CAT (QA) PYQs

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

If $x$ and $y$ are real numbers such that $4x^2 + 4y^2 - 4xy - 6y + 3 = 0$, then the value of $(4x + 5y)$ 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