Let $f:R-\left\{0\right\}\to R$ be a function satisfying $f\left(\frac{x}{y}\right)=\frac{f\left(x\right)}{f\left(y\right)}$ for all $x,y,f\left(y\right)\ne 0$. If $f^{\text{'}}\left(1\right)=2024$, then

A function $f\left(x\right)$ is given by $f\left(x\right)=\frac{5^x}{5^x+5}$, then the sum of the series $f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\dots +f\left(\frac{39}{20}\right)$ is equal to:

Let $R$ be the set of all real numbers and let $f$ be a function $R$ to $R$ such that $f\left(x\right)+\left(x+\frac{1}{2}\right)f\left(1-x\right)=1$, for all $x\in R$. Then $2f\left(0\right)+3f\left(1\right)$ is equal to.

The real valued function $f\left(x\right)=\frac{{cosec}^{-1}x}{\sqrt{x-\left[x\right]}},$ where $\left[x\right]$ denotes the greatest integer less than or equal to $x,$ is defined for all $x$ belonging to: