CAT (QA) PYQs

Let $m$ and $n$ be positive integers, If $x^2 + mx + 2n = 0$ and $x^2 + 2nx + m = 0$ have real roots, then the smallest possible value of $m + n$ is

A quadratic equation $x^{2}+b x+c=0$ has two real roots. It the difference between the reciprocals of the roots is $1 / 3$ and the sum of the reciprocals of the squares of the roots is $5 / 9$, then the largest possible value of ( $b+c$ ) is

The equation $x^3 + (2r + 1)x^2 + (4r - 1)x + 2 = 0$ has $- 2$ as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of $r$ is