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 $a^2+b^2=$

If Rolle's theorem is applicable for the function $f(x)=x(x+3) e^{-x / 2}$ on $[3,0]$, then the value of $c$ is

For all $x \in[0,2024]$ assume that $f(x)$ is differentiable, $f(0)=-2$ and $f^{\prime}(x) \geq 5$. Then, the least possible value of $f(2024)$ is