The angle made by the tangent at $\theta = \pi / 3$ on the curve $x = a(\theta + \sin \theta)$,$y = a(1 - \cos \theta)$ with the $X$-axis is

If $x = 4t^3 + 3$, $y = 3t^4 + 4$ and $\frac{d^2x}{dy^2} = (\frac{dx}{dy})^n$ is constant, then the value of $n$ is

If $x$ is the displacement,$v$ is the velocity,and $f$ is the acceleration of a particle at time $t$,which of the following is true?

If $x=3 \tan t$ and $y=3 \sec t$,then the value of $\frac{d^2 y}{d x^2}$ at $t=\frac{\pi}{4}$ 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

The slope of the tangent to the curve $x = t^2 + 3t - 8$,$y = 2t^2 - 2t - 5$ at the point $(2, -1)$ is