Consider the following partial differential equation (PDE)

$$a{{{\partial ^2}f(x,y)} \over {\partial {x^2}}} + b{{{\partial ^2}f(x,y)} \over {\partial {y^2}}} = f(x,y)$$,

where a and b are distinct positive real numbers. Select the combination(s) of values of the real parameters $$\xi $$ and $$\eta $$ such that $$f(x,y) = {e^{\xi x + \eta y}}$$ is a solution of the given PDE.

The bar graph shown the frequency of the number of wickets taken in a match by a bowler in her career. For example, in 17 of her matches, the bowler has taken 5 wickets each. The median number of wickets taken by the bowler in a match is ____________ (rounded off to one decimal place).

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.)