CAT (QA) PYQs

Suppose $k$ is any integer such that the equation $2 x^{2}+k x+5=0$ has no real roots and the equation $x^{2}+(k-5) x+1=0$ has two distinct real roots for $x$. Then, the number of possible values of $k$ is

Let $f(x)=x^{2}+a x+b$ and $g(x)=f(x+1)-f(x-1)$. If $f(x) \geq 0$ for all real $x$, and $g(20)=72$, then the smallest possible value of $b$ 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