If the function $$f(x) = {{\cos (\sin x) - \cos x} \over {{x^4}}}$$ is continuous at each point in its domain and $$f(0) = {1 \over k}$$, then k is ____________.

The minimum distance between any two points P₁ and P₂ while considering point P₁ on one circle and point P₂ on the other circle for the given circles' equations

x² + y² $$-$$ 10x $$-$$ 10y + 41 = 0

x² + y² $$-$$ 24x $$-$$ 10y + 160 = 0 is ___________.

A boy is rolling a 0.5 kg ball on the frictionless floor with the speed of 20 ms^-1. The ball gets deflected by an obstacle on the way. After deflection it moves with 5% of its initial kinetic energy. What is the speed of the ball now?

A polyatomic ideal gas has 24 vibrational modes. What is the value of $$\gamma$$?