If $\sin^{-1}\left(\tan\dfrac{\pi}{4}\right) - \sin^{-1}\left(\sqrt{\dfrac{3}{x}}\right) = \dfrac{\pi}{6}$ then $x$ is a root of the equation

If $f(x) = x^2$ and $g(x) = [x^2]$ where $[\cdot]$ represents the greatest integer function then, $(f \circ g)\left(\dfrac{3}{2}\right) + (g \circ f)\left(\dfrac{3}{2}\right)$ is equal to ...

If $y = \sin(2\sin^{-1} x)$ then $\dfrac{dy}{dx} =$..

Let $x = at^2 - 1$, where $a > 0$ and $y = t^3 + 1$. If at $t = 1$, $\dfrac{d^2y}{dx^2} = \dfrac{3}{16}$, then the value of $a$ is...