Algebra of complex numbers

Let $s$, $t$, $r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + iy$ $(x, y \in \mathbb{R}, i = \sqrt{-1})$ of the equation $sz + t\bar{z} + r = 0$, where $\bar{z} = x - iy$. Then, which of the following statement(s) is (are) TRUE?

Let $a$, $b$, $x$ and $y$ be real numbers such that $a - b = 1$ and $y \ne 0$. If the complex number $z = x + iy$ satisfies $\mathrm{Im}\left(\frac{az + b}{z + 1}\right) = y$, then which of the following is(are) possible value(s) of $x$?

A value of θ for which $\dfrac{2+3i\sin\theta}{1-2i\sin\theta}$ is purely imaginary, is :

Let z = 1 + ai be a complex number, a > 0, such that $z^3$ is a real number. Then the sum $1+z+z^2+\dots+z^{11}$ is equal to :

If $2+3i$ is one of the roots of the equation $2x^3-9x^2+kx-13=0$, $k\in\mathbb R$, then the real root of this equation :

If $z$ is a non-real complex number, then the minimum value of $\dfrac{\text{Im }z^5}{(\text{Im }z)^5}$ is :

The quadratic equation $p(x)=0$ with real coefficients has purely imaginary roots. Then the equation
$$p(p(x))=0$$
has

Let $w$ ($\operatorname{Im}w\ne0$) be a complex number. Then the set of all complex numbers $z$ satisfying the equation $w-\bar wz=k(1-z)$, for some real number $k$, is :

For all complex numbers $z$ of the form $1+i\alpha$, $\alpha\in\mathbb R$, if $z^2=x+iy$, then :

Let $z$ be a complex number such that the imaginary part of $z$ is nonzero and $a=z^2+z+1$ is real. Then $a$ cannot take the value

Match the statements given in Column I with the intervals/union of intervals given in Column II

Let $z=x+iy$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation
$$z\bar z^3+\bar zz^3=350$$
is