CAT (QA) PYQs

Consider the pair of equations: $x^{2}-x y-x=22$ and $y^{2}-x y+y=34$. If $x>y$, then $x-y$ equals

Let x, y, and z be real numbers satisfying

$4(x^2 + y^2 + z^2) = a$

$4(x - y - z) = 3+ a$

Then a equals

If $(a+b \sqrt{n})$ is the positive square root of $(29-12 \sqrt{5})$, where $a$ and $b$ are integers, and $n$ is a natural number, then the maximum possible value of $(a+b+n)$ is