CAT (QA) PYQs

How many different pairs $(a, b)$ of positive integers are there such that $a ≤ b$ and $\frac{1}{a} + \frac{1}{b} = \frac{1}{9}$

If $m$ and $n$ are integers such that $(m+2n)(2m+n) = 27$, then the maximum possible value of $2m-3n$ is

How many pairs $(m, n)$ of positive integers satisfy the equation $m^2 + 105 = n^2$?