CAT (QA) PYQs

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

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

Suppose one of the roots of the equation $ax^2 - bx + c = 0$ is $2+\sqrt{3}$ where a, b and c are rational numbers and $a \neq 0$. If $b = c^3$ then $|a|$ equals