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

If $x=a \cos ^{3} \theta$ and $y=a \sin ^{3} \theta$,then $\frac{d y}{d x}$ is

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

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 $x=t-\sin t, y=1-\cos t$ and $\frac{d^2 y}{d x^2}=-1$ at $t=K, K>0$, then $\lim_{t \rightarrow K} \frac{y}{x}=$

If $x=\sec \theta-\cos \theta$,$y=\sec ^{10} \theta-\cos ^{10} \theta$ and $(x^2+4)(\frac{dy}{dx})^2=k(y^2+4)$,then $k=$