Let f : R $$ \to $$ R be defined as
$$f\left( x \right) = \left\{ {\matrix{ {{x^5}\sin \left( {{1 \over x}} \right) + 5{x^2},} & {x < 0} \cr {0,} & {x = 0} \cr {{x^5}\cos \left( {{1 \over x}} \right) + \lambda {x^2},} & {x > 0} \cr } } \right.$$

The value of $$\lambda $$ for which f ''(0) exists, is _______.

Let AD and BC be two vertical poles
at A and B respectively on a horizontal ground.
If AD = 8 m, BC = 11 m and AB = 10 m; then the distance
(in meters) of a point M on AB from the point A such
that MD² + MC² is minimum is ______.

Set A has m elements and set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of m.n is ______.

An electron, a doubly ionized helium ion (He⁺⁺) and a proton are having the same kinetic energy. The relation between their respective de-Broglie wavelengths $$\lambda $$_e, $$\lambda $$_He⁺⁺ and $$\lambda $$_p is :