The sum of the intercepts on the coordinate axes made by the tangent to the curve $\sqrt{x} + \sqrt{y} = \sqrt{a}$ is

If the tangent to the curve $y=x^{3}$ at the point $P(t, t^{3})$ meets the curve again at $Q$,then the ordinate of the point which divides $PQ$ internally in the ratio $1:2$ is

Consider $f(x) = \tan^{-1}\left(\sqrt{\frac{1 + \sin x}{1 - \sin x}}\right)$,where $x \in (0, \frac{\pi}{2})$. $A$ normal to $y = f(x)$ at $x = \frac{\pi}{6}$ also passes through the point:

The point$(s)$ on the curve $y^3 + 3x^2 = 12y$ where the tangent is vertical (parallel to $y$-axis) is (are):

Find the slope of the tangent to the curve $y = x^{3} - 3x + 2$ at the point whose $x$-coordinate is $3$.

If the straight line $x \cos \alpha + y \sin \alpha = p$ touches the curve $(\frac{x}{a})^n + (\frac{y}{b})^n = 2$ at the point $(a, b)$ on it and $\frac{1}{a^2} + \frac{1}{b^2} = \frac{k}{p^2}$,then $k =$