CAT (QA) PYQs

For real $x,$ the maximum possible value of $\frac{x}{\sqrt{1+x^4}}$ is

Among $100$ students, $x_{1}$ have birthdays in January, $x_{2}$ have birthday in February, and so on.

If $x_{0}=\max \left(x_{1}, x_{2}, \ldots , x_{12}\right)$, then the smallest possible value of $x_{0}$ is

For natural numbers $x$, $y$, and $z$, if $xy + yz = 19$ and $yz + xz = 51$, then the minimum possible value of $xyz$ is