CAT (QA) PYQs

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

Let $f(x) = x^2$ and $g(x) = 2^x$, for all real $x$. Then the value of $f(f(g(x)) + g(f(x)))$ at $x = 1$ is

For any positive integer $n$, let $f(n)=n(n+1)$ if $n$ is even, and $f(n)=n+3$ if $n$ is odd. If $m$ is a positive integer such that $8 f(m+1)-f(m)=2$, then $m$ equals