If $x=\operatorname{cosec} \theta-\sin \theta$,$y=\operatorname{cosec}^{2022} \theta-\sin ^{2022} \theta$ and $\left(\frac{d y}{d x}\right)^2=\frac{k\left(y^2+4\right)}{g(x)}$ where $k \in R$,then $10+k-g(2022)=$

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 $x = a \sin 2\theta (1 + \cos 2\theta )$ and $y = b \cos 2\theta (1 - \cos 2\theta )$,then $\frac{dy}{dx} = $

If $a \neq 0$,$x=a(t+\sin t)$ and $y=a(1-\cos t)$,then $\frac{d^2 y}{d x^2}$ at $t=\frac{2 \pi}{3}$ is

If $x=\log _e\left(\frac{\cos \frac{y}{2}-\sin \frac{y}{2}}{\cos \frac{y}{2}+\sin \frac{y}{2}}\right)$ and $\tan \frac{y}{2}=\sqrt{\frac{1-t}{1+t}}$,then the value of $\left(\frac{dy}{dx}\right)_{t=\frac{1}{2}}$ is

If $x = t^2$ and $y = t^3$,then $\frac{d^2y}{dx^2} =$