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

If $x=3 \sqrt{2} \cos ^3 \theta$ and $y=4 \tan ^2 \theta$,then $\left(\frac{d y}{d x}\right)_{\theta=\frac{\pi}{4}} = $

If $x$ and $y$ are connected parametrically by the equations,without eliminating the parameter,find $\frac{dy}{dx}$ for $x=2at^{2}$ and $y=at^{4}$.

At any two points of the curve represented parametrically by $x = a(2 \cos t - \cos 2t)$ and $y = a(2 \sin t - \sin 2t)$,the tangents are parallel to the $x$-axis. The values of the parameter $t$ corresponding to these points differ from each other by:

If $x = \sqrt{10^{\sin^{-1} t}}$ and $y = \sqrt{10^{\cos^{-1} t}}$,then $\frac{dy}{dx} = $ . . . . . .

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: