Rolle's theorem holds for monic quadratic polynomial $f(x)$ on the interval $[\alpha, \alpha + 3]$ where $f(\alpha) = 0$. Similarly, $g(x) = f(x) + 2$ also follows Rolle's theorem in the interval $[\beta, 3]$ where $g(3) = 0$, such that the value of $c$ is the same for both $f(x)$ and $g(x)$. Then the value of $(f \circ g)(\alpha)$ is...

If the rate of increase of surface area of a spherical balloon is $5\,\text{cm}^2/\text{sec}$ and rate of increase of volume of a spherical balloon is $10\,\text{cm}^3/\text{sec}$, then the radius of the balloon at that time is...

The value of integral $\int x^3 \cos x\,dx$ is...

If $u$ and $v$ are functions of $x$, then $\int \dfrac{1}{v^3}\left(uv\dfrac{du}{dx} - u^2\dfrac{dv}{dx}\right)dx = $