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

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

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