CAT (QA) PYQs

The number of distinct real roots of the equation $(x + \frac{1}{x})^2 - 3(x + \frac{1}{x}) + 2 = 0$ equals

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