Algebra of complex numbers

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 ?

Let $x$ and $y$ be real numbers such that $50\left(\dfrac{2x}{1+3i}-\dfrac{y}{1-2i}\right)=31+17i$, $i=\sqrt{-1}$. Then the value of $10(x-3y)$ is :

Let $a,b\in\mathbb C$. Let $\alpha,\beta$ be the roots of the equation $x^2+ax+b=0$. If $\beta-\alpha=\sqrt{11}$ and $\beta^2-\alpha^2=3i\sqrt{11}$, then $\left(\beta^3-\alpha^3\right)^2$ is equal to:

Let $z_1,z_2\in\mathbb C$ be the distinct solutions of the equation $z^2+4z-(1+12i)=0$. Then $|z_1|^2+|z_2|^2$ is equal to :

Let $A = \left\{\dfrac{1967 + 1686i\sin\theta}{7 - 3i\cos\theta} : \theta \in \mathbb{R}\right\}$. If $A$ contains exactly one positive integer $n$, then the value of $n$ is _____.

Let $z$ be a complex number satisfying $|z|^3 + 2z^2 + 4\bar z - 8 = 0$, where $\bar z$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.

Match each entry in List-I to the correct entries in List-II.

The correct option is:

Let $z$ be a complex number with non-zero imaginary part. If $$\frac{2+3z+4z^2}{2-3z+4z^2}$$ is a real number, then the value of $|z|^2$ 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 _____.

Let $\bar z$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
$$(\bar z)^2 + \frac{1}{z^2}$$
are integers, then which of the following is/are possible value(s) of $|z|$?

Let $A=\begin{pmatrix}1+i & 1\\ -i & 0\end{pmatrix}$ where $i=\sqrt{-1}$. Then, the number of elements in the set $\left\{n\in\{1,2,\dots,100\}:A^n=A\right\}$ is ______.

Sum of squares of modulus of all the complex numbers $z$ satisfying $\bar z=iz^2+z^2-z$ is equal to ______.

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+(2i-1)=0$. Then, the value of $\left|\alpha^8+\beta^8\right|$ is equal to :

The real part of the complex number $\dfrac{(1+2i)^8\cdot(1-2i)^2}{(3+2i)\cdot\overline{(4-6i)}}$ is equal to :

Let the minimum value $v_0$ of $v=|z|^2+|z-3|^2+|z-6i|^2$, $z\in\mathbf C$ is attained at $z=z_0$. Then $\left|2z_0^2-\bar z_0^3+3\right|^2+v_0^2$ is equal to

Let $S=\left\{z\in\mathbf C:z^2+\bar z=0\right\}$. Then $\displaystyle\sum_{z\in S}\left(\operatorname{Re}(z)+\operatorname{Im}(z)\right)$ is equal to ______.

Let S be the set of all $(\alpha,\beta)$, $\pi<\alpha,\beta<2\pi$, for which the complex number $\dfrac{1-i\sin\alpha}{1+2i\sin\alpha}$ is purely imaginary and $\dfrac{1+i\cos\beta}{1-2i\cos\beta}$ is purely real. Let $Z_{\alpha\beta}=\sin2\alpha+i\cos2\beta$, $(\alpha,\beta)\in S$. Then $\displaystyle\sum_{(\alpha,\beta)\in S}\left(iZ_{\alpha\beta}+\frac1{i\bar Z_{\alpha\beta}}\right)$ is equal to :

If $z=2+3i$, then $z^5+(\bar z)^5$ is equal to :

If the least and the largest real values of α, for which the equation $z+\alpha|z-1|+2i=0$ ($z\in\mathbb C$ and $i=\sqrt{-1}$) has a solution, are p and q respectively; then $4(p^2+q^2)$ is equal to _______.

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^2 + 20x - 2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^2 - 20x + 2020$. Then the value of

$$ac(a-c) + ad(a-d) + bc(b-c) + bd(b-d)$$

is

For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4 - |z|^4 = 4iz^2$, where $i = \sqrt{-1}$. Then the minimum possible value of $|z_1 - z_2|^2$, where $z_1, z_2 \in S$ with $\operatorname{Re}(z_1) > 0$ and $\operatorname{Re}(z_2) < 0$, is ______