CAT (QA) PYQs

Let r be a real number and $f(x) = \begin{cases} 2x-r & \text{if } x \ge r \\ 1 & \text{if } x < r \end{cases}$. Then, the equation $f(x) = f(f(x))$ holds for all real values of x where.

If $f(ab) = f(a)f(b)$ for all positive integers $a$ and $b$, then the largest possible value of $f(1)$ 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