Let $\alpha = 8 - 14i,A = \left\{ {z \in c:{{\alpha z - \overline \alpha \overline z } \over {{z^2} - {{\left( {\overline z } \right)}^2} - 112i}}=1} \right\}$ and $B = \left\{ {z \in c:\left| {z + 3i} \right| = 4} \right\}$. Then $\sum\limits_{z \in A \cap B} {({\mathop{\rm Re}\nolimits} z - {\mathop{\rm Im}\nolimits} z)}$ is equal to ____________.

Sum of squares of modulus of all the complex numbers z satisfying $\overline z = i{z^2} + {z^2} - z$ is equal to ___________.

For $\alpha, \beta, z \in \mathbb{C}$ and $\lambda > 1$, if is the radius of the circle , then is equal to __________.

If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ and , where , then the distance of the point from the line is __________.

$\begin{aligned} &\text { If } \lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)] \\ &=33 \cdot \lim _{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot\left[1^{k}+2^{k}+3^{k}+\ldots+n^{k}\right] \end{aligned}$, then the integral value of is equal to _____________

If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of ${\left( {{x^n} + {2 \over {{x^5}}}} \right)^7}$ is 939, then the sum of all the possible integral values of n is _________.

Let $\mathrm{S}=\left\{z \in \mathbb{C}-\{i, 2 i\}: \frac{z^{2}+8 i z-15}{z^{2}-3 i z-2} \in \mathbb{R}\right\}$. If , then is equal to _________.

If the circles ${x^2} + {y^2} + 6x + 8y + 16 = 0$ and , , touch internally at the point , then is equal to ________________.

Let $S=\left\{z \in \mathbb{C}: z^{2}+\bar{z}=0\right\}$. Then is equal to ______________.