Find $\frac{dy}{dx}$,if $x = a \cos \theta$ and $y = a \sin \theta$.

The differential coefficient of $\sin^{-1}x$ with respect to $\cos^{-1}\sqrt{1-x^2}$ is

If $y = t^2 + t^3$ and $x = t - t^4$,then the value of $\frac{d^2 y}{d x^2}$ at $t = 1$ is equal to

If $x = \sqrt{2} e^t(\sin t - \cos t)$ and $y = \sqrt{2} e^t(\sin t + \cos t)$,then $\left(\frac{d^2 y}{d x^2}\right)_{t = \pi/4} = $

If $x = a \cos^4 \theta$ and $y = a \sin^4 \theta$,then $\frac{dy}{dx}$ at $\theta = \frac{3\pi}{4}$ is

Match the values of $\frac{dy}{dx}$ at $\theta = \frac{\pi}{3}$ for the following system of curves in parametric form given in List-$I$ with those of the items in List-$II$.

List-$I$	List-$II$
$(i)$ $x = a(\theta - \sin \theta), y = a(1 - \cos \theta)$	$(A)$ $4\sqrt{3}$
(ii) $x = 3\cos \theta - 2\cos^3 \theta, y = 3\sin \theta - 2\sin^3 \theta$	$(B)$ $-\frac{1}{3\sqrt{3}}$
(iii) $x = 3\cos \theta - \cos^3 \theta, y = 3\sin \theta - \sin^3 \theta$	$(C)$ $\sqrt{3}$
(iv) $x = a \log \sin \theta, y = a \tan \theta$	$(D)$ $\frac{1}{\sqrt{3}}$
	$(E)$ $\frac{1}{3\sqrt{3}}$