,\gamma,\delta$ are the roots of the equation $x^4+x^3+x^2+x+1=0$, then $\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}$ is equal to :

Let $A=\begin{pmatrix}2&-1&-1\\1&0&-1\\1&-1&0\end{pmatrix}$ and $B=A-I$. If $\omega=\dfrac{\sqrt3i-1}2$, then the number of elements in the set $\left\{n\in\{1,2,\dots,100\}:A^n+(\omega B)^n=A+B\right\}$ is equal to ______.

Let $z=a+ib$, $b\ne0$ be complex numbers satisfying $z^2=\bar z\cdot2^{1-|z|}$. Then the least value of $n\in\mathbf N$, such that $z^n=(z+1)^n$, is equal to ______.

Let $\omega \ne 1$ be a cube root of unity. Then the minimum of the set

$$\left\{|a + b\omega + c\omega^2|^2 : a, b, c \text{ distinct non-zero integers}\right\}$$

equals ______

Let ω be a complex number such that $2\omega+1=z$ where $z=\sqrt{-3}$. If
$$\begin{vmatrix}1&1&1\\1&-\omega^2-1&\omega^2\\1&\omega^2&\omega^7\end{vmatrix}=3k,$$
then k is equal to :

Let $z=\dfrac{-1+\sqrt3\,i}{2}$, where $i=\sqrt{-1}$, and $r,s\in\{1,2,3\}$. Let $P=\begin{bmatrix}(-z)^r&z^{2s}\\z^{2s}&z^r\end{bmatrix}$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r,s)$ for which $P^2=-I$ is

For any integer $k$, let $\alpha_k=\cos\left(\dfrac{k\pi}7\right)+i\sin\left(\dfrac{k\pi}7\right)$, where $i=\sqrt{-1}$. The value of the expression $\dfrac{\sum_{k=1}^{12}|\alpha_{k+1}-\alpha_k|}{\sum_{k=1}^{3}|\alpha_{4k-1}-\alpha_{4k-2}|}$ is

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

Let $\omega$ be a complex cube root of unity with $\omega\ne1$ and $P=[p_{ij}]$ be a $n\times n$ matrix with $p_{ij}=\omega^{i+j}$. Then $P^2\ne0$, when $n=$

Let $\omega$ be a solution of $x^3-1=0$ with $\operatorname{Im}(\omega)>0$. If $a=2$ with $b$ and $c$ satisfying (E), then the value of
$$\frac3{\omega^a}+\frac1{\omega^b}+\frac3{\omega^c}$$
is equal to

Let $\omega\ne1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form
$$\begin{bmatrix}1&a&b\\\omega&1&c\\\omega^2&\omega&1\end{bmatrix},$$
where each of $a$, $b$, and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is

Let $\omega=e^{i\pi/3}$, and $a$, $b$, $c$, $x$, $y$, $z$ be non-zero complex numbers such that
$$a+b+c=x,\qquad a+b\omega+c\omega^2=y,\qquad a+b\omega^2+c\omega=z.$$
Then the value of $\dfrac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}$ is