let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit

$$\mathop {\lim }\limits_{x \to {0^ + }} {{{{(1 - x)}^{1/x}} - {e^{ - 1}}} \over {{x^a}}}$$

is equal to a non-zero real number, is .............

As shown schematically in the figure, two vessels contain water solutions (at temperature T) of potassium permanganate (KMnO₄) of different concentrations n₁ and n₂ (n₁ > n₂) molecules per unit volume with $$\Delta $$n = (n₁ − n₂) << n₁. When they are connected by a tube of small length l and cross-sectional area S, KMnO₄ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed v of the molecules is limited by the viscous force −$$\beta $$v on each molecule, where $$\beta $$ is a constant. Neglecting all terms of the order ($$\Delta $$n)², which of the following is/are correct? (k_B is the Boltzmann constant)

A football of radius R is kept on a hole of radius r (r < R) made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle $$\theta $$ from the horizontal as shown in the figure below. The maximum value of $$\theta $$ so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale)

A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of 2 ms⁻¹ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is 320 ms⁻¹, the smallest value of the percentage change required in the length of the pipe is ____________.