If $x = a\sin^3 t$ and $y = a\cos^3 t$, then the value of $\dfrac{d^2y}{dx^2}$ at $t = \dfrac{\pi}{3}$ is equal to

If $\sec^{-1}\left(\dfrac{x^2 + y^2}{x^2 - y^2}\right) = 2a$ such that $y\dfrac{dy}{dx} = x \cdot f(a)$ then the value of $f\left(\dfrac{2\pi}{3}\right)$ is

If $\sqrt{y + x} + \sqrt{y - x} = c$ and $\dfrac{dy}{dx} = K - \sqrt{\dfrac{y^2}{x^2} - 1}$ , then the value of $K$ is

If $y = (x^2 + 1)^{\sin x}$ for $x > 0$ such that $\dfrac{dy}{dx} = y\left[\dfrac{2x \sin x}{g(x)} + \cos x \cdot \log[g(x)]\right]$, then the function $\dfrac{1}{g(x)}$ is...