CAT (QA) PYQs

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

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 $x$ is a positive real number such that $x^8 + \frac{1}{x^8} = 47$, then the value of $x^9 + \frac{1}{x^9}$ is