Let $$f(x)=\frac{x}{\left(1+x^{n}\right)^{\frac{1}{n}}}, x \in \mathbb{R}-\{-1\}, n \in \mathbb{N}, n > 2$$.

If $$f^{n}(x)=\left(f \circ f \circ f \ldots .\right.$$. upto $$n$$ times) $$(x)$$, then

$$\lim _\limits{n \rightarrow \infty} \int_\limits{0}^{1} x^{n-2}\left(f^{n}(x)\right) d x$$ is equal to ____________.

A particle starts with an initial velocity of $$10.0 \mathrm{~ms}^{-1}$$ along $$x$$-direction and accelerates uniformly at the rate of $$2.0 \mathrm{~ms}^{-2}$$. The time taken by the particle to reach the velocity of $$60.0 \mathrm{~ms}^{-1}$$ is __________.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R

Assertion A: The phase difference of two light waves change if they travel through different media having same thickness, but different indices of refraction.

Reason R: The wavelengths of waves are different in different media.

In the light of the above statements, choose the most appropriate answer from the options given below

The temperature of an ideal gas is increased from $$200 \mathrm{~K}$$ to $$800 \mathrm{~K}$$. If r.m.s. speed of gas at $$200 \mathrm{~K}$$ is $$v_{0}$$. Then, r.m.s. speed of the gas at $$800 \mathrm{~K}$$ will be: