$\cos (x + iy)$ is equal to

If the set $R = \{(a, b) : a + 5b = 42, a, b \in N\}$ has $m$ elements and $\sum_{n=1}^m (1 - i^{n!}) = x + iy$,where $i = \sqrt{-1}$,then the value of $m + x + y$ is:

Let $z$ and $w$ be two complex numbers such that $|z| \le 1$,$|w| \le 1$ and $|z + iw| = |z - i\overline{w}| = 2$. Then $z$ is equal to

Let $f(x) = x^4 + ax^3 + bx^2 + c$ be a polynomial with real coefficients such that $f(1) = -9$. Suppose that $i\sqrt{3}$ is a root of the equation $4x^3 + 3ax^2 + 2bx = 0$,where $i = \sqrt{-1}$. If $\alpha_1, \alpha_2, \alpha_3$,and $\alpha_4$ are all the roots of the equation $f(x) = 0$,then $|\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2$ is equal to:

Let a complex number $z$,$|z| \neq 1$,satisfy $\log_{\frac{1}{\sqrt{2}}} \left( \frac{|z|+11}{(|z|-1)^2} \right) \leq 2$. Then,the largest value of $|z|$ is equal to ............

Let the complex number $z = x + iy$ be such that $\frac{2z - 3i}{2z + i}$ is purely imaginary. If $x + y^2 = 0$,then $y^4 + y^2 - y$ is equal to: