A simple closed path C in the complex plane is shown in the figure. If

$$\oint\limits_c {{{{2^z}} \over {{z^2} - 1}}dz = - i\pi A} $$,

where $$i = \sqrt { - 1} $$, then the value of A is ___________ (rounded off to two decimal places).

The function f(x) = 8log_e x $$-$$ x² + 3 attains its minimum over the interval [1, e] at x = __________.

(Here log_e x is the natural logarithm of x.)

Let $$\alpha$$, $$\beta$$ two non-zero real numbers and v₁, v₂ be two non-zero real vectors of size 3 $$\times$$ 1. Suppose that v₁ and v₂ satisfy $$v_1^T{v_2} = 0$$, $$v_1^T{v_1} = 1$$ and $$v_2^T{v_2} = 1$$. Let A be the 3 $$\times$$ 3 matrix given by :

A = $$\alpha$$v₁$$v_1^T$$ + $$\beta$$v₂$$v_2^T$$

The eigen values of A are __________.

Consider the following series:

$$\sum\limits_{n = 1}^\infty {{{{n^d}} \over {{c^n}}}} $$

For which of the following combinations of c, d values does this series converge?