For any complex number w = c + id, let $$\arg (w) \in ( - \pi ,\pi ]$$, where $$i = \sqrt { - 1} $$. Let $$\alpha$$ and $$\beta$$ be real numbers such that for all complex numbers z = x + iy satisfying $$\arg \left( {{{z + \alpha } \over {z + \beta }}} \right) = {\pi \over 4}$$, the ordered pair (x, y) lies on the circle $${x^2} + {y^2} + 5x - 3y + 4 = 0$$, Then which of the following statements is (are) TRUE?

Let S be the set of all complex numbers z
satisfying |z² + z + 1| = 1. Then which of the following statements is/are TRUE?

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

For a non-zero complex number z, let arg(z) denote the principal argument with $$-$$ $$\pi $$ < arg(z) $$ \le $$ $$\pi $$. Then, which of the following statement(s) is (are) FALSE?