CAT (QA) PYQs

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

Suppose $f(x, y)$ is a real valued function such that $f(3x + 2y, 2x- 5y) = 19x$, for all real numbers $x$ and $y$. The value of $x$ for which $f(x, 2x) = 27$, is

Suppose for all integers x, there are two function f and g such that $f(x) + f(x-1) - 1 = 0$ and $g(x) = x^2$. If $f(x^2 - x) = 5$, then the value of the sum $f(g(5)) + g(f(5))$ is