If $x=a(t+\sin t)$ and $y=a(1-\cos t)$,then $\frac{dy}{dx}=$

If $x = a \cos \theta$ and $y = a \sin \theta$, then $\frac{d^2 y}{d x^2} = \rule{1cm}{0.15mm} \, (a \neq 0; \theta \neq k\pi, k \in Z)$.

If $x=a t^2$ and $y=2 a t$,then $\frac{d^2 y}{d x^2}$ at $t=\frac{1}{2}$ is

If $x = a \sin 2t (1 + \cos 2t)$ and $y = b \cos 2t (1 - \cos 2t)$,then $\frac{dy}{dx}$ is equal to

If $x=f(t)$ and $y=g(t)$,then the value of $\frac{d^{2} y}{d x^{2}}$ is

Consider the curve represented parametrically by the equations $x = t^3 - 4t^2 - 3t$ and $y = 2t^2 + 3t - 5$,where $t \in \mathbb{R}$. If $H$ denotes the number of points on the curve where the tangent is horizontal and $V$ denotes the number of points where the tangent is vertical,then: