If  $$\int \, $$x⁵.e^$$-$$4x3 dx = $${1 \over {48}}$$e^$$-$$4x3 f(x) + C, where C is a constant of inegration, then f(x) is equal to -

The length of the chord of the parabola x² $$=$$ 4y having equation x – $$\sqrt 2 y + 4\sqrt 2 = 0$$  is -

Let N be the set of natural numbers and two functions f and g be defined as f, g : N $$ \to $$ N such that

f(n) = $$\left\{ {\matrix{ {{{n + 1} \over 2};} & {if\,\,n\,\,is\,\,odd} \cr {{n \over 2};} & {if\,\,n\,\,is\,\,even} \cr } \,\,} \right.$$;

      and g(n) = n $$-$$($$-$$ 1)ⁿ.

Then fog is -

Two identical spherical balls of mass M and radius R each are stuck on two ends of a rod of length 2R and mass M (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is :