CAT (QA) PYQs

Let $A$ be a real number. Then the roots of the equation $x^2 - 4x - \log_2A = 0$ are real and distinct if and only if

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