If the function $f(x)=\sqrt{x^2-4}$ satisfies the Lagrange's mean value theorem on $[2,4]$, then the value of $C$ is

If $x, y$ are two positive integers such that $x+y=20$ and the maximum value of $x^3 y$ is $k$ at $x=\alpha$ and $y=\beta$, then $\frac{k}{\alpha^2 \beta^2}=$

If $y=\left(1+\alpha+\alpha^2+\ldots\right) e^{\eta x}$, where $\alpha$ and $n$ are constants, then the relative error in $y$ is

If the equation of tangent at $(2,3)$ on $y^2=a x^3+b$ is $y=4 x-5$, then the value of $a^2+b^2=$