CAT (QA) PYQs

A function $f$ maps the set of natural numbers to whole numbers, such that $f (xy) = f (x) f (y) + f (x) + f (y)$ for all $x, y$ and $f (p) = 1$ for every prime number $p$. Then, the value of $f (160000)$ is

For all real values of x, the range of the function f(x) = $\frac{x^2+2x+4}{2x^2+4x+9}$ is

Consider a function $f$ satisfying $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) \mathrm{f}(\mathrm{y})$ where $x , y$ are positive integers, and $f(1)=2$. If $f(\mathrm{a}+1)+f(\mathrm{a}+2)+\ldots \ldots+f(\mathrm{a}+\mathrm{n})=$ $16\left(2^{n}-1\right)$ then a is equal to