CAT (QA) PYQs

If $f_{1}(x)=x^{2}+11 x+n$ and $f_{2}(x)=x$, then the largest positive integer $n$ for which the equation $f_{1}(x)=f_{2}(x)$ has two distinct real roots, is

If the equations $x^2 + mx + 9 = 0$, $x^2 + nx + 17 = 0$, and $x^2 + (m + n)x + 35 = 0$ have a common negative root, then the value of $2m + 3n$ is

Let $a, b, c$ be non-zero real numbers such that $b^2 < 4ac$, and $f(x) = ax^2 + bx + c$. If the set S consists of all integers m such that $f(m) < 0$, then the set S must necessarily be