If the radius of a circle increases from $3 \, cm$ to $3.2 \, cm$,then the increase in the area of the circle is

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

The diameter of a sphere is measured as $42 \text{ cm}$. If there is an error of $1/77 \text{ cm}$ in measuring it, then the error involved in the volume of that sphere (in cubic centimeters) is

The nearest approximate value of $\sqrt{2023}$ is

If an error of $0.02 \text{ cm}^2$ is found in the surface area of a sphere when its radius is measured as $10 \text{ cm}$, then the approximate error that occurs in the volume of the sphere, in cubic centimetres, is

The approximate value of $x^3-2x^2+3x+2$ at $x=2.01$ is