Let $f$ be a twice differentiable non-negative function such that $(f(x))^2=25+\int_0^x\left((f(\mathrm{t}))^2+\left(f^{\prime}(\mathrm{t})\right)^2\right) \mathrm{dt}$. Then the mean of $f\left(\log _{\mathrm{e}}(1)\right), f\left(\log _{\mathrm{e}}(2)\right), \ldots . ., f\left(\log _{\mathrm{e}}(625)\right)$ is equal to $\_\_\_\_$ .

From the first 100 natural numbers, two numbers first $a$ and then $b$ are selected randomly without replacement. If the probability that $\mathrm{a}-\mathrm{b} \geqslant 10$ is $\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to

$\_\_\_\_$ .

The number of 4 -letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is $\_\_\_\_$ .

A simple pendulum of string length 30 cm performs 20 oscillations in 10 s . The length of the string required for the pendulum to perform 40 oscillations in the same time duration is

$\_\_\_\_$ cm . [Assume that the mass of the pendulum remains same.]