CAT (QA) PYQs

Let $f(x)=$ max{$5x$, $52-2x^2$}, where $x$ is any positive real number. Then the minimum possible value of $f(x)$ is

The minimum possible value of the sum of the squares of the roots of the equation $x^{2}+(a+3) x-(a+5)=0$ 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