The second order derivative of $a \sin^3 t$ with respect to $a \cos^3 t$ at $t = \frac{\pi}{4}$ is

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

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) $x = a \log \sin \theta, y = a \tan \theta$	$(D)$ $\frac{1}{\sqrt{3}}$
	$(E)$ $\frac{1}{3\sqrt{3}}$

If $x = e^t \sin t$ and $y = e^t \cos t$,where $t$ is a parameter,then $\frac{d^2y}{dx^2}$ at $(1, 1)$ is equal to

If $x$ and $y$ are connected parametrically by the equations,without eliminating the parameter,find $\frac{dy}{dx}$ for $x = a \cos \theta$ and $y = b \cos \theta$.

For the curve $y = 3 \sin \theta \cos \theta$,$x = e^{\theta} \sin \theta$,$0 \leq \theta \leq \pi$,the tangent is parallel to the $x-$axis when $\theta$ is