The number of ordered pairs (r, k) for which
6.³⁵C_r = (k² - 3). ³⁶C_{r + 1}, where k is an integer, is :

The value of $$\alpha $$ for which
$$4\alpha \int\limits_{ - 1}^2 {{e^{ - \alpha \left| x \right|}}dx} = 5$$, is:

The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line x = y is :

If the function ƒ defined on $$\left( { - {1 \over 3},{1 \over 3}} \right)$$ by

f(x) = $$\left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + 3x} \over {1 - 2x}}} \right),} & {when\,x \ne 0} \cr {k,} & {when\,x = 0} \cr } } \right.$$

is continuous, then k is equal to_______.